Line 3: |
Line 3: |
| ===Ascii Tables=== | | ===Ascii Tables=== |
| | | |
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
| + | | |
| <pre> | | <pre> |
| o-------------------o | | o-------------------o |
Line 41: |
Line 43: |
| | | | | | | |
| o-------------------o | | o-------------------o |
− | | |
| |
| | | | | | | |
| | a b | | | | a b | |
Line 83: |
Line 84: |
| | | | | | | |
| o-------------------o | | o-------------------o |
− | | |
| |
| | | | | | | |
| | b c | | | | b c | |
Line 95: |
Line 95: |
| o-------------------o | | o-------------------o |
| </pre> | | </pre> |
| + | |} |
| | | |
− | <br>
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
− | | + | | |
| <pre> | | <pre> |
| Table 13. The Existential Interpretation | | Table 13. The Existential Interpretation |
Line 194: |
Line 195: |
| o----o-------------------o-------------------o-------------------o | | o----o-------------------o-------------------o-------------------o |
| </pre> | | </pre> |
| + | |} |
| | | |
− | <br>
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
− | | + | | |
| <pre> | | <pre> |
| Table 14. The Entitative Interpretation | | Table 14. The Entitative Interpretation |
Line 294: |
Line 296: |
| o----o-------------------o-------------------o-------------------o | | o----o-------------------o-------------------o-------------------o |
| </pre> | | </pre> |
| + | |} |
| | | |
− | <br>
| + | {| align="center" cellpadding="6" style="text-align:center; width:90%" |
− | | + | | |
| <pre> | | <pre> |
| Table 15. Existential & Entitative Interpretations of Cactus Structures | | Table 15. Existential & Entitative Interpretations of Cactus Structures |
Line 329: |
Line 332: |
| o-----------------o-----------------o-----------------o-----------------o | | o-----------------o-----------------o-----------------o-----------------o |
| </pre> | | </pre> |
| + | |} |
| | | |
− | ==Differential Logic== | + | ===Wiki TeX Tables=== |
| | | |
− | ===Ascii Tables===
| + | <br> |
| | | |
− | <pre> | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
− | Table A1. Propositional Forms On Two Variables | + | |+ <math>\text{Table A.}~~\text{Existential Interpretation}</math> |
− | o---------o---------o---------o----------o------------------o----------o
| + | |- style="background:#f0f0ff" |
− | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | + | | <math>\text{Cactus Graph}\!</math> |
− | | | | | | | | | + | | <math>\text{Cactus Expression}\!</math> |
− | | Decimal | Binary | Vector | Cactus | English | Ordinary | | + | | <math>\text{Interpretation}\!</math> |
− | o---------o---------o---------o----------o------------------o----------o
| + | |- |
− | | | x : 1 1 0 0 | | | | | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | | | y : 1 0 1 0 | | | | | + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
− | o---------o---------o---------o----------o------------------o----------o
| + | | <math>\operatorname{true}.</math> |
− | | | | | | | | | + | |- |
− | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | | | | | | | | | + | | <math>\texttt{(~)}</math> |
− | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | + | | <math>\operatorname{false}.</math> |
− | | | | | | | | | + | |- |
− | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | | | | | | | |
| + | | <math>a\!</math> |
− | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| + | | <math>a.\!</math> |
− | | | | | | | |
| + | |- |
− | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | | | | | | | | | + | | <math>\texttt{(} a \texttt{)}</math> |
− | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | + | | |
− | | | | | | | | | + | <math>\begin{matrix} |
− | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| + | \tilde{a} |
− | | | | | | | |
| + | \\[2pt] |
− | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| + | a^\prime |
− | | | | | | | |
| + | \\[2pt] |
− | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | + | \lnot a |
− | | | | | | | | | + | \\[2pt] |
− | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | + | \operatorname{not}~ a. |
− | | | | | | | |
| + | \end{matrix}</math> |
− | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | + | | <math>a~b~c</math> |
− | | | | | | | |
| + | | |
− | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | + | <math>\begin{matrix} |
− | | | | | | | | | + | a \land b \land c |
− | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | + | \\[6pt] |
− | | | | | | | | | + | a ~\operatorname{and}~ b ~\operatorname{and}~ c. |
− | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | + | \end{matrix}</math> |
− | | | | | | | | | + | |- |
− | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | | | | | | | |
| + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
− | o---------o---------o---------o----------o------------------o----------o
| + | | |
− | </pre> | + | <math>\begin{matrix} |
| + | a \lor b \lor c |
| + | \\[6pt] |
| + | a ~\operatorname{or}~ b ~\operatorname{or}~ c. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
| + | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a \Rightarrow b |
| + | \\[2pt] |
| + | a ~\operatorname{implies}~ b. |
| + | \\[2pt] |
| + | \operatorname{if}~ a ~\operatorname{then}~ b. |
| + | \\[2pt] |
| + | \operatorname{not}~ a ~\operatorname{without}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a + b |
| + | \\[2pt] |
| + | a \neq b |
| + | \\[2pt] |
| + | a ~\operatorname{exclusive-or}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{not~equal~to}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a = b |
| + | \\[2pt] |
| + | a \iff b |
| + | \\[2pt] |
| + | a ~\operatorname{equals}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{if~and~only~if}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{genus}~ a ~\operatorname{of~species}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{partition}~ a ~\operatorname{into}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{pie}~ a ~\operatorname{of~slices}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| | | |
− | <pre> | + | <br> |
− | Table A2. Propositional Forms On Two Variables
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| |
− | | | | | | | |
| |
− | | Decimal | Binary | Vector | Cactus | English | Ordinary |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | x : 1 1 0 0 | | | |
| |
− | | | y : 1 0 1 0 | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| |
− | | | | | | | |
| |
− | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| |
− | | | | | | | |
| |
− | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| |
− | | | | | | | |
| |
− | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| |
− | | | | | | | |
| |
− | | f_12 | f_1100 | 1 1 0 0 | x | x | x |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| |
− | | | | | | | |
| |
− | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| |
− | | | | | | | |
| |
− | | f_10 | f_1010 | 1 0 1 0 | y | y | y |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| |
− | | | | | | | |
| |
− | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| |
− | | | | | | | |
| |
− | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| |
− | | | | | | | |
| |
− | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | | | | | | | |
| |
− | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| |
− | | | | | | | |
| |
− | o---------o---------o---------o----------o------------------o----------o
| |
− | </pre>
| |
| | | |
− | <pre> | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
− | Table A3. Ef Expanded Over Differential Features {dx, dy} | + | |+ <math>\text{Table B.}~~\text{Entitative Interpretation}</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- style="background:#f0f0ff" |
− | | | | | | | | | + | | <math>\text{Cactus Graph}\!</math> |
− | | | f | T_11 f | T_10 f | T_01 f | T_00 f | | + | | <math>\text{Cactus Expression}\!</math> |
− | | | | | | | | | + | | <math>\text{Interpretation}\!</math> |
− | | | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)| | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | o------o------------o------------o------------o------------o------------o
| + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
− | | | | | | | | | + | | <math>\operatorname{false}.</math> |
− | | f_0 | () | () | () | () | () | | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | o------o------------o------------o------------o------------o------------o
| + | | <math>\texttt{(~)}</math> |
− | | | | | | | | | + | | <math>\operatorname{true}.</math> |
− | | f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) | | + | |- |
− | | | | | | | | | + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| + | | <math>a\!</math> |
− | | | | | | | | | + | | <math>a.\!</math> |
− | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) | | + | |- |
− | | | | | | | | | + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | | f_8 | x y | (x)(y) | (x) y | x (y) | x y | | + | | <math>\texttt{(} a \texttt{)}</math> |
− | | | | | | | |
| + | | |
− | o------o------------o------------o------------o------------o------------o
| + | <math>\begin{matrix} |
− | | | | | | | |
| + | \tilde{a} |
− | | f_3 | (x) | x | x | (x) | (x) |
| + | \\[2pt] |
− | | | | | | | |
| + | a^\prime |
− | | f_12 | x | (x) | (x) | x | x |
| + | \\[2pt] |
− | | | | | | | |
| + | \lnot a |
− | o------o------------o------------o------------o------------o------------o
| + | \\[2pt] |
− | | | | | | | | | + | \operatorname{not}~ a. |
− | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | | + | \end{matrix}</math> |
− | | | | | | | | | + | |- |
− | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | | | | | | | |
| + | | <math>a~b~c</math> |
− | o------o------------o------------o------------o------------o------------o
| + | | |
− | | | | | | | |
| + | <math>\begin{matrix} |
− | | f_5 | (y) | y | (y) | y | (y) |
| + | a \lor b \lor c |
− | | | | | | | |
| + | \\[6pt] |
− | | f_10 | y | (y) | y | (y) | y |
| + | a ~\operatorname{or}~ b ~\operatorname{or}~ c. |
− | | | | | | | |
| + | \end{matrix}</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- |
− | | | | | | | | | + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | | + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
− | | | | | | | |
| + | | |
− | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| + | <math>\begin{matrix} |
− | | | | | | | |
| + | a \land b \land c |
− | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| + | \\[6pt] |
− | | | | | | | | | + | a ~\operatorname{and}~ b ~\operatorname{and}~ c. |
− | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | | + | \end{matrix}</math> |
− | | | | | | | |
| + | |- |
− | o------o------------o------------o------------o------------o------------o
| + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
− | | | | | | | |
| + | | <math>\texttt{(} a \texttt{)} b</math> |
− | | f_15 | (()) | (()) | (()) | (()) | (()) |
| + | | |
− | | | | | | | |
| + | <math>\begin{matrix} |
− | o------o------------o------------o------------o------------o------------o
| + | a \Rightarrow b |
− | | | | | | |
| + | \\[2pt] |
− | | Fixed Point Total | 4 | 4 | 4 | 16 |
| + | a ~\operatorname{implies}~ b. |
− | | | | | | | | + | \\[2pt] |
− | o-------------------o------------o------------o------------o------------o
| + | \operatorname{if}~ a ~\operatorname{then}~ b. |
− | </pre> | + | \\[2pt] |
| + | \operatorname{not}~ a, ~\operatorname{or}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a = b |
| + | \\[2pt] |
| + | a \iff b |
| + | \\[2pt] |
| + | a ~\operatorname{equals}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{if~and~only~if}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | a + b |
| + | \\[2pt] |
| + | a \neq b |
| + | \\[2pt] |
| + | a ~\operatorname{exclusive-or}~ b. |
| + | \\[2pt] |
| + | a ~\operatorname{not~equal~to}~ b. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{((} a, b, c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one~of} |
| + | \\ |
| + | a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{genus}~ a ~\operatorname{of~species}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{partition}~ a ~\operatorname{into}~ b, c. |
| + | \\[6pt] |
| + | \operatorname{pie}~ a ~\operatorname{of~slices}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| | | |
− | <pre> | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%" |
− | Table A4. Df Expanded Over Differential Features {dx, dy} | + | |+ <math>\text{Table C.}~~\text{Dualing Interpretations}</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- style="background:#f0f0ff" |
− | | | | | | | | | + | | <math>\text{Graph}\!</math> |
− | | | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)| | + | | <math>\text{String}\!</math> |
− | | | | | | | | | + | | <math>\text{Existential}\!</math> |
− | o------o------------o------------o------------o------------o------------o
| + | | <math>\text{Entitative}\!</math> |
− | | | | | | | | | + | |- |
− | | f_0 | () | () | () | () | () | | + | | height="100px" | [[Image:Cactus Node Big Fat.jpg|20px]] |
− | | | | | | | | | + | | <math>{}^{\backprime\backprime}\texttt{~}{}^{\prime\prime}</math> |
− | o------o------------o------------o------------o------------o------------o
| + | | <math>\operatorname{true}.</math> |
− | | | | | | | | | + | | <math>\operatorname{false}.</math> |
− | | f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| + | |- |
− | | | | | | | |
| + | | height="100px" | [[Image:Cactus Spike Big Fat.jpg|20px]] |
− | | f_2 | (x) y | (x, y) | y | (x) | () | | + | | <math>\texttt{(~)}</math> |
− | | | | | | | | | + | | <math>\operatorname{false}.</math> |
− | | f_4 | x (y) | (x, y) | (y) | x | () |
| + | | <math>\operatorname{true}.</math> |
− | | | | | | | |
| + | |- |
− | | f_8 | x y | ((x, y)) | y | x | () |
| + | | height="100px" | [[Image:Cactus A Big.jpg|20px]] |
− | | | | | | | |
| + | | <math>a\!</math> |
− | o------o------------o------------o------------o------------o------------o
| + | | <math>a.\!</math> |
− | | | | | | | |
| + | | <math>a.\!</math> |
− | | f_3 | (x) | (()) | (()) | () | () | | + | |- |
− | | | | | | | | | + | | height="120px" | [[Image:Cactus (A) Big.jpg|20px]] |
− | | f_12 | x | (()) | (()) | () | () | | + | | <math>\texttt{(} a \texttt{)}</math> |
− | | | | | | | | | + | | <math>\lnot a</math> |
− | o------o------------o------------o------------o------------o------------o | + | | <math>\lnot a</math> |
− | | | | | | | | | + | |- |
− | | f_6 | (x, y) | () | (()) | (()) | () | | + | | height="100px" | [[Image:Cactus ABC Big.jpg|50px]] |
− | | | | | | | |
| + | | <math>a~b~c</math> |
− | | f_9 | ((x, y)) | () | (()) | (()) | () |
| + | | <math>a \land b \land c</math> |
− | | | | | | | |
| + | | <math>a \lor b \lor c</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- |
− | | | | | | | |
| + | | height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]] |
− | | f_5 | (y) | (()) | () | (()) | () | | + | | <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math> |
− | | | | | | | | | + | | <math>a \lor b \lor c</math> |
− | | f_10 | y | (()) | () | (()) | () | | + | | <math>a \land b \land c</math> |
− | | | | | | | | | + | |- |
− | o------o------------o------------o------------o------------o------------o
| + | | height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]] |
− | | | | | | | | | + | | <math>\texttt{(} a \texttt{(} b \texttt{))}</math> |
− | | f_7 | (x y) | ((x, y)) | y | x | () | | + | | <math>a \Rightarrow b</math> |
− | | | | | | | | | + | | |
− | | f_11 | (x (y)) | (x, y) | (y) | x | () | | + | |- |
− | | | | | | | | | + | | height="120px" | [[Image:Cactus (A)B Big.jpg|35px]] |
− | | f_13 | ((x) y) | (x, y) | y | (x) | () | | + | | <math>\texttt{(} a \texttt{)} b</math> |
− | | | | | | | | | + | | |
− | | f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () | | + | | <math>a \Rightarrow b</math> |
− | | | | | | | | | + | |- |
− | o------o------------o------------o------------o------------o------------o
| + | | height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]] |
− | | | | | | | | | + | | <math>\texttt{(} a, b \texttt{)}</math> |
− | | f_15 | (()) | () | () | () | () | | + | | <math>a \neq b</math> |
− | | | | | | | | | + | | <math>a = b\!</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- |
− | </pre>
| + | | height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]] |
− | | + | | <math>\texttt{((} a, b \texttt{))}</math> |
− | <pre>
| + | | <math>a = b\!</math> |
− | Table A5. Ef Expanded Over Ordinary Features {x, y}
| + | | <math>a \neq b\!</math> |
− | o------o------------o------------o------------o------------o------------o
| + | |- |
− | | | | | | | |
| + | | height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]] |
− | | | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| + | | <math>\texttt{(} a, b, c \texttt{)}</math> |
− | | | | | | | |
| + | | |
− | o------o------------o------------o------------o------------o------------o
| + | <math>\begin{matrix} |
− | | | | | | | |
| + | \operatorname{just~one} |
− | | f_0 | () | () | () | () | () |
| + | \\ |
− | | | | | | | |
| + | \operatorname{of}~ a, b, c |
− | o------o------------o------------o------------o------------o------------o
| + | \\ |
− | | | | | | | |
| + | \operatorname{is~false}. |
− | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| + | \end{matrix}</math> |
− | | | | | | | |
| + | | |
− | | f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| + | <math>\begin{matrix} |
− | | | | | | | |
| + | \operatorname{not~just~one} |
− | | f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| + | \\ |
− | | | | | | | |
| + | \operatorname{of}~ a, b, c |
− | | f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| + | \\ |
− | | | | | | | |
| + | \operatorname{is~true}. |
− | o------o------------o------------o------------o------------o------------o
| + | \end{matrix}</math> |
− | | | | | | | |
| + | |- |
− | | f_3 | (x) | dx | dx | (dx) | (dx) |
| + | | height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]] |
− | | | | | | | |
| + | | <math>\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
− | | f_12 | x | (dx) | (dx) | dx | dx |
| + | | |
− | | | | | | | |
| + | <math>\begin{matrix} |
− | o------o------------o------------o------------o------------o------------o
| + | \operatorname{just~one} |
− | | | | | | | |
| + | \\ |
− | | f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| + | \operatorname{of}~ a, b, c |
− | | | | | | | |
| + | \\ |
− | | f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| + | \operatorname{is~true}. |
− | | | | | | | |
| + | \end{matrix}</math> |
− | o------o------------o------------o------------o------------o------------o
| + | | |
− | | | | | | | |
| + | <math>\begin{matrix} |
− | | f_5 | (y) | dy | (dy) | dy | (dy) |
| + | \operatorname{not~just~one} |
− | | | | | | | |
| + | \\ |
− | | f_10 | y | (dy) | dy | (dy) | dy |
| + | \operatorname{of}~ a, b, c |
− | | | | | | | |
| + | \\ |
− | o------o------------o------------o------------o------------o------------o
| + | \operatorname{is~false}. |
− | | | | | | | |
| + | \end{matrix}</math> |
− | | f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| + | |- |
− | | | | | | | |
| + | | height="160px" | [[Image:Cactus ((A,B,C)) Big.jpg|65px]] |
− | | f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| + | | <math>\texttt{((} a, b, c \texttt{))}</math> |
− | | | | | | | |
| + | | |
− | | f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| + | <math>\begin{matrix} |
− | | | | | | | |
| + | \operatorname{not~just~one} |
− | | f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| + | \\ |
− | | | | | | | |
| + | \operatorname{of}~ a, b, c |
− | o------o------------o------------o------------o------------o------------o
| + | \\ |
− | | | | | | | |
| + | \operatorname{is~false}. |
− | | f_15 | (()) | (()) | (()) | (()) | (()) |
| + | \end{matrix}</math> |
− | | | | | | | |
| + | | |
− | o------o------------o------------o------------o------------o------------o
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),(B),(C))) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{)))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{not~just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~true}. |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{just~one} |
| + | \\ |
| + | \operatorname{of}~ a, b, c |
| + | \\ |
| + | \operatorname{is~false}. |
| + | \end{matrix}</math> |
| + | |- |
| + | | height="160px" | [[Image:Cactus (A,(B),(C)) Big.jpg|65px]] |
| + | | <math>\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{partition}~ a |
| + | \\ |
| + | \operatorname{into}~ b, c. |
| + | \end{matrix}</math> |
| + | | |
| + | |- |
| + | | height="200px" | [[Image:Cactus (((A),B,C)) Big.jpg|65px]] |
| + | | <math>\texttt{(((} a \texttt{)}, b, c \texttt{))}</math> |
| + | | |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{partition}~ a |
| + | \\ |
| + | \operatorname{into}~ b, c. |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ==Differential Logic== |
| + | |
| + | ===Ascii Tables=== |
| + | |
| + | <pre> |
| + | Table A1. Propositional Forms On Two Variables |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
| + | | | | | | | | |
| + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | x : 1 1 0 0 | | | | |
| + | | | y : 1 0 1 0 | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
| + | | | | | | | | |
| + | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | |
| + | | | | | | | | |
| + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
| + | | | | | | | | |
| + | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | |
| + | | | | | | | | |
| + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
| + | | | | | | | | |
| + | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | |
| + | | | | | | | | |
| + | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | |
| + | | | | | | | | |
| + | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | |
| + | | | | | | | | |
| + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
| + | | | | | | | | |
| + | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | |
| + | | | | | | | | |
| + | | f_10 | f_1010 | 1 0 1 0 | y | y | y | |
| + | | | | | | | | |
| + | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | |
| + | | | | | | | | |
| + | | f_12 | f_1100 | 1 1 0 0 | x | x | x | |
| + | | | | | | | | |
| + | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | |
| + | | | | | | | | |
| + | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | |
| + | | | | | | | | |
| + | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| </pre> | | </pre> |
| | | |
| <pre> | | <pre> |
− | Table A6. Df Expanded Over Ordinary Features {x, y} | + | Table A2. Propositional Forms On Two Variables |
− | o------o------------o------------o------------o------------o------------o | + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | | | + | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | |
− | | | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)| | + | | | | | | | | |
− | | | | | | | | | + | | Decimal | Binary | Vector | Cactus | English | Ordinary | |
− | o------o------------o------------o------------o------------o------------o | + | o---------o---------o---------o----------o------------------o----------o |
− | | | | | | | | | + | | | x : 1 1 0 0 | | | | |
− | | f_0 | () | () | () | () | () | | + | | | y : 1 0 1 0 | | | | |
− | | | | | | | | | + | o---------o---------o---------o----------o------------------o----------o |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | | | |
− | | | | | | | | | + | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | |
− | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | | + | | | | | | | | |
− | | | | | | | | | + | o---------o---------o---------o----------o------------------o----------o |
− | | f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | | + | | | | | | | | |
− | | | | | | | | | + | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | |
− | | f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | | + | | | | | | | | |
− | | | | | | | | | + | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | |
− | | f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | | + | | | | | | | | |
− | | | | | | | | | + | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | | | |
− | | | | | | | | | + | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | |
− | | f_3 | (x) | dx | dx | dx | dx | | + | | | | | | | | |
− | | | | | | | | | + | o---------o---------o---------o----------o------------------o----------o |
− | | f_12 | x | dx | dx | dx | dx | | + | | | | | | | | |
− | | | | | | | | | + | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | |
− | o------o------------o------------o------------o------------o------------o | + | | | | | | | | |
− | | | | | | | | | + | | f_12 | f_1100 | 1 1 0 0 | x | x | x | |
− | | f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | | + | | | | | | | | |
− | | | | | | | | | + | o---------o---------o---------o----------o------------------o----------o |
− | | f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | | + | | | | | | | | |
| + | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | |
| + | | | | | | | | |
| + | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | |
| + | | | | | | | | |
| + | | f_10 | f_1010 | 1 0 1 0 | y | y | y | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | |
| + | | | | | | | | |
| + | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | |
| + | | | | | | | | |
| + | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | |
| + | | | | | | | | |
| + | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | | | | | | | | |
| + | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | |
| + | | | | | | | | |
| + | o---------o---------o---------o----------o------------------o----------o |
| + | </pre> |
| + | |
| + | <pre> |
| + | Table A3. Ef Expanded Over Differential Features {dx, dy} |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | | f | T_11 f | T_10 f | T_01 f | T_00 f | |
| + | | | | | | | | |
| + | | | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)| |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_5 | (y) | dy | dy | dy | dy | | + | | f_0 | () | () | () | () | () | |
− | | | | | | | |
| |
− | | f_10 | y | dy | dy | dy | dy |
| |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | | + | | f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) | |
| | | | | | | | | | | | | | | | | |
− | | f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | | + | | f_2 | (x) y | x (y) | x y | (x)(y) | (x) y | |
| | | | | | | | | | | | | | | | | |
− | | f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | | + | | f_4 | x (y) | (x) y | (x)(y) | x y | x (y) | |
| | | | | | | | | | | | | | | | | |
− | | f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | | + | | f_8 | x y | (x)(y) | (x) y | x (y) | x y | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| | | | | | | | | | | | | | | | | |
− | | f_15 | (()) | () | () | () | () | | + | | f_3 | (x) | x | x | (x) | (x) | |
| + | | | | | | | | |
| + | | f_12 | x | (x) | (x) | x | x | |
| | | | | | | | | | | | | | | | | |
| o------o------------o------------o------------o------------o------------o | | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) | |
| + | | | | | | | | |
| + | | f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_5 | (y) | y | (y) | y | (y) | |
| + | | | | | | | | |
| + | | f_10 | y | (y) | y | (y) | y | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) | |
| + | | | | | | | | |
| + | | f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) | |
| + | | | | | | | | |
| + | | f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) | |
| + | | | | | | | | |
| + | | f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | | |
| + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
| + | | | | | | | | |
| + | o------o------------o------------o------------o------------o------------o |
| + | | | | | | | |
| + | | Fixed Point Total | 4 | 4 | 4 | 16 | |
| + | | | | | | | |
| + | o-------------------o------------o------------o------------o------------o |
| </pre> | | </pre> |
| | | |
| <pre> | | <pre> |
− | o----------o----------o----------o----------o----------o | + | Table A4. Df Expanded Over Differential Features {dx, dy} |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | | · % T_00 | T_01 | T_10 | T_11 | | + | | | | | | | | |
− | | % | | | | | + | | | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)| |
− | o==========o==========o==========o==========o==========o
| + | | | | | | | | |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | | T_00 % T_00 | T_01 | T_10 | T_11 | | + | | | | | | | | |
− | | % | | | | | + | | f_0 | () | () | () | () | () | |
− | o----------o----------o----------o----------o----------o | + | | | | | | | | |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | | T_01 % T_01 | T_00 | T_11 | T_10 | | + | | | | | | | | |
− | | % | | | | | + | | f_1 | (x)(y) | ((x, y)) | (y) | (x) | () | |
− | o----------o----------o----------o----------o----------o | + | | | | | | | | |
− | | % | | | | | + | | f_2 | (x) y | (x, y) | y | (x) | () | |
− | | T_10 % T_10 | T_11 | T_00 | T_01 | | + | | | | | | | | |
− | | % | | | | | + | | f_4 | x (y) | (x, y) | (y) | x | () | |
− | o----------o----------o----------o----------o----------o
| + | | | | | | | | |
− | | % | | | | | + | | f_8 | x y | ((x, y)) | y | x | () | |
− | | T_11 % T_11 | T_10 | T_01 | T_00 | | + | | | | | | | | |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | o----------o----------o----------o----------o----------o | + | | | | | | | | |
− | </pre>
| + | | f_3 | (x) | (()) | (()) | () | () | |
− | | + | | | | | | | | |
− | <pre>
| + | | f_12 | x | (()) | (()) | () | () | |
− | o---------o---------o---------o---------o---------o
| + | | | | | | | | |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | | · % e | f | g | h | | + | | | | | | | | |
− | | % | | | | | + | | f_6 | (x, y) | () | (()) | (()) | () | |
− | o=========o=========o=========o=========o=========o
| + | | | | | | | | |
− | | % | | | | | + | | f_9 | ((x, y)) | () | (()) | (()) | () | |
− | | e % e | f | g | h | | + | | | | | | | | |
− | | % | | | | | + | o------o------------o------------o------------o------------o------------o |
− | o---------o---------o---------o---------o---------o | + | | | | | | | | |
− | | % | | | | | + | | f_5 | (y) | (()) | () | (()) | () | |
− | | f % f | e | h | g | | + | | | | | | | | |
− | | % | | | | | + | | f_10 | y | (()) | () | (()) | () | |
− | o---------o---------o---------o---------o---------o | + | | | | | | | | |
− | | % | | | |
| + | o------o------------o------------o------------o------------o------------o |
− | | g % g | h | e | f |
| + | | | | | | | | |
− | | % | | | |
| + | | f_7 | (x y) | ((x, y)) | y | x | () | |
− | o---------o---------o---------o---------o---------o
| + | | | | | | | | |
− | | % | | | | | + | | f_11 | (x (y)) | (x, y) | (y) | x | () | |
− | | h % h | g | f | e | | + | | | | | | | | |
− | | % | | | | | + | | f_13 | ((x) y) | (x, y) | y | (x) | () | |
− | o---------o---------o---------o---------o---------o | + | | | | | | | | |
− | </pre>
| + | | f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () | |
− | | + | | | | | | | | |
− | <pre>
| + | o------o------------o------------o------------o------------o------------o |
− | Permutation Substitutions in Sym {A, B, C}
| + | | | | | | | | |
− | o---------o---------o---------o---------o---------o---------o
| + | | f_15 | (()) | () | () | () | () | |
− | | | | | | | | | + | | | | | | | | |
− | | e | f | g | h | i | j | | + | o------o------------o------------o------------o------------o------------o |
− | | | | | | | | | |
− | o=========o=========o=========o=========o=========o=========o
| |
− | | | | | | | | | |
− | | A B C | A B C | A B C | A B C | A B C | A B C | | |
− | | | | | | | | | |
− | | | | | | | | | | | | | | | | | | | | | | | | | | | |
− | | v v v | v v v | v v v | v v v | v v v | v v v | | |
− | | | | | | | | | |
− | | A B C | C A B | B C A | A C B | C B A | B A C | | |
− | | | | | | | | | |
− | o---------o---------o---------o---------o---------o---------o | |
| </pre> | | </pre> |
| | | |
| <pre> | | <pre> |
− | Matrix Representations of Permutations in Sym(3)
| + | Table A5. Ef Expanded Over Ordinary Features {x, y} |
− | o---------o---------o---------o---------o---------o---------o | + | o------o------------o------------o------------o------------o------------o |
− | | | | | | | | | + | | | | | | | | |
− | | e | f | g | h | i | j | | + | | | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)| |
− | | | | | | | | | + | | | | | | | | |
− | o=========o=========o=========o=========o=========o=========o | + | o------o------------o------------o------------o------------o------------o |
− | | | | | | | | | + | | | | | | | | |
− | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 | | + | | f_0 | () | () | () | () | () | |
− | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 | | + | | | | | | | | |
− | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 | | + | o------o------------o------------o------------o------------o------------o |
− | | | | | | | | | + | | | | | | | | |
− | o---------o---------o---------o---------o---------o---------o | + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) | |
− | </pre>
| + | | | | | | | | |
− | | + | | f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy | |
− | <pre>
| + | | | | | | | | |
− | Symmetric Group S_3
| + | | f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) | |
− | o-------------------------------------------------o | + | | | | | | | | |
− | | | | + | | f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy | |
− | | ^ | | + | | | | | | | | |
− | | e / \ e | | + | o------o------------o------------o------------o------------o------------o |
− | | / \ | | + | | | | | | | | |
− | | / e \ | | + | | f_3 | (x) | dx | dx | (dx) | (dx) | |
− | | f / \ / \ f | | + | | | | | | | | |
− | | / \ / \ | | + | | f_12 | x | (dx) | (dx) | dx | dx | |
− | | / f \ f \ | | + | | | | | | | | |
− | | g / \ / \ / \ g | | + | o------o------------o------------o------------o------------o------------o |
− | | / \ / \ / \ | | + | | | | | | | | |
− | | / g \ g \ g \ | | + | | f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) | |
− | | h / \ / \ / \ / \ h | | + | | | | | | | | |
− | | / \ / \ / \ / \ | | + | | f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) | |
− | | / h \ e \ e \ h \ | | + | | | | | | | | |
− | | i / \ / \ / \ / \ / \ i | | + | o------o------------o------------o------------o------------o------------o |
− | | / \ / \ / \ / \ / \ | | + | | | | | | | | |
− | | / i \ i \ f \ j \ i \ | | + | | f_5 | (y) | dy | (dy) | dy | (dy) | |
− | | j / \ / \ / \ / \ / \ / \ j | | + | | | | | | | | |
− | | / \ / \ / \ / \ / \ / \ | | + | | f_10 | y | (dy) | dy | (dy) | dy | |
− | | ( j \ j \ j \ i \ h \ j ) | | + | | | | | | | | |
− | | \ / \ / \ / \ / \ / \ / | | + | o------o------------o------------o------------o------------o------------o |
− | | \ / \ / \ / \ / \ / \ / | | + | | | | | | | | |
− | | \ h \ h \ e \ j \ i / | | + | | f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) | |
− | | \ / \ / \ / \ / \ / | | + | | | | | | | | |
− | | \ / \ / \ / \ / \ / | | + | | f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) | |
− | | \ i \ g \ f \ h / | | + | | | | | | | | |
− | | \ / \ / \ / \ / | | + | | f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) | |
− | | \ / \ / \ / \ / | | + | | | | | | | | |
− | | \ f \ e \ g / | | + | | f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) | |
− | | \ / \ / \ / | | + | | | | | | | | |
− | | \ / \ / \ / | | + | o------o------------o------------o------------o------------o------------o |
− | | \ g \ f / | | + | | | | | | | | |
− | | \ / \ / | | + | | f_15 | (()) | (()) | (()) | (()) | (()) | |
− | | \ / \ / | | + | | | | | | | | |
− | | \ e / | | + | o------o------------o------------o------------o------------o------------o |
− | | \ / | | |
− | | \ / | | |
− | | v | | |
− | | |
| |
− | o-------------------------------------------------o | |
| </pre> | | </pre> |
| | | |
− | ===Wiki Tables : New Versions===
| + | <pre> |
− | | + | Table A6. Df Expanded Over Ordinary Features {x, y} |
− | ====Propositional Forms on Two Variables====
| + | o------o------------o------------o------------o------------o------------o |
− | | + | | | | | | | | |
− | <br> | + | | | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)| |
− | | + | | | | | | | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
| + | o------o------------o------------o------------o------------o------------o |
− | |+ '''Table A1. Propositional Forms on Two Variables''' | + | | | | | | | | |
− | |- style="background:#f0f0ff" | + | | f_0 | () | () | () | () | () | |
− | ! width="15%" | L<sub>1</sub>
| + | | | | | | | | |
− | ! width="15%" | L<sub>2</sub>
| + | o------o------------o------------o------------o------------o------------o |
− | ! width="15%" | L<sub>3</sub>
| + | | | | | | | | |
− | ! width="15%" | L<sub>4</sub>
| + | | f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | |
− | ! width="25%" | L<sub>5</sub>
| + | | | | | | | | |
− | ! width="15%" | L<sub>6</sub>
| + | | f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | |
− | |- style="background:#f0f0ff" | + | | | | | | | | |
− | | | + | | f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | |
− | | align="right" | x : | + | | | | | | | | |
− | | 1 1 0 0 | + | | f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | |
− | | | + | | | | | | | | |
− | | | + | o------o------------o------------o------------o------------o------------o |
− | | | + | | | | | | | | |
− | |- style="background:#f0f0ff"
| + | | f_3 | (x) | dx | dx | dx | dx | |
− | | | + | | | | | | | | |
− | | align="right" | y : | + | | f_12 | x | dx | dx | dx | dx | |
− | | 1 0 1 0 | + | | | | | | | | |
− | | | + | o------o------------o------------o------------o------------o------------o |
− | | | + | | | | | | | | |
− | | | + | | f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
− | |- | + | | | | | | | | |
− | | f<sub>0</sub> | + | | f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) | |
− | | f<sub>0000</sub> | + | | | | | | | | |
− | | 0 0 0 0 | + | o------o------------o------------o------------o------------o------------o |
− | | ( ) | + | | | | | | | | |
− | | false | + | | f_5 | (y) | dy | dy | dy | dy | |
− | | 0 | + | | | | | | | | |
− | |-
| + | | f_10 | y | dy | dy | dy | dy | |
− | | f<sub>1</sub> | + | | | | | | | | |
− | | f<sub>0001</sub> | + | o------o------------o------------o------------o------------o------------o |
− | | 0 0 0 1 | + | | | | | | | | |
− | | (x)(y) | + | | f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy | |
− | | neither x nor y | + | | | | | | | | |
− | | ¬x ∧ ¬y | + | | f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) | |
− | |- | + | | | | | | | | |
− | | f<sub>2</sub> | + | | f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy | |
− | | f<sub>0010</sub> | + | | | | | | | | |
− | | 0 0 1 0 | + | | f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) | |
− | | (x) y | + | | | | | | | | |
− | | y and not x | + | o------o------------o------------o------------o------------o------------o |
− | | ¬x ∧ y | + | | | | | | | | |
− | |- | + | | f_15 | (()) | () | () | () | () | |
− | | f<sub>3</sub> | + | | | | | | | | |
− | | f<sub>0011</sub> | + | o------o------------o------------o------------o------------o------------o |
− | | 0 0 1 1 | + | </pre> |
− | | (x) | + | |
− | | not x | + | <pre> |
− | | ¬x | + | o----------o----------o----------o----------o----------o |
− | |- | + | | % | | | | |
− | | f<sub>4</sub> | + | | · % T_00 | T_01 | T_10 | T_11 | |
− | | f<sub>0100</sub> | + | | % | | | | |
− | | 0 1 0 0 | + | o==========o==========o==========o==========o==========o |
− | | x (y) | + | | % | | | | |
− | | x and not y | + | | T_00 % T_00 | T_01 | T_10 | T_11 | |
− | | x ∧ ¬y | + | | % | | | | |
− | |- | + | o----------o----------o----------o----------o----------o |
− | | f<sub>5</sub> | + | | % | | | | |
− | | f<sub>0101</sub> | + | | T_01 % T_01 | T_00 | T_11 | T_10 | |
− | | 0 1 0 1 | + | | % | | | | |
− | | (y) | + | o----------o----------o----------o----------o----------o |
− | | not y | + | | % | | | | |
− | | ¬y | + | | T_10 % T_10 | T_11 | T_00 | T_01 | |
− | |- | + | | % | | | | |
− | | f<sub>6</sub> | + | o----------o----------o----------o----------o----------o |
− | | f<sub>0110</sub> | + | | % | | | | |
− | | 0 1 1 0 | + | | T_11 % T_11 | T_10 | T_01 | T_00 | |
− | | (x, y) | + | | % | | | | |
− | | x not equal to y | + | o----------o----------o----------o----------o----------o |
− | | x ≠ y | + | </pre> |
− | |- | + | |
− | | f<sub>7</sub> | + | <pre> |
− | | f<sub>0111</sub> | + | o---------o---------o---------o---------o---------o |
− | | 0 1 1 1 | + | | % | | | | |
− | | (x y) | + | | · % e | f | g | h | |
− | | not both x and y | + | | % | | | | |
− | | ¬x ∨ ¬y | + | o=========o=========o=========o=========o=========o |
− | |- | + | | % | | | | |
− | | f<sub>8</sub> | + | | e % e | f | g | h | |
− | | f<sub>1000</sub> | + | | % | | | | |
− | | 1 0 0 0 | + | o---------o---------o---------o---------o---------o |
− | | x y | + | | % | | | | |
− | | x and y | + | | f % f | e | h | g | |
− | | x ∧ y | + | | % | | | | |
− | |-
| + | o---------o---------o---------o---------o---------o |
− | | f<sub>9</sub>
| + | | % | | | | |
− | | f<sub>1001</sub>
| + | | g % g | h | e | f | |
− | | 1 0 0 1 | + | | % | | | | |
− | | ((x, y)) | + | o---------o---------o---------o---------o---------o |
− | | x equal to y | + | | % | | | | |
− | | x = y
| + | | h % h | g | f | e | |
− | |- | + | | % | | | | |
− | | f<sub>10</sub> | + | o---------o---------o---------o---------o---------o |
− | | f<sub>1010</sub> | + | </pre> |
− | | 1 0 1 0 | + | |
− | | y | + | <pre> |
− | | y | + | Permutation Substitutions in Sym {A, B, C} |
− | | y | + | o---------o---------o---------o---------o---------o---------o |
− | |-
| + | | | | | | | | |
− | | f<sub>11</sub> | + | | e | f | g | h | i | j | |
− | | f<sub>1011</sub> | + | | | | | | | | |
− | | 1 0 1 1 | + | o=========o=========o=========o=========o=========o=========o |
− | | (x (y)) | + | | | | | | | | |
− | | not x without y | + | | A B C | A B C | A B C | A B C | A B C | A B C | |
− | | x ⇒ y | + | | | | | | | | |
− | |-
| + | | | | | | | | | | | | | | | | | | | | | | | | | | |
− | | f<sub>12</sub>
| + | | v v v | v v v | v v v | v v v | v v v | v v v | |
− | | f<sub>1100</sub> | + | | | | | | | | |
− | | 1 1 0 0 | + | | A B C | C A B | B C A | A C B | C B A | B A C | |
− | | x | + | | | | | | | | |
− | | x | + | o---------o---------o---------o---------o---------o---------o |
− | | x | + | </pre> |
− | |-
| |
− | | f<sub>13</sub> | |
− | | f<sub>1101</sub> | |
− | | 1 1 0 1 | |
− | | ((x) y) | |
− | | not y without x | |
− | | x ⇐ y | |
− | |- | |
− | | f<sub>14</sub> | |
− | | f<sub>1110</sub>
| |
− | | 1 1 1 0 | |
− | | ((x)(y)) | |
− | | x or y | |
− | | x ∨ y | |
− | |- | |
− | | f<sub>15</sub> | |
− | | f<sub>1111</sub> | |
− | | 1 1 1 1 | |
− | | (( )) | |
− | | true || 1 | |
− | |} | |
| | | |
− | <br> | + | <pre> |
− | | + | Matrix Representations of Permutations in Sym(3) |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" | + | o---------o---------o---------o---------o---------o---------o |
− | |+ '''Table A2. Propositional Forms on Two Variables''' | + | | | | | | | | |
− | |- style="background:#f0f0ff" | + | | e | f | g | h | i | j | |
− | ! width="15%" | L<sub>1</sub> | + | | | | | | | | |
− | ! width="15%" | L<sub>2</sub> | + | o=========o=========o=========o=========o=========o=========o |
− | ! width="15%" | L<sub>3</sub> | + | | | | | | | | |
− | ! width="15%" | L<sub>4</sub> | + | | 1 0 0 | 0 0 1 | 0 1 0 | 1 0 0 | 0 0 1 | 0 1 0 | |
− | ! width="25%" | L<sub>5</sub> | + | | 0 1 0 | 1 0 0 | 0 0 1 | 0 0 1 | 0 1 0 | 1 0 0 | |
− | ! width="15%" | L<sub>6</sub> | + | | 0 0 1 | 0 1 0 | 1 0 0 | 0 1 0 | 1 0 0 | 0 0 1 | |
− | |- style="background:#f0f0ff" | + | | | | | | | | |
− | | | + | o---------o---------o---------o---------o---------o---------o |
− | | align="right" | x : | + | </pre> |
− | | 1 1 0 0 | + | |
− | | | + | <pre> |
− | | | + | Symmetric Group S_3 |
− | | | + | o-------------------------------------------------o |
− | |- style="background:#f0f0ff" | + | | | |
− | | | + | | ^ | |
− | | align="right" | y : | + | | e / \ e | |
− | | 1 0 1 0 | + | | / \ | |
− | | | + | | / e \ | |
− | | | + | | f / \ / \ f | |
− | | | + | | / \ / \ | |
− | |-
| + | | / f \ f \ | |
− | | f<sub>0</sub>
| + | | g / \ / \ / \ g | |
− | | f<sub>0000</sub>
| + | | / \ / \ / \ | |
− | | 0 0 0 0
| + | | / g \ g \ g \ | |
− | | ( )
| + | | h / \ / \ / \ / \ h | |
− | | false
| + | | / \ / \ / \ / \ | |
− | | 0
| + | | / h \ e \ e \ h \ | |
− | |-
| + | | i / \ / \ / \ / \ / \ i | |
− | |
| + | | / \ / \ / \ / \ / \ | |
− | {| align="center"
| + | | / i \ i \ f \ j \ i \ | |
− | |
| + | | j / \ / \ / \ / \ / \ / \ j | |
− | <p>f<sub>1</sub></p>
| + | | / \ / \ / \ / \ / \ / \ | |
− | <p>f<sub>2</sub></p>
| + | | ( j \ j \ j \ i \ h \ j ) | |
− | <p>f<sub>4</sub></p>
| + | | \ / \ / \ / \ / \ / \ / | |
− | <p>f<sub>8</sub></p>
| + | | \ / \ / \ / \ / \ / \ / | |
− | |}
| + | | \ h \ h \ e \ j \ i / | |
− | |
| + | | \ / \ / \ / \ / \ / | |
− | {| align="center"
| + | | \ / \ / \ / \ / \ / | |
− | |
| + | | \ i \ g \ f \ h / | |
− | <p>f<sub>0001</sub></p>
| + | | \ / \ / \ / \ / | |
− | <p>f<sub>0010</sub></p>
| + | | \ / \ / \ / \ / | |
− | <p>f<sub>0100</sub></p>
| + | | \ f \ e \ g / | |
− | <p>f<sub>1000</sub></p>
| + | | \ / \ / \ / | |
− | |}
| + | | \ / \ / \ / | |
− | |
| + | | \ g \ f / | |
− | {| align="center"
| + | | \ / \ / | |
− | |
| + | | \ / \ / | |
− | <p>0 0 0 1</p>
| + | | \ e / | |
− | <p>0 0 1 0</p>
| + | | \ / | |
− | <p>0 1 0 0</p>
| + | | \ / | |
− | <p>1 0 0 0</p>
| + | | v | |
− | |}
| + | | | |
− | |
| + | o-------------------------------------------------o |
− | {| align="center"
| + | </pre> |
− | |
| + | |
− | <p>(x)(y)</p>
| + | ===Wiki Tables : New Versions=== |
− | <p>(x) y </p>
| + | |
− | <p> x (y)</p>
| + | ====Propositional Forms on Two Variables==== |
− | <p> x y </p>
| + | |
− | |}
| + | <br> |
− | |
| + | |
− | {| align="center"
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
− | |
| + | |+ '''Table A1. Propositional Forms on Two Variables''' |
− | <p>neither x nor y</p>
| + | |- style="background:#f0f0ff" |
− | <p>not x but y</p>
| + | ! width="15%" | L<sub>1</sub> |
− | <p>x but not y</p>
| + | ! width="15%" | L<sub>2</sub> |
− | <p>x and y</p>
| + | ! width="15%" | L<sub>3</sub> |
− | |}
| + | ! width="15%" | L<sub>4</sub> |
− | |
| + | ! width="25%" | L<sub>5</sub> |
− | {| align="center"
| + | ! width="15%" | L<sub>6</sub> |
− | |
| + | |- style="background:#f0f0ff" |
− | <p>¬x ∧ ¬y</p>
| + | | |
− | <p>¬x ∧ y</p>
| + | | align="right" | x : |
− | <p>x ∧ ¬y</p>
| + | | 1 1 0 0 |
− | <p>x ∧ y</p>
| + | | |
− | |}
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | y : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| |- | | |- |
− | | | + | | f<sub>0</sub> |
− | {| align="center"
| + | | f<sub>0000</sub> |
− | |
| + | | 0 0 0 0 |
− | <p>f<sub>3</sub></p>
| + | | ( ) |
− | <p>f<sub>12</sub></p>
| + | | false |
− | |} | + | | 0 |
− | | | + | |- |
− | {| align="center"
| + | | f<sub>1</sub> |
− | | | + | | f<sub>0001</sub> |
− | <p>f<sub>0011</sub></p>
| + | | 0 0 0 1 |
− | <p>f<sub>1100</sub></p>
| + | | (x)(y) |
− | |} | + | | neither x nor y |
− | |
| + | | ¬x ∧ ¬y |
− | {| align="center"
| |
− | |
| |
− | <p>0 0 1 1</p>
| |
− | <p>1 1 0 0</p>
| |
− | |}
| |
− | | | |
− | {| align="center"
| |
− | |
| |
− | <p>(x)</p>
| |
− | <p> x </p>
| |
− | |} | |
− | |
| |
− | {| align="center"
| |
− | |
| |
− | <p>not x</p>
| |
− | <p>x</p>
| |
− | |}
| |
− | |
| |
− | {| align="center"
| |
− | |
| |
− | <p>¬x</p>
| |
− | <p>x</p>
| |
− | |}
| |
| |- | | |- |
− | | | + | | f<sub>2</sub> |
− | {| align="center"
| + | | f<sub>0010</sub> |
− | |
| + | | 0 0 1 0 |
− | <p>f<sub>6</sub></p>
| + | | (x) y |
− | <p>f<sub>9</sub></p>
| + | | y and not x |
− | |} | + | | ¬x ∧ y |
− | | | + | |- |
− | {| align="center"
| + | | f<sub>3</sub> |
− | | | + | | f<sub>0011</sub> |
− | <p>f<sub>0110</sub></p>
| + | | 0 0 1 1 |
− | <p>f<sub>1001</sub></p>
| + | | (x) |
− | |}
| + | | not x |
− | |
| + | | ¬x |
− | {| align="center"
| + | |- |
− | | | + | | f<sub>4</sub> |
− | <p>0 1 1 0</p>
| + | | f<sub>0100</sub> |
− | <p>1 0 0 1</p>
| + | | 0 1 0 0 |
− | |}
| + | | x (y) |
− | |
| + | | x and not y |
− | {| align="center"
| + | | x ∧ ¬y |
− | |
| |
− | <p> (x, y) </p>
| |
− | <p>((x, y))</p>
| |
− | |} | |
− | | | |
− | {| align="center"
| |
− | |
| |
− | <p>x not equal to y</p> | |
− | <p>x equal to y</p> | |
− | |} | |
− | | | |
− | {| align="center"
| |
− | | | |
− | <p>x ≠ y</p>
| |
− | <p>x = y</p>
| |
− | |}
| |
| |- | | |- |
− | | | + | | f<sub>5</sub> |
− | {| align="center"
| + | | f<sub>0101</sub> |
− | |
| + | | 0 1 0 1 |
− | <p>f<sub>5</sub></p>
| + | | (y) |
− | <p>f<sub>10</sub></p>
| + | | not y |
− | |} | + | | ¬y |
− | | | + | |- |
− | {| align="center"
| + | | f<sub>6</sub> |
− | | | + | | f<sub>0110</sub> |
− | <p>f<sub>0101</sub></p>
| + | | 0 1 1 0 |
− | <p>f<sub>1010</sub></p>
| + | | (x, y) |
− | |}
| + | | x not equal to y |
− | |
| + | | x ≠ y |
− | {| align="center"
| + | |- |
− | | | + | | f<sub>7</sub> |
− | <p>0 1 0 1</p>
| + | | f<sub>0111</sub> |
− | <p>1 0 1 0</p>
| + | | 0 1 1 1 |
− | |} | + | | (x y) |
− | |
| + | | not both x and y |
− | {| align="center"
| + | | ¬x ∨ ¬y |
− | |
| |
− | <p>(y)</p>
| |
− | <p> y </p>
| |
− | |} | |
− | | | |
− | {| align="center"
| |
− | |
| |
− | <p>not y</p> | |
− | <p>y</p> | |
− | |} | |
− | | | |
− | {| align="center"
| |
− | | | |
− | <p>¬y</p>
| |
− | <p>y</p>
| |
− | |}
| |
| |- | | |- |
− | | | + | | f<sub>8</sub> |
− | {| align="center"
| + | | f<sub>1000</sub> |
− | |
| + | | 1 0 0 0 |
− | <p>f<sub>7</sub></p>
| + | | x y |
− | <p>f<sub>11</sub></p>
| + | | x and y |
− | <p>f<sub>13</sub></p>
| + | | x ∧ y |
− | <p>f<sub>14</sub></p>
| + | |- |
− | |} | + | | f<sub>9</sub> |
− | | | + | | f<sub>1001</sub> |
− | {| align="center"
| + | | 1 0 0 1 |
− | | | + | | ((x, y)) |
− | <p>f<sub>0111</sub></p>
| + | | x equal to y |
− | <p>f<sub>1011</sub></p>
| + | | x = y |
− | <p>f<sub>1101</sub></p>
| |
− | <p>f<sub>1110</sub></p>
| |
− | |} | |
− | |
| |
− | {| align="center"
| |
− | |
| |
− | <p>0 1 1 1</p>
| |
− | <p>1 0 1 1</p>
| |
− | <p>1 1 0 1</p>
| |
− | <p>1 1 1 0</p>
| |
− | |}
| |
− | |
| |
− | {| align="center"
| |
− | | | |
− | <p>(x y)</p>
| |
− | <p>(x (y))</p>
| |
− | <p>((x) y)</p>
| |
− | <p>((x)(y))</p>
| |
− | |} | |
− | |
| |
− | {| align="center"
| |
− | |
| |
− | <p>not both x and y</p>
| |
− | <p>not x without y</p>
| |
− | <p>not y without x</p>
| |
− | <p>x or y</p>
| |
− | |}
| |
− | |
| |
− | {| align="center"
| |
− | |
| |
− | <p>¬x ∨ ¬y</p>
| |
− | <p>x ⇒ y</p>
| |
− | <p>x ⇐ y</p>
| |
− | <p>x ∨ y</p>
| |
− | |}
| |
| |- | | |- |
− | | f<sub>15</sub> | + | | f<sub>10</sub> |
− | | f<sub>1111</sub> | + | | f<sub>1010</sub> |
− | | 1 1 1 1 | + | | 1 0 1 0 |
− | | (( )) | + | | y |
− | | true | + | | y |
− | | 1 | + | | y |
− | |} | + | |- |
| + | | f<sub>11</sub> |
| + | | f<sub>1011</sub> |
| + | | 1 0 1 1 |
| + | | (x (y)) |
| + | | not x without y |
| + | | x ⇒ y |
| + | |- |
| + | | f<sub>12</sub> |
| + | | f<sub>1100</sub> |
| + | | 1 1 0 0 |
| + | | x |
| + | | x |
| + | | x |
| + | |- |
| + | | f<sub>13</sub> |
| + | | f<sub>1101</sub> |
| + | | 1 1 0 1 |
| + | | ((x) y) |
| + | | not y without x |
| + | | x ⇐ y |
| + | |- |
| + | | f<sub>14</sub> |
| + | | f<sub>1110</sub> |
| + | | 1 1 1 0 |
| + | | ((x)(y)) |
| + | | x or y |
| + | | x ∨ y |
| + | |- |
| + | | f<sub>15</sub> |
| + | | f<sub>1111</sub> |
| + | | 1 1 1 1 |
| + | | (( )) |
| + | | true || 1 |
| + | |} |
| | | |
| <br> | | <br> |
| | | |
− | ====Differential Propositions====
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
− | | + | |+ '''Table A2. Propositional Forms on Two Variables''' |
− | <br>
| + | |- style="background:#f0f0ff" |
− | | + | ! width="15%" | L<sub>1</sub> |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" | + | ! width="15%" | L<sub>2</sub> |
− | |+ '''Table 14. Differential Propositions''' | + | ! width="15%" | L<sub>3</sub> |
| + | ! width="15%" | L<sub>4</sub> |
| + | ! width="25%" | L<sub>5</sub> |
| + | ! width="15%" | L<sub>6</sub> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | | | | |
− | | align="right" | A : | + | | align="right" | x : |
| | 1 1 0 0 | | | 1 1 0 0 |
| | | | | |
Line 1,199: |
Line 1,367: |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | | | | |
− | | align="right" | dA : | + | | align="right" | y : |
| | 1 0 1 0 | | | 1 0 1 0 |
| | | | | |
Line 1,206: |
Line 1,374: |
| |- | | |- |
| | f<sub>0</sub> | | | f<sub>0</sub> |
− | | g<sub>0</sub> | + | | f<sub>0000</sub> |
| | 0 0 0 0 | | | 0 0 0 0 |
| | ( ) | | | ( ) |
− | | False | + | | false |
| | 0 | | | 0 |
| |- | | |- |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | <br>
| + | <p>f<sub>1</sub></p> |
− | <br>
| + | <p>f<sub>2</sub></p> |
− | <br>
| + | <p>f<sub>4</sub></p> |
− |
| + | <p>f<sub>8</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | g<sub>1</sub><br>
| + | <p>f<sub>0001</sub></p> |
− | g<sub>2</sub><br>
| + | <p>f<sub>0010</sub></p> |
− | g<sub>4</sub><br>
| + | <p>f<sub>0100</sub></p> |
− | g<sub>8</sub>
| + | <p>f<sub>1000</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | 0 0 0 1<br> | + | <p>0 0 0 1</p> |
− | 0 0 1 0<br> | + | <p>0 0 1 0</p> |
− | 0 1 0 0<br> | + | <p>0 1 0 0</p> |
− | 1 0 0 0 | + | <p>1 0 0 0</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | (A)(dA)<br> | + | <p>(x)(y)</p> |
− | (A) dA <br> | + | <p>(x) y </p> |
− | A (dA)<br>
| + | <p> x (y)</p> |
− | A dA
| + | <p> x y </p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | Neither A nor dA<br>
| + | <p>neither x nor y</p> |
− | Not A but dA<br>
| + | <p>not x but y</p> |
− | A but not dA<br>
| + | <p>x but not y</p> |
− | A and dA
| + | <p>x and y</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | ¬A ∧ ¬dA<br> | + | <p>¬x ∧ ¬y</p> |
− | ¬A ∧ dA<br> | + | <p>¬x ∧ y</p> |
− | A ∧ ¬dA<br>
| + | <p>x ∧ ¬y</p> |
− | A ∧ dA
| + | <p>x ∧ y</p> |
| |} | | |} |
| |- | | |- |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | f<sub>1</sub><br> | + | <p>f<sub>3</sub></p> |
− | f<sub>2</sub> | + | <p>f<sub>12</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | g<sub>3</sub><br>
| + | <p>f<sub>0011</sub></p> |
− | g<sub>12</sub>
| + | <p>f<sub>1100</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | 0 0 1 1<br> | + | <p>0 0 1 1</p> |
− | 1 1 0 0 | + | <p>1 1 0 0</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | (A)<br> | + | <p>(x)</p> |
− | A
| + | <p> x </p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | Not A<br>
| + | <p>not x</p> |
− | A
| + | <p>x</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | ¬A<br> | + | <p>¬x</p> |
− | A
| + | <p>x</p> |
| |} | | |} |
| |- | | |- |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | <br>
| + | <p>f<sub>6</sub></p> |
− |
| + | <p>f<sub>9</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | g<sub>6</sub><br>
| + | <p>f<sub>0110</sub></p> |
− | g<sub>9</sub>
| + | <p>f<sub>1001</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | 0 1 1 0<br> | + | <p>0 1 1 0</p> |
− | 1 0 0 1 | + | <p>1 0 0 1</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | (A, dA)<br> | + | <p> (x, y) </p> |
− | ((A, dA)) | + | <p>((x, y))</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | A not equal to dA<br>
| + | <p>x not equal to y</p> |
− | A equal to dA
| + | <p>x equal to y</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | A ≠ dA<br>
| + | <p>x ≠ y</p> |
− | A = dA
| + | <p>x = y</p> |
| |} | | |} |
| |- | | |- |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | <br>
| + | <p>f<sub>5</sub></p> |
− |
| + | <p>f<sub>10</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | g<sub>5</sub><br>
| + | <p>f<sub>0101</sub></p> |
− | g<sub>10</sub>
| + | <p>f<sub>1010</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | 0 1 0 1<br> | + | <p>0 1 0 1</p> |
− | 1 0 1 0 | + | <p>1 0 1 0</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | (dA)<br> | + | <p>(y)</p> |
− | dA
| + | <p> y </p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | Not dA<br>
| + | <p>not y</p> |
− | dA
| + | <p>y</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | ¬dA<br> | + | <p>¬y</p> |
− | dA
| + | <p>y</p> |
| |} | | |} |
| |- | | |- |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | <br>
| + | <p>f<sub>7</sub></p> |
− | <br>
| + | <p>f<sub>11</sub></p> |
− | <br>
| + | <p>f<sub>13</sub></p> |
− |
| + | <p>f<sub>14</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | g<sub>7</sub><br>
| + | <p>f<sub>0111</sub></p> |
− | g<sub>11</sub><br>
| + | <p>f<sub>1011</sub></p> |
− | g<sub>13</sub><br>
| + | <p>f<sub>1101</sub></p> |
− | g<sub>14</sub>
| + | <p>f<sub>1110</sub></p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | 0 1 1 1<br> | + | <p>0 1 1 1</p> |
− | 1 0 1 1<br> | + | <p>1 0 1 1</p> |
− | 1 1 0 1<br> | + | <p>1 1 0 1</p> |
− | 1 1 1 0 | + | <p>1 1 1 0</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | (A dA)<br> | + | <p>(x y)</p> |
− | (A (dA))<br> | + | <p>(x (y))</p> |
− | ((A) dA)<br> | + | <p>((x) y)</p> |
− | ((A)(dA)) | + | <p>((x)(y))</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | Not both A and dA<br>
| + | <p>not both x and y</p> |
− | Not A without dA<br>
| + | <p>not x without y</p> |
− | Not dA without A<br>
| + | <p>not y without x</p> |
− | A or dA
| + | <p>x or y</p> |
| |} | | |} |
| | | | | |
− | {| | + | {| align="center" |
| | | | | |
− | ¬A ∨ ¬dA<br> | + | <p>¬x ∨ ¬y</p> |
− | A ⇒ dA<br>
| + | <p>x ⇒ y</p> |
− | A ⇐ dA<br>
| + | <p>x ⇐ y</p> |
− | A ∨ dA
| + | <p>x ∨ y</p> |
| |} | | |} |
| |- | | |- |
− | | f<sub>3</sub> | + | | f<sub>15</sub> |
− | | g<sub>15</sub> | + | | f<sub>1111</sub> |
| | 1 1 1 1 | | | 1 1 1 1 |
| | (( )) | | | (( )) |
− | | True | + | | true |
| | 1 | | | 1 |
| |} | | |} |
Line 1,431: |
Line 1,599: |
| <br> | | <br> |
| | | |
− | ===Wiki Tables : Old Versions=== | + | ====Differential Propositions==== |
− | | |
− | ====Propositional Forms on Two Variables====
| |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%" |
− | |+ '''Table 1. Propositional Forms on Two Variables''' | + | |+ '''Table 14. Differential Propositions''' |
− | |- style="background:paleturquoise" | + | |- style="background:#f0f0ff" |
− | ! width="15%" | L<sub>1</sub>
| |
− | ! width="15%" | L<sub>2</sub>
| |
− | ! width="15%" | L<sub>3</sub>
| |
− | ! width="15%" | L<sub>4</sub>
| |
− | ! width="25%" | L<sub>5</sub>
| |
− | ! width="15%" | L<sub>6</sub>
| |
− | |- style="background:paleturquoise"
| |
| | | | | |
− | | align="right" | x : | + | | align="right" | A : |
| | 1 1 0 0 | | | 1 1 0 0 |
| | | | | |
| | | | | |
| | | | | |
− | |- style="background:paleturquoise" | + | |- style="background:#f0f0ff" |
| | | | | |
− | | align="right" | y : | + | | align="right" | dA : |
| | 1 0 1 0 | | | 1 0 1 0 |
| | | | | |
Line 1,461: |
Line 1,620: |
| | | | | |
| |- | | |- |
− | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 | + | | f<sub>0</sub> |
| + | | g<sub>0</sub> |
| + | | 0 0 0 0 |
| + | | ( ) |
| + | | False |
| + | | 0 |
| |- | | |- |
− | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y
| + | | |
− | |-
| + | {| |
− | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y
| + | | |
− | |-
| + | <br> |
− | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x
| + | <br> |
− | |-
| + | <br> |
− | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y
| + | |
− | |-
| + | |} |
− | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y
| + | | |
− | |-
| + | {| |
− | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
| + | | |
− | |-
| + | g<sub>1</sub><br> |
− | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y
| + | g<sub>2</sub><br> |
− | |-
| + | g<sub>4</sub><br> |
− | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y
| + | g<sub>8</sub> |
− | |-
| |
− | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
| |
− | |-
| |
− | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
| |
− | |-
| |
− | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y
| |
− | |-
| |
− | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
| |
− | |-
| |
− | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y
| |
− | |-
| |
− | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y
| |
− | |-
| |
− | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
| |
− | |}
| |
− | | |
− | <br>
| |
− | | |
− | ====Differential Propositions====
| |
− | | |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
| |
− | |+ '''Table 14. Differential Propositions'''
| |
− | |- style="background:ghostwhite"
| |
− | |
| |
− | | align="right" | A :
| |
− | | 1 1 0 0
| |
− | |
| |
− | |
| |
− | |
| |
− | |- style="background:ghostwhite"
| |
− | |
| |
− | | align="right" | dA :
| |
− | | 1 0 1 0
| |
− | |
| |
− | |
| |
− | |
| |
− | |-
| |
− | | f<sub>0</sub>
| |
− | | g<sub>0</sub>
| |
− | | 0 0 0 0
| |
− | | ( )
| |
− | | False
| |
− | | 0
| |
− | |-
| |
− | | | |
− | {| | |
− | | | |
− | <br> | |
− | <br> | |
− | <br> | |
− | | |
− | |} | |
− | | | |
− | {| | |
− | | | |
− | g<sub>1</sub><br> | |
− | g<sub>2</sub><br> | |
− | g<sub>4</sub><br> | |
− | g<sub>8</sub> | |
| |} | | |} |
| | | | | |
Line 1,728: |
Line 1,831: |
| | | | | |
| ¬A ∨ ¬dA<br> | | ¬A ∨ ¬dA<br> |
− | A → dA<br> | + | A ⇒ dA<br> |
− | A ← dA<br> | + | A ⇐ dA<br> |
| A ∨ dA | | A ∨ dA |
| |} | | |} |
Line 1,743: |
Line 1,846: |
| <br> | | <br> |
| | | |
− | ===Wiki TeX Tables : PQ=== | + | ===Wiki Tables : Old Versions=== |
| + | |
| + | ====Propositional Forms on Two Variables==== |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |
− | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> | + | |+ '''Table 1. Propositional Forms on Two Variables''' |
− | |- style="background:#f0f0ff" | + | |- style="background:paleturquoise" |
− | | width="15%" |
| + | ! width="15%" | L<sub>1</sub> |
− | <p><math>\mathcal{L}_1</math></p>
| + | ! width="15%" | L<sub>2</sub> |
− | <p><math>\text{Decimal}</math></p>
| + | ! width="15%" | L<sub>3</sub> |
− | | width="15%" |
| + | ! width="15%" | L<sub>4</sub> |
− | <p><math>\mathcal{L}_2</math></p>
| + | ! width="25%" | L<sub>5</sub> |
− | <p><math>\text{Binary}</math></p>
| + | ! width="15%" | L<sub>6</sub> |
− | | width="15%" |
| + | |- style="background:paleturquoise" |
− | <p><math>\mathcal{L}_3</math></p>
| |
− | <p><math>\text{Vector}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_4</math></p>
| |
− | <p><math>\text{Cactus}</math></p>
| |
− | | width="25%" |
| |
− | <p><math>\mathcal{L}_5</math></p>
| |
− | <p><math>\text{English}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_6</math></p>
| |
− | <p><math>\text{Ordinary}</math></p>
| |
− | |- style="background:#f0f0ff" | |
| | | | | |
− | | align="right" | <math>p\colon\!</math> | + | | align="right" | x : |
− | | <math>1~1~0~0\!</math> | + | | 1 1 0 0 |
| | | | | |
| | | | | |
| | | | | |
− | |- style="background:#f0f0ff" | + | |- style="background:paleturquoise" |
| | | | | |
− | | align="right" | <math>q\colon\!</math> | + | | align="right" | y : |
− | | <math>1~0~1~0\!</math> | + | | 1 0 1 0 |
| | | | | |
| | | | | |
| | | | | |
| |- | | |- |
| + | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0 |
| + | |- |
| + | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |
| + | |- |
| + | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |
| + | |- |
| + | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x |
| + | |- |
| + | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |
| + | |- |
| + | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y |
| + | |- |
| + | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |
| + | |- |
| + | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |
| + | |- |
| + | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y |
| + | |- |
| + | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y |
| + | |- |
| + | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y |
| + | |- |
| + | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y |
| + | |- |
| + | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x |
| + | |- |
| + | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y |
| + | |- |
| + | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |
| + | |- |
| + | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1 |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ====Differential Propositions==== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%" |
| + | |+ '''Table 14. Differential Propositions''' |
| + | |- style="background:ghostwhite" |
| + | | |
| + | | align="right" | A : |
| + | | 1 1 0 0 |
| + | | |
| + | | |
| + | | |
| + | |- style="background:ghostwhite" |
| + | | |
| + | | align="right" | dA : |
| + | | 1 0 1 0 |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | f<sub>0</sub> |
| + | | g<sub>0</sub> |
| + | | 0 0 0 0 |
| + | | ( ) |
| + | | False |
| + | | 0 |
| + | |- |
| + | | |
| + | {| |
| | | | | |
− | <math>\begin{matrix} | + | <br> |
− | f_0
| + | <br> |
− | \\[4pt]
| + | <br> |
− | f_1
| + | |
− | \\[4pt]
| + | |} |
− | f_2
| |
− | \\[4pt]
| |
− | f_3
| |
− | \\[4pt]
| |
− | f_4
| |
− | \\[4pt]
| |
− | f_5
| |
− | \\[4pt]
| |
− | f_6
| |
− | \\[4pt]
| |
− | f_7
| |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix} | |
− | f_{0000}
| |
− | \\[4pt]
| |
− | f_{0001}
| |
− | \\[4pt]
| |
− | f_{0010}
| |
− | \\[4pt]
| |
− | f_{0011}
| |
− | \\[4pt]
| |
− | f_{0100}
| |
− | \\[4pt]
| |
− | f_{0101}
| |
− | \\[4pt]
| |
− | f_{0110}
| |
− | \\[4pt]
| |
− | f_{0111}
| |
− | \end{matrix}</math>
| |
− | | | |
− | <math>\begin{matrix}
| |
− | 0~0~0~0
| |
− | \\[4pt]
| |
− | 0~0~0~1
| |
− | \\[4pt]
| |
− | 0~0~1~0
| |
− | \\[4pt]
| |
− | 0~0~1~1
| |
− | \\[4pt]
| |
− | 0~1~0~0
| |
− | \\[4pt]
| |
− | 0~1~0~1
| |
− | \\[4pt]
| |
− | 0~1~1~0
| |
− | \\[4pt]
| |
− | 0~1~1~1
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | (~)
| |
− | \\[4pt]
| |
− | (p)(q)
| |
− | \\[4pt]
| |
− | (p)~q~
| |
− | \\[4pt]
| |
− | (p)~~~
| |
− | \\[4pt]
| |
− | ~p~(q)
| |
− | \\[4pt]
| |
− | ~~~(q)
| |
− | \\[4pt]
| |
− | (p,~q)
| |
− | \\[4pt]
| |
− | (p~~q)
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | g<sub>1</sub><br> |
− | \text{false}
| + | g<sub>2</sub><br> |
− | \\[4pt]
| + | g<sub>4</sub><br> |
− | \text{neither}~ p ~\text{nor}~ q
| + | g<sub>8</sub> |
− | \\[4pt]
| + | |} |
− | q ~\text{without}~ p
| |
− | \\[4pt]
| |
− | \text{not}~ p
| |
− | \\[4pt]
| |
− | p ~\text{without}~ q
| |
− | \\[4pt]
| |
− | \text{not}~ q
| |
− | \\[4pt]
| |
− | p ~\text{not equal to}~ q
| |
− | \\[4pt]
| |
− | \text{not both}~ p ~\text{and}~ q
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | {| |
− | 0 | + | | |
− | \\[4pt]
| + | 0 0 0 1<br> |
− | \lnot p \land \lnot q
| + | 0 0 1 0<br> |
− | \\[4pt]
| + | 0 1 0 0<br> |
− | \lnot p \land q
| + | 1 0 0 0 |
− | \\[4pt]
| + | |} |
− | \lnot p
| + | | |
− | \\[4pt]
| + | {| |
− | p \land \lnot q
| + | | |
− | \\[4pt]
| + | (A)(dA)<br> |
− | \lnot q
| + | (A) dA <br> |
− | \\[4pt]
| + | A (dA)<br> |
− | p \ne q
| + | A dA |
− | \\[4pt]
| + | |} |
− | \lnot p \lor \lnot q
| |
− | \end{matrix}</math>
| |
− | |- | |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_8
| |
− | \\[4pt]
| |
− | f_9
| |
− | \\[4pt]
| |
− | f_{10}
| |
− | \\[4pt]
| |
− | f_{11}
| |
− | \\[4pt]
| |
− | f_{12}
| |
− | \\[4pt]
| |
− | f_{13}
| |
− | \\[4pt]
| |
− | f_{14}
| |
− | \\[4pt]
| |
− | f_{15}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | Neither A nor dA<br> |
− | f_{1000}
| + | Not A but dA<br> |
− | \\[4pt]
| + | A but not dA<br> |
− | f_{1001}
| + | A and dA |
− | \\[4pt]
| + | |} |
− | f_{1010}
| |
− | \\[4pt]
| |
− | f_{1011}
| |
− | \\[4pt]
| |
− | f_{1100}
| |
− | \\[4pt]
| |
− | f_{1101}
| |
− | \\[4pt]
| |
− | f_{1110}
| |
− | \\[4pt]
| |
− | f_{1111}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | 1~0~0~0
| |
− | \\[4pt]
| |
− | 1~0~0~1
| |
− | \\[4pt]
| |
− | 1~0~1~0
| |
− | \\[4pt]
| |
− | 1~0~1~1
| |
− | \\[4pt]
| |
− | 1~1~0~0
| |
− | \\[4pt]
| |
− | 1~1~0~1
| |
− | \\[4pt]
| |
− | 1~1~1~0
| |
− | \\[4pt]
| |
− | 1~1~1~1
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | ¬A ∧ ¬dA<br> |
− | ~~p~~q~~
| + | ¬A ∧ dA<br> |
− | \\[4pt]
| + | A ∧ ¬dA<br> |
− | ((p,~q))
| + | A ∧ dA |
− | \\[4pt]
| + | |} |
− | ~~~~~q~~
| + | |- |
− | \\[4pt]
| |
− | ~(p~(q))
| |
− | \\[4pt]
| |
− | ~~p~~~~~
| |
− | \\[4pt]
| |
− | ((p)~q)~
| |
− | \\[4pt]
| |
− | ((p)(q))
| |
− | \\[4pt]
| |
− | ((~))
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | p ~\text{and}~ q
| + | | |
− | \\[4pt]
| + | f<sub>1</sub><br> |
− | p ~\text{equal to}~ q
| + | f<sub>2</sub> |
− | \\[4pt]
| + | |} |
− | q
| + | | |
− | \\[4pt]
| + | {| |
− | \text{not}~ p ~\text{without}~ q
| |
− | \\[4pt]
| |
− | p
| |
− | \\[4pt]
| |
− | \text{not}~ q ~\text{without}~ p
| |
− | \\[4pt]
| |
− | p ~\text{or}~ q
| |
− | \\[4pt]
| |
− | \text{true}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | g<sub>3</sub><br> |
− | p \land q
| + | g<sub>12</sub> |
− | \\[4pt]
| + | |} |
− | p = q
| + | | |
− | \\[4pt]
| + | {| |
− | q
| + | | |
− | \\[4pt]
| + | 0 0 1 1<br> |
− | p \Rightarrow q
| + | 1 1 0 0 |
− | \\[4pt]
| + | |} |
− | p
| + | | |
− | \\[4pt]
| + | {| |
− | p \Leftarrow q
| + | | |
− | \\[4pt]
| + | (A)<br> |
− | p \lor q
| + | A |
− | \\[4pt]
| |
− | 1
| |
− | \end{matrix}</math>
| |
| |} | | |} |
− |
| |
− | <br>
| |
− |
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
| |
− | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_1</math></p>
| |
− | <p><math>\text{Decimal}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_2</math></p>
| |
− | <p><math>\text{Binary}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_3</math></p>
| |
− | <p><math>\text{Vector}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_4</math></p>
| |
− | <p><math>\text{Cactus}</math></p>
| |
− | | width="25%" |
| |
− | <p><math>\mathcal{L}_5</math></p>
| |
− | <p><math>\text{English}</math></p>
| |
− | | width="15%" |
| |
− | <p><math>\mathcal{L}_6</math></p>
| |
− | <p><math>\text{Ordinary}</math></p>
| |
− | |- style="background:#f0f0ff"
| |
− | |
| |
− | | align="right" | <math>p\colon\!</math>
| |
− | | <math>1~1~0~0\!</math>
| |
− | |
| |
− | |
| |
− | |
| |
− | |- style="background:#f0f0ff"
| |
− | |
| |
− | | align="right" | <math>q\colon\!</math>
| |
− | | <math>1~0~1~0\!</math>
| |
− | |
| |
− | |
| |
− | |
| |
− | |-
| |
− | | <math>f_0\!</math>
| |
− | | <math>f_{0000}\!</math>
| |
− | | <math>0~0~0~0</math>
| |
− | | <math>(~)</math>
| |
− | | <math>\text{false}\!</math>
| |
− | | <math>0\!</math>
| |
− | |-
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_1
| |
− | \\[4pt]
| |
− | f_2
| |
− | \\[4pt]
| |
− | f_4
| |
− | \\[4pt]
| |
− | f_8
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | Not A<br> |
− | f_{0001}
| + | A |
− | \\[4pt]
| + | |} |
− | f_{0010}
| |
− | \\[4pt]
| |
− | f_{0100}
| |
− | \\[4pt]
| |
− | f_{1000}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | 0~0~0~1
| |
− | \\[4pt]
| |
− | 0~0~1~0
| |
− | \\[4pt]
| |
− | 0~1~0~0
| |
− | \\[4pt]
| |
− | 1~0~0~0
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | ¬A<br> |
− | (p)(q)
| + | A |
− | \\[4pt]
| + | |} |
− | (p)~q~
| + | |- |
− | \\[4pt]
| |
− | ~p~(q)
| |
− | \\[4pt]
| |
− | ~p~~q~
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | \text{neither}~ p ~\text{nor}~ q
| |
− | \\[4pt]
| |
− | q ~\text{without}~ p
| |
− | \\[4pt]
| |
− | p ~\text{without}~ q
| |
− | \\[4pt]
| |
− | p ~\text{and}~ q
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | <br> |
− | \lnot p \land \lnot q
| + | |
− | \\[4pt]
| + | |} |
− | \lnot p \land q
| |
− | \\[4pt]
| |
− | p \land \lnot q
| |
− | \\[4pt]
| |
− | p \land q
| |
− | \end{matrix}</math>
| |
− | |-
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_3
| + | | |
− | \\[4pt]
| + | g<sub>6</sub><br> |
− | f_{12}
| + | g<sub>9</sub> |
− | \end{matrix}</math>
| + | |} |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_{0011}
| |
− | \\[4pt]
| |
− | f_{1100}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | 0 1 1 0<br> |
− | 0~0~1~1 | + | 1 0 0 1 |
− | \\[4pt]
| + | |} |
− | 1~1~0~0
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | (p)
| |
− | \\[4pt]
| |
− | ~p~
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | (A, dA)<br> |
− | \text{not}~ p
| + | ((A, dA)) |
− | \\[4pt]
| + | |} |
− | p
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | \lnot p
| |
− | \\[4pt]
| |
− | p
| |
− | \end{matrix}</math>
| |
− | |- | |
| | | | | |
− | <math>\begin{matrix} | + | A not equal to dA<br> |
− | f_6
| + | A equal to dA |
− | \\[4pt]
| + | |} |
− | f_9
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_{0110}
| |
− | \\[4pt]
| |
− | f_{1001}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | A ≠ dA<br> |
− | 0~1~1~0
| + | A = dA |
− | \\[4pt]
| + | |} |
− | 1~0~0~1
| + | |- |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | ~(p,~q)~
| |
− | \\[4pt]
| |
− | ((p,~q))
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | <br> |
− | p ~\text{not equal to}~ q
| + | |
− | \\[4pt]
| + | |} |
− | p ~\text{equal to}~ q
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | p \ne q
| |
− | \\[4pt]
| |
− | p = q
| |
− | \end{matrix}</math>
| |
− | |- | |
| | | | | |
− | <math>\begin{matrix} | + | g<sub>5</sub><br> |
− | f_5
| + | g<sub>10</sub> |
− | \\[4pt]
| + | |} |
− | f_{10}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_{0101}
| |
− | \\[4pt]
| |
− | f_{1010}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | 0 1 0 1<br> |
− | 0~1~0~1 | + | 1 0 1 0 |
− | \\[4pt]
| + | |} |
− | 1~0~1~0 | |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | (q)
| |
− | \\[4pt]
| |
− | ~q~
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | (dA)<br> |
− | \text{not}~ q
| + | dA |
− | \\[4pt]
| + | |} |
− | q
| + | | |
− | \end{matrix}</math>
| + | {| |
| + | | |
| + | Not dA<br> |
| + | dA |
| + | |} |
| + | | |
| + | {| |
| | | | | |
− | <math>\begin{matrix} | + | ¬dA<br> |
− | \lnot q
| + | dA |
− | \\[4pt]
| + | |} |
− | q
| |
− | \end{matrix}</math>
| |
| |- | | |- |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_7
| |
− | \\[4pt]
| |
− | f_{11}
| |
− | \\[4pt]
| |
− | f_{13}
| |
− | \\[4pt]
| |
− | f_{14}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | <br> |
− | f_{0111}
| + | <br> |
− | \\[4pt]
| + | <br> |
− | f_{1011}
| + | |
− | \\[4pt]
| + | |} |
− | f_{1101}
| |
− | \\[4pt]
| |
− | f_{1110}
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | 0~1~1~1
| |
− | \\[4pt]
| |
− | 1~0~1~1
| |
− | \\[4pt]
| |
− | 1~1~0~1
| |
− | \\[4pt]
| |
− | 1~1~1~0
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | g<sub>7</sub><br> |
− | ~(p~~q)~
| + | g<sub>11</sub><br> |
− | \\[4pt]
| + | g<sub>13</sub><br> |
− | ~(p~(q))
| + | g<sub>14</sub> |
− | \\[4pt]
| + | |} |
− | ((p)~q)~
| |
− | \\[4pt]
| |
− | ((p)(q))
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | \text{not both}~ p ~\text{and}~ q
| |
− | \\[4pt]
| |
− | \text{not}~ p ~\text{without}~ q
| |
− | \\[4pt]
| |
− | \text{not}~ q ~\text{without}~ p
| |
− | \\[4pt]
| |
− | p ~\text{or}~ q
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | 0 1 1 1<br> |
− | \lnot p \lor \lnot q
| + | 1 0 1 1<br> |
− | \\[4pt]
| + | 1 1 0 1<br> |
− | p \Rightarrow q
| + | 1 1 1 0 |
− | \\[4pt]
| + | |} |
− | p \Leftarrow q
| + | | |
− | \\[4pt]
| + | {| |
− | p \lor q
| + | | |
− | \end{matrix}</math>
| + | (A dA)<br> |
− | |-
| + | (A (dA))<br> |
− | | <math>f_{15}\!</math> | + | ((A) dA)<br> |
− | | <math>f_{1111}\!</math> | + | ((A)(dA)) |
− | | <math>1~1~1~1</math> | |
− | | <math>((~))</math>
| |
− | | <math>\text{true}\!</math>
| |
− | | <math>1\!</math>
| |
| |} | | |} |
− |
| |
− | <br>
| |
− |
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
| |
− | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="10%" |
| |
− | | width="18%" | <math>f\!</math>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{11} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{10} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{01} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{00} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
| |
− | |-
| |
− | | <math>f_0\!</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | |-
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | f_1
| |
− | \\[4pt]
| |
− | f_2
| |
− | \\[4pt]
| |
− | f_4
| |
− | \\[4pt]
| |
− | f_8
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | Not both A and dA<br> |
− | (p)(q)
| + | Not A without dA<br> |
− | \\[4pt]
| + | Not dA without A<br> |
− | (p)~q~
| + | A or dA |
− | \\[4pt]
| + | |} |
− | ~p~(q)
| |
− | \\[4pt]
| |
− | ~p~~q~
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix}
| + | {| |
− | ~p~~q~
| |
− | \\[4pt]
| |
− | ~p~(q)
| |
− | \\[4pt]
| |
− | (p)~q~
| |
− | \\[4pt]
| |
− | (p)(q)
| |
− | \end{matrix}</math>
| |
| | | | | |
− | <math>\begin{matrix} | + | ¬A ∨ ¬dA<br> |
− | ~p~(q)
| + | A → dA<br> |
− | \\[4pt]
| + | A ← dA<br> |
− | ~p~~q~
| + | A ∨ dA |
− | \\[4pt]
| + | |} |
− | (p)(q) | + | |- |
− | \\[4pt] | + | | f<sub>3</sub> |
− | (p)~q~
| + | | g<sub>15</sub> |
− | \end{matrix}</math> | + | | 1 1 1 1 |
− | | | + | | (( )) |
− | <math>\begin{matrix} | + | | True |
− | (p)~q~
| + | | 1 |
− | \\[4pt] | + | |} |
− | (p)(q)
| + | |
− | \\[4pt] | + | <br> |
− | ~p~~q~
| + | |
− | \\[4pt] | + | ===Wiki TeX Tables : PQ=== |
− | ~p~(q)
| + | |
− | \end{matrix}</math> | + | <br> |
− | | | + | |
− | <math>\begin{matrix} | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | (p)(q)
| + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
− | \\[4pt] | + | |- style="background:#f0f0ff" |
− | (p)~q~
| + | | width="15%" | |
− | \\[4pt]
| + | <p><math>\mathcal{L}_1</math></p> |
− | ~p~(q)
| + | <p><math>\text{Decimal}</math></p> |
− | \\[4pt] | + | | width="15%" | |
− | ~p~~q~ | + | <p><math>\mathcal{L}_2</math></p> |
− | \end{matrix}</math> | + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>p\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>q\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_3
| + | f_0 |
− | \\[4pt] | + | \\[4pt] |
− | f_{12}
| + | f_1 |
− | \end{matrix}</math> | + | \\[4pt] |
− | |
| + | f_2 |
− | <math>\begin{matrix}
| + | \\[4pt] |
− | (p)
| + | f_3 |
− | \\[4pt] | + | \\[4pt] |
− | ~p~
| + | f_4 |
− | \end{matrix}</math> | + | \\[4pt] |
− | |
| + | f_5 |
− | <math>\begin{matrix}
| + | \\[4pt] |
− | ~p~
| + | f_6 |
| \\[4pt] | | \\[4pt] |
− | (p)
| + | f_7 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~p~
| + | f_{0000} |
| \\[4pt] | | \\[4pt] |
− | (p)
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0011} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{0101} |
| + | \\[4pt] |
| + | f_{0110} |
| + | \\[4pt] |
| + | f_{0111} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (p)
| + | 0~0~0~0 |
| + | \\[4pt] |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 0~1~0~1 |
| + | \\[4pt] |
| + | 0~1~1~0 |
| \\[4pt] | | \\[4pt] |
− | ~p~ | + | 0~1~1~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (p) | + | (~) |
| + | \\[4pt] |
| + | (p)(q) |
| + | \\[4pt] |
| + | (p)~q~ |
| + | \\[4pt] |
| + | (p)~~~ |
| + | \\[4pt] |
| + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | ~p~ | + | ~~~(q) |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_6
| |
| \\[4pt] | | \\[4pt] |
− | f_9
| + | (p,~q) |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~(p,~q)~
| |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | (p~~q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(p,~q)~
| + | \text{false} |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | \text{neither}~ p ~\text{nor}~ q |
| + | \\[4pt] |
| + | q ~\text{without}~ p |
| + | \\[4pt] |
| + | \text{not}~ p |
| + | \\[4pt] |
| + | p ~\text{without}~ q |
| + | \\[4pt] |
| + | \text{not}~ q |
| + | \\[4pt] |
| + | p ~\text{not equal to}~ q |
| + | \\[4pt] |
| + | \text{not both}~ p ~\text{and}~ q |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((p,~q))
| + | 0 |
| + | \\[4pt] |
| + | \lnot p \land \lnot q |
| + | \\[4pt] |
| + | \lnot p \land q |
| + | \\[4pt] |
| + | \lnot p |
| + | \\[4pt] |
| + | p \land \lnot q |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~
| + | \lnot q |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ((p,~q))
| |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~
| + | p \ne q |
− | \end{matrix}</math> | |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~(p,~q)~
| |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | \lnot p \lor \lnot q |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_5
| + | f_8 |
| + | \\[4pt] |
| + | f_9 |
| \\[4pt] | | \\[4pt] |
| f_{10} | | f_{10} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (q)
| |
| \\[4pt] | | \\[4pt] |
− | ~q~
| + | f_{11} |
| + | \\[4pt] |
| + | f_{12} |
| + | \\[4pt] |
| + | f_{13} |
| + | \\[4pt] |
| + | f_{14} |
| + | \\[4pt] |
| + | f_{15} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~q~
| + | f_{1000} |
| \\[4pt] | | \\[4pt] |
− | (q)
| + | f_{1001} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (q)
| |
| \\[4pt] | | \\[4pt] |
− | ~q~
| + | f_{1010} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~q~
| |
| \\[4pt] | | \\[4pt] |
− | (q)
| + | f_{1011} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (q)
| |
| \\[4pt] | | \\[4pt] |
− | ~q~
| + | f_{1100} |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_7
| |
| \\[4pt] | | \\[4pt] |
− | f_{11} | + | f_{1101} |
| \\[4pt] | | \\[4pt] |
− | f_{13} | + | f_{1110} |
| \\[4pt] | | \\[4pt] |
− | f_{14} | + | f_{1111} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~p~~q~)
| + | 1~0~0~0 |
| + | \\[4pt] |
| + | 1~0~0~1 |
| \\[4pt] | | \\[4pt] |
− | (~p~(q))
| + | 1~0~1~0 |
| \\[4pt] | | \\[4pt] |
− | ((p)~q~)
| + | 1~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | ((p)(q))
| + | 1~1~0~0 |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ((p)(q))
| |
| \\[4pt] | | \\[4pt] |
− | ((p)~q~)
| + | 1~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | (~p~(q))
| + | 1~1~1~0 |
| \\[4pt] | | \\[4pt] |
− | (~p~~q~)
| + | 1~1~1~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((p)~q~)
| + | ~~p~~q~~ |
| \\[4pt] | | \\[4pt] |
− | ((p)(q)) | + | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | (~p~~q~)
| + | ~~~~~q~~ |
| \\[4pt] | | \\[4pt] |
− | (~p~(q)) | + | ~(p~(q)) |
| + | \\[4pt] |
| + | ~~p~~~~~ |
| + | \\[4pt] |
| + | ((p)~q)~ |
| + | \\[4pt] |
| + | ((p)(q)) |
| + | \\[4pt] |
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~p~(q))
| + | p ~\text{and}~ q |
| + | \\[4pt] |
| + | p ~\text{equal to}~ q |
| + | \\[4pt] |
| + | q |
| + | \\[4pt] |
| + | \text{not}~ p ~\text{without}~ q |
| + | \\[4pt] |
| + | p |
| \\[4pt] | | \\[4pt] |
− | (~p~~q~)
| + | \text{not}~ q ~\text{without}~ p |
| \\[4pt] | | \\[4pt] |
− | ((p)(q))
| + | p ~\text{or}~ q |
| \\[4pt] | | \\[4pt] |
− | ((p)~q~)
| + | \text{true} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~p~~q~)
| + | p \land q |
| + | \\[4pt] |
| + | p = q |
| + | \\[4pt] |
| + | q |
| + | \\[4pt] |
| + | p \Rightarrow q |
| + | \\[4pt] |
| + | p |
| \\[4pt] | | \\[4pt] |
− | (~p~(q))
| + | p \Leftarrow q |
| \\[4pt] | | \\[4pt] |
− | ((p)~q~)
| + | p \lor q |
| \\[4pt] | | \\[4pt] |
− | ((p)(q))
| + | 1 |
| \end{matrix}</math> | | \end{matrix}</math> |
− | |-
| |
− | | <math>f_{15}\!</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | colspan="2" | <math>\text{Fixed Point Total}\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>16\!</math>
| |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> | + | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
− | | width="10%" | | + | | width="15%" | |
− | | width="18%" | <math>f\!</math> | + | <p><math>\mathcal{L}_1</math></p> |
− | | width="18%" | | + | <p><math>\text{Decimal}</math></p> |
− | <math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math> | + | | width="15%" | |
− | | width="18%" | | + | <p><math>\mathcal{L}_2</math></p> |
− | <math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math> | + | <p><math>\text{Binary}</math></p> |
− | | width="18%" | | + | | width="15%" | |
− | <math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math> | + | <p><math>\mathcal{L}_3</math></p> |
− | | width="18%" | | + | <p><math>\text{Vector}</math></p> |
− | <math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math> | + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>p\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>q\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| |- | | |- |
| | <math>f_0\!</math> | | | <math>f_0\!</math> |
| + | | <math>f_{0000}\!</math> |
| + | | <math>0~0~0~0</math> |
| | <math>(~)</math> | | | <math>(~)</math> |
− | | <math>(~)</math> | + | | <math>\text{false}\!</math> |
− | | <math>(~)</math> | + | | <math>0\!</math> |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
| |- | | |- |
| | | | | |
Line 2,597: |
Line 2,474: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (p)(q)
| + | f_{0001} |
| \\[4pt] | | \\[4pt] |
− | (p)~q~
| + | f_{0010} |
| \\[4pt] | | \\[4pt] |
− | ~p~(q)
| + | f_{0100} |
| \\[4pt] | | \\[4pt] |
− | ~p~~q~
| + | f_{1000} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((p,~q))
| + | 0~0~0~1 |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~ | + | 0~0~1~0 |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~ | + | 0~1~0~0 |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | 1~0~0~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (q) | + | (p)(q) |
| \\[4pt] | | \\[4pt] |
− | ~q~ | + | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | (q) | + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | ~q~ | + | ~p~~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (p)
| + | \text{neither}~ p ~\text{nor}~ q |
| \\[4pt] | | \\[4pt] |
− | (p)
| + | q ~\text{without}~ p |
| \\[4pt] | | \\[4pt] |
− | ~p~ | + | p ~\text{without}~ q |
| \\[4pt] | | \\[4pt] |
− | ~p~ | + | p ~\text{and}~ q |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \lnot p \land \lnot q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | \lnot p \land q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p \land \lnot q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p \land q |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,654: |
Line 2,531: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (p)
| + | f_{0011} |
| \\[4pt] | | \\[4pt] |
− | ~p~
| + | f_{1100} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | 0~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | 1~1~0~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~)) | + | (p) |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | ~p~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \text{not}~ p |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \lnot p |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,691: |
Line 2,568: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(p,~q)~
| + | f_{0110} |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | f_{1001} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | 0~1~1~0 |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | 1~0~0~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~)) | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ((~)) | + | ((p,~q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | p ~\text{not equal to}~ q |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | p ~\text{equal to}~ q |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | p \ne q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p = q |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,728: |
Line 2,605: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (q)
| + | f_{0101} |
| \\[4pt] | | \\[4pt] |
− | ~q~
| + | f_{1010} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | 0~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | 1~0~1~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~) | + | (q) |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | ~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | \text{not}~ q |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | q |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \lnot q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | q |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,769: |
Line 2,646: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(p~~q)~
| + | f_{0111} |
| \\[4pt] | | \\[4pt] |
− | ~(p~(q))
| + | f_{1011} |
| \\[4pt] | | \\[4pt] |
− | ((p)~q)~
| + | f_{1101} |
| \\[4pt] | | \\[4pt] |
− | ((p)(q))
| + | f_{1110} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((p,~q))
| + | 0~1~1~1 |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~ | + | 1~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | ~(p,~q)~ | + | 1~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | ((p,~q))
| + | 1~1~1~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~q~ | + | ~(p~~q)~ |
| \\[4pt] | | \\[4pt] |
− | (q) | + | ~(p~(q)) |
| \\[4pt] | | \\[4pt] |
− | ~q~ | + | ((p)~q)~ |
| \\[4pt] | | \\[4pt] |
− | (q) | + | ((p)(q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~p~ | + | \text{not both}~ p ~\text{and}~ q |
| \\[4pt] | | \\[4pt] |
− | ~p~ | + | \text{not}~ p ~\text{without}~ q |
| \\[4pt] | | \\[4pt] |
− | (p)
| + | \text{not}~ q ~\text{without}~ p |
| \\[4pt] | | \\[4pt] |
− | (p)
| + | p ~\text{or}~ q |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \lnot p \lor \lnot q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p \Rightarrow q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p \Leftarrow q |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | p \lor q |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
| + | | <math>f_{1111}\!</math> |
| + | | <math>1~1~1~1</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
− | | <math>(~)</math> | + | | <math>\text{true}\!</math> |
− | | <math>(~)</math> | + | | <math>1\!</math> |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> | + | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="10%" | | | | width="10%" | |
| | width="18%" | <math>f\!</math> | | | width="18%" | <math>f\!</math> |
− | | width="18%" | <math>\operatorname{E}f|_{xy}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{E}f|_{p(q)}</math> | + | <p><math>\operatorname{T}_{11} f</math></p> |
− | | width="18%" | <math>\operatorname{E}f|_{(p)q}</math> | + | <p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p> |
− | | width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math> | + | | width="18%" | |
| + | <p><math>\operatorname{T}_{10} f</math></p> |
| + | <p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{01} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{00} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p> |
| |- | | |- |
| | <math>f_0\!</math> | | | <math>f_0\!</math> |
Line 2,867: |
Line 2,752: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}p~~\operatorname{d}q~ | + | ~p~~q~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~(\operatorname{d}q) | + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)~\operatorname{d}q~ | + | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)(\operatorname{d}q) | + | (p)(q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}p~(\operatorname{d}q) | + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~~\operatorname{d}q~ | + | ~p~~q~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)(\operatorname{d}q) | + | (p)(q) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)~\operatorname{d}q~ | + | (p)~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p)~\operatorname{d}q~ | + | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)(\operatorname{d}q) | + | (p)(q) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~~\operatorname{d}q~ | + | ~p~~q~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~(\operatorname{d}q) | + | ~p~(q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p)(\operatorname{d}q) | + | (p)(q) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p)~\operatorname{d}q~ | + | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~(\operatorname{d}q) | + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~~\operatorname{d}q~ | + | ~p~~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,920: |
Line 2,805: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}p~ | + | ~p~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p) | + | (p) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}p~ | + | ~p~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p) | + | (p) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p) | + | (p) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~ | + | ~p~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p) | + | (p) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}p~ | + | ~p~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,957: |
Line 2,842: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}p,~\operatorname{d}q)~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p,~\operatorname{d}q)) | + | ((p,~q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p,~\operatorname{d}q)) | + | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p,~\operatorname{d}q)~ | + | ~(p,~q)~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p,~\operatorname{d}q)) | + | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p,~\operatorname{d}q)~ | + | ~(p,~q)~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}p,~\operatorname{d}q)~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p,~\operatorname{d}q)) | + | ((p,~q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 2,994: |
Line 2,879: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}q~ | + | ~q~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}q) | + | (q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}q) | + | (q) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}q~ | + | ~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}q~ | + | ~q~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}q) | + | (q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}q) | + | (q) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}q~ | + | ~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,039: |
Line 2,924: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p)(q)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)~\operatorname{d}q~) | + | ((p)~q~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~(\operatorname{d}q)) | + | (~p~(q)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~~\operatorname{d}q~) | + | (~p~~q~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p)~\operatorname{d}q~) | + | ((p)~q~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p)(q)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~~\operatorname{d}q~) | + | (~p~~q~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~(\operatorname{d}q)) | + | (~p~(q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~\operatorname{d}p~(\operatorname{d}q)) | + | (~p~(q)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~~\operatorname{d}q~) | + | (~p~~q~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p)(q)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)~\operatorname{d}q~) | + | ((p)~q~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~\operatorname{d}p~~\operatorname{d}q~) | + | (~p~~q~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}p~(\operatorname{d}q)) | + | (~p~(q)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)~\operatorname{d}q~) | + | ((p)~q~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p)(q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,084: |
Line 2,969: |
| | <math>((~))</math> | | | <math>((~))</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
| + | |- style="background:#f0f0ff" |
| + | | colspan="2" | <math>\text{Fixed Point Total}\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>16\!</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> | + | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="10%" | | | | width="10%" | |
| | width="18%" | <math>f\!</math> | | | width="18%" | <math>f\!</math> |
− | | width="18%" | <math>\operatorname{D}f|_{xy}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{D}f|_{p(q)}</math> | + | <math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math> |
− | | width="18%" | <math>\operatorname{D}f|_{(p)q}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math> | + | <math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math> |
| |- | | |- |
| | <math>f_0\!</math> | | | <math>f_0\!</math> |
Line 3,127: |
Line 3,022: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}p~~\operatorname{d}q~~
| + | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p,~q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}p~(\operatorname{d}q)~
| + | (q) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | ~q~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (q) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | ~q~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}p)~\operatorname{d}q~~
| + | (p) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (p) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | ~p~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | ~p~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,180: |
Line 3,075: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}p
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}p
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}p
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}p
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}p
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}p
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}p
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}p
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,217: |
Line 3,112: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p,~\operatorname{d}q) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p,~\operatorname{d}q) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p,~\operatorname{d}q) | + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p,~\operatorname{d}q) | + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p,~\operatorname{d}q) | + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p,~\operatorname{d}q) | + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}p,~\operatorname{d}q) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}p,~\operatorname{d}q) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,254: |
Line 3,149: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}q
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}q
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}q
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}q
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}q
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}q
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}q
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}q
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 3,289: |
Line 3,184: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~p~~q~) | + | ~(p~~q)~ |
| \\[4pt] | | \\[4pt] |
− | (~p~(q)) | + | ~(p~(q)) |
| \\[4pt] | | \\[4pt] |
− | ((p)~q~) | + | ((p)~q)~ |
| \\[4pt] | | \\[4pt] |
| ((p)(q)) | | ((p)(q)) |
Line 3,299: |
Line 3,194: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | ((p,~q)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~
| + | ((p,~q)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | ~q~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (q) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | ~q~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~
| + | (q) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | ~p~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | ~p~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (p) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~
| + | (p) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}p~~\operatorname{d}q~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}p~(\operatorname{d}q)~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}p)~\operatorname{d}q~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}p)(\operatorname{d}q)) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | ===Wiki TeX Tables : XY===
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | | + | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | |
− | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> | |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
− | | width="15%" | | + | | width="10%" | |
− | <p><math>\mathcal{L}_1</math></p>
| + | | width="18%" | <math>f\!</math> |
− | <p><math>\text{Decimal}</math></p>
| + | | width="18%" | <math>\operatorname{E}f|_{xy}</math> |
− | | width="15%" | | + | | width="18%" | <math>\operatorname{E}f|_{p(q)}</math> |
− | <p><math>\mathcal{L}_2</math></p>
| + | | width="18%" | <math>\operatorname{E}f|_{(p)q}</math> |
− | <p><math>\text{Binary}</math></p>
| + | | width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math> |
− | | width="15%" | | + | |- |
− | <p><math>\mathcal{L}_3</math></p>
| + | | <math>f_0\!</math> |
− | <p><math>\text{Vector}</math></p>
| + | | <math>(~)</math> |
− | | width="15%" | | + | | <math>(~)</math> |
− | <p><math>\mathcal{L}_4</math></p>
| + | | <math>(~)</math> |
− | <p><math>\text{Cactus}</math></p>
| + | | <math>(~)</math> |
− | | width="25%" | | + | | <math>(~)</math> |
− | <p><math>\mathcal{L}_5</math></p> | + | |- |
− | <p><math>\text{English}</math></p>
| + | | |
− | | width="15%" | | + | <math>\begin{matrix} |
− | <p><math>\mathcal{L}_6</math></p>
| + | f_1 |
− | <p><math>\text{Ordinary}</math></p>
| + | \\[4pt] |
− | |- style="background:#f0f0ff" | + | f_2 |
− | | | + | \\[4pt] |
− | | align="right" | <math>x\colon\!</math>
| + | f_4 |
− | | <math>1~1~0~0\!</math> | + | \\[4pt] |
− | |
| + | f_8 |
− | |
| + | \end{matrix}</math> |
− | |
| + | | |
− | |- style="background:#f0f0ff"
| + | <math>\begin{matrix} |
− | |
| + | (p)(q) |
− | | align="right" | <math>y\colon\!</math>
| + | \\[4pt] |
− | | <math>1~0~1~0\!</math> | + | (p)~q~ |
− | |
| + | \\[4pt] |
− | |
| + | ~p~(q) |
− | |
| + | \\[4pt] |
− | |-
| + | ~p~~q~ |
− | | <math>f_{0}\!</math>
| + | \end{matrix}</math> |
− | | <math>f_{0000}\!</math> | + | | |
− | | <math>0~0~0~0\!</math>
| + | <math>\begin{matrix} |
− | | <math>(~)\!</math>
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
− | | <math>\text{false}\!</math>
| + | \\[4pt] |
− | | <math>0\!</math>
| + | ~\operatorname{d}p~(\operatorname{d}q) |
− | |- | + | \\[4pt] |
− | | <math>f_{1}\!</math>
| + | (\operatorname{d}p)~\operatorname{d}q~ |
− | | <math>f_{0001}\!</math>
| + | \\[4pt] |
− | | <math>0~0~0~1\!</math>
| + | (\operatorname{d}p)(\operatorname{d}q) |
− | | <math>(x)(y)\!</math> | + | \end{matrix}</math> |
− | | <math>\text{neither}~ x ~\text{nor}~ y\!</math>
| + | | |
− | | <math>\lnot x \land \lnot y\!</math>
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \\[4pt] |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p)(\operatorname{d}q) |
| + | \\[4pt] |
| + | (\operatorname{d}p)~\operatorname{d}q~ |
| + | \\[4pt] |
| + | ~\operatorname{d}p~(\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | <math>f_{2}\!</math> | + | | |
− | | <math>f_{0010}\!</math> | + | <math>\begin{matrix} |
− | | <math>0~0~1~0\!</math> | + | f_3 |
− | | <math>(x)~y\!</math>
| + | \\[4pt] |
− | | <math>y ~\text{without}~ x\!</math> | + | f_{12} |
− | | <math>\lnot x \land y\!</math>
| + | \end{matrix}</math> |
− | |- | + | | |
− | | <math>f_{3}\!</math>
| + | <math>\begin{matrix} |
− | | <math>f_{0011}\!</math>
| + | (p) |
− | | <math>0~0~1~1\!</math>
| + | \\[4pt] |
− | | <math>(x)\!</math> | + | ~p~ |
− | | <math>\text{not}~ x\!</math>
| + | \end{matrix}</math> |
− | | <math>\lnot x\!</math>
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}p~ |
| + | \\[4pt] |
| + | (\operatorname{d}p) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}p~ |
| + | \\[4pt] |
| + | (\operatorname{d}p) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p) |
| + | \\[4pt] |
| + | ~\operatorname{d}p~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | <math>f_{4}\!</math> | + | | |
− | | <math>f_{0100}\!</math> | + | <math>\begin{matrix} |
− | | <math>0~1~0~0\!</math>
| + | f_6 |
− | | <math>x~(y)\!</math>
| + | \\[4pt] |
− | | <math>x ~\text{without}~ y\!</math> | + | f_9 |
− | | <math>x \land \lnot y\!</math>
| + | \end{matrix}</math> |
− | |- | + | | |
− | | <math>f_{5}\!</math>
| + | <math>\begin{matrix} |
− | | <math>f_{0101}\!</math>
| + | ~(p,~q)~ |
− | | <math>0~1~0~1\!</math>
| + | \\[4pt] |
− | | <math>(y)\!</math> | + | ((p,~q)) |
− | | <math>\text{not}~ y\!</math>
| + | \end{matrix}</math> |
− | | <math>\lnot y\!</math>
| + | | |
− | |-
| + | <math>\begin{matrix} |
− | | <math>f_{6}\!</math>
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
− | | <math>f_{0110}\!</math>
| + | \\[4pt] |
− | | <math>0~1~1~0\!</math> | + | ((\operatorname{d}p,~\operatorname{d}q)) |
− | | <math>(x,~y)\!</math>
| + | \end{matrix}</math> |
− | | <math>x ~\text{not equal to}~ y\!</math>
| + | | |
− | | <math>x \ne y\!</math>
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}p,~\operatorname{d}q)~ |
| + | \\[4pt] |
| + | ((\operatorname{d}p,~\operatorname{d}q)) |
| + | \end{matrix}</math> |
| |- | | |- |
− | | <math>f_{7}\!</math> | + | | |
− | | <math>f_{0111}\!</math>
| + | <math>\begin{matrix} |
− | | <math>0~1~1~1\!</math> | + | f_5 |
− | | <math>(x~y)\!</math>
| + | \\[4pt] |
− | | <math>\text{not both}~ x ~\text{and}~ y\!</math> | + | f_{10} |
− | | <math>\lnot x \lor \lnot y\!</math>
| + | \end{matrix}</math> |
− | |- | + | | |
− | | <math>f_{8}\!</math>
| + | <math>\begin{matrix} |
− | | <math>f_{1000}\!</math>
| + | (q) |
− | | <math>1~0~0~0\!</math>
| + | \\[4pt] |
− | | <math>x~y\!</math> | + | ~q~ |
− | | <math>x ~\text{and}~ y\!</math>
| + | \end{matrix}</math> |
− | | <math>x \land y\!</math>
| + | | |
− | |-
| + | <math>\begin{matrix} |
− | | <math>f_{9}\!</math>
| + | ~\operatorname{d}q~ |
− | | <math>f_{1001}\!</math>
| + | \\[4pt] |
− | | <math>1~0~0~1\!</math> | + | (\operatorname{d}q) |
− | | <math>((x,~y))\!</math>
| + | \end{matrix}</math> |
− | | <math>x ~\text{equal to}~ y\!</math>
| + | | |
− | | <math>x = y\!</math>
| + | <math>\begin{matrix} |
| + | (\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~\operatorname{d}q~ |
| + | \\[4pt] |
| + | (\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}q) |
| + | \\[4pt] |
| + | ~\operatorname{d}q~ |
| + | \end{matrix}</math> |
| |- | | |- |
− | | <math>f_{10}\!</math> | + | | |
− | | <math>f_{1010}\!</math>
| + | <math>\begin{matrix} |
− | | <math>1~0~1~0\!</math> | + | f_7 |
− | | <math>y\!</math>
| + | \\[4pt] |
− | | <math>y\!</math>
| + | f_{11} |
− | | <math>y\!</math>
| + | \\[4pt] |
− | |- | + | f_{13} |
− | | <math>f_{11}\!</math>
| + | \\[4pt] |
− | | <math>f_{1011}\!</math>
| + | f_{14} |
− | | <math>1~0~1~1\!</math>
| + | \end{matrix}</math> |
− | | <math>(x~(y))\!</math>
| + | | |
− | | <math>\text{not}~ x ~\text{without}~ y\!</math>
| + | <math>\begin{matrix} |
− | | <math>x \Rightarrow y\!</math> | + | (~p~~q~) |
− | |-
| + | \\[4pt] |
− | | <math>f_{12}\!</math>
| + | (~p~(q)) |
− | | <math>f_{1100}\!</math>
| + | \\[4pt] |
− | | <math>1~1~0~0\!</math>
| + | ((p)~q~) |
− | | <math>x\!</math>
| + | \\[4pt] |
− | | <math>x\!</math>
| + | ((p)(q)) |
− | | <math>x\!</math>
| + | \end{matrix}</math> |
− | |- | + | | |
− | | <math>f_{13}\!</math>
| + | <math>\begin{matrix} |
− | | <math>f_{1101}\!</math>
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | | <math>1~1~0~1\!</math>
| + | \\[4pt] |
− | | <math>((x)~y)\!</math>
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
− | | <math>\text{not}~ y ~\text{without}~ x\!</math>
| + | \\[4pt] |
− | | <math>x \Leftarrow y\!</math>
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
− | |- | + | \\[4pt] |
− | | <math>f_{14}\!</math>
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
− | | <math>f_{1110}\!</math>
| + | \end{matrix}</math> |
− | | <math>1~1~1~0\!</math>
| + | | |
− | | <math>((x)(y))\!</math>
| + | <math>\begin{matrix} |
− | | <math>x ~\text{or}~ y\!</math>
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
− | | <math>x \lor y\!</math>
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (~\operatorname{d}p~~\operatorname{d}q~) |
| + | \\[4pt] |
| + | (~\operatorname{d}p~(\operatorname{d}q)) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)~\operatorname{d}q~) |
| + | \\[4pt] |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| + | \end{matrix}</math> |
| |- | | |- |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
− | | <math>f_{1111}\!</math> | + | | <math>((~))</math> |
− | | <math>1~1~1~1\!</math> | + | | <math>((~))</math> |
− | | <math>((~))\!</math> | + | | <math>((~))</math> |
− | | <math>\text{true}\!</math> | + | | <math>((~))</math> |
− | | <math>1\!</math> | + | | <math>((~))</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> | + | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
− | | width="15%" | | + | | width="10%" | |
− | <p><math>\mathcal{L}_1</math></p>
| + | | width="18%" | <math>f\!</math> |
− | <p><math>\text{Decimal}</math></p>
| + | | width="18%" | <math>\operatorname{D}f|_{xy}</math> |
− | | width="15%" | | + | | width="18%" | <math>\operatorname{D}f|_{p(q)}</math> |
− | <p><math>\mathcal{L}_2</math></p>
| + | | width="18%" | <math>\operatorname{D}f|_{(p)q}</math> |
− | <p><math>\text{Binary}</math></p>
| + | | width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math> |
− | | width="15%" | | + | |- |
− | <p><math>\mathcal{L}_3</math></p>
| + | | <math>f_0\!</math> |
− | <p><math>\text{Vector}</math></p>
| + | | <math>(~)</math> |
− | | width="15%" | | + | | <math>(~)</math> |
− | <p><math>\mathcal{L}_4</math></p>
| + | | <math>(~)</math> |
− | <p><math>\text{Cactus}</math></p>
| + | | <math>(~)</math> |
− | | width="25%" | | + | | <math>(~)</math> |
− | <p><math>\mathcal{L}_5</math></p>
| |
− | <p><math>\text{English}</math></p>
| |
− | | width="15%" | | |
− | <p><math>\mathcal{L}_6</math></p>
| |
− | <p><math>\text{Ordinary}</math></p>
| |
− | |- style="background:#f0f0ff"
| |
− | |
| |
− | | align="right" | <math>x\colon\!</math>
| |
− | | <math>1~1~0~0\!</math> | |
− | |
| |
− | |
| |
− | |
| |
− | |- style="background:#f0f0ff"
| |
− | |
| |
− | | align="right" | <math>y\colon\!</math>
| |
− | | <math>1~0~1~0\!</math> | |
− | |
| |
− | |
| |
− | |
| |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_0
| |
− | \\[4pt]
| |
| f_1 | | f_1 |
| \\[4pt] | | \\[4pt] |
| f_2 | | f_2 |
− | \\[4pt]
| |
− | f_3
| |
| \\[4pt] | | \\[4pt] |
| f_4 | | f_4 |
| \\[4pt] | | \\[4pt] |
− | f_5
| + | f_8 |
− | \\[4pt]
| |
− | f_6
| |
− | \\[4pt]
| |
− | f_7
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_{0000}
| + | (p)(q) |
| \\[4pt] | | \\[4pt] |
− | f_{0001}
| + | (p)~q~ |
| \\[4pt] | | \\[4pt] |
− | f_{0010}
| + | ~p~(q) |
| \\[4pt] | | \\[4pt] |
− | f_{0011}
| + | ~p~~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | f_{0100}
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | f_{0101}
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | f_{0110}
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | \\[4pt]
| |
− | f_{0111}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0~0~0~0
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | 0~0~0~1
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | 0~0~1~0
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | 0~0~1~1
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | 0~1~0~0
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | 0~1~0~1
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | 0~1~1~0
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
− | \\[4pt]
| |
− | 0~1~1~1
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~) | + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | (x)(y) | + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | (x)~y~ | + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | (x)~~~
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_3 |
| \\[4pt] | | \\[4pt] |
− | ~x~(y)
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (p) |
| \\[4pt] | | \\[4pt] |
− | ~~~(y) | + | ~p~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| \\[4pt] | | \\[4pt] |
− | (x,~y)
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| \\[4pt] | | \\[4pt] |
− | (x~~y)
| + | \operatorname{d}p |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \text{false} | + | \operatorname{d}p |
| \\[4pt] | | \\[4pt] |
− | \text{neither}~ x ~\text{nor}~ y | + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}p |
| \\[4pt] | | \\[4pt] |
− | y ~\text{without}~ x
| + | \operatorname{d}p |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_6 |
| \\[4pt] | | \\[4pt] |
− | \text{not}~ x | + | f_9 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~(p,~q)~ |
| \\[4pt] | | \\[4pt] |
− | x ~\text{without}~ y
| + | ((p,~q)) |
− | \\[4pt]
| |
− | \text{not}~ y
| |
− | \\[4pt]
| |
− | x ~\text{not equal to}~ y
| |
− | \\[4pt]
| |
− | \text{not both}~ x ~\text{and}~ y
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0
| + | (\operatorname{d}p,~\operatorname{d}q) |
| \\[4pt] | | \\[4pt] |
− | \lnot x \land \lnot y | + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| \\[4pt] | | \\[4pt] |
− | \lnot x \land y | + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| \\[4pt] | | \\[4pt] |
− | \lnot x | + | (\operatorname{d}p,~\operatorname{d}q) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (\operatorname{d}p,~\operatorname{d}q) |
| \\[4pt] | | \\[4pt] |
− | x \land \lnot y
| + | (\operatorname{d}p,~\operatorname{d}q) |
− | \\[4pt]
| |
− | \lnot y
| |
− | \\[4pt]
| |
− | x \ne y
| |
− | \\[4pt]
| |
− | \lnot x \lor \lnot y
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_8
| + | f_5 |
− | \\[4pt]
| |
− | f_9
| |
| \\[4pt] | | \\[4pt] |
| f_{10} | | f_{10} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | (q) |
| \\[4pt] | | \\[4pt] |
− | f_{11}
| + | ~q~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| \\[4pt] | | \\[4pt] |
− | f_{12}
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| \\[4pt] | | \\[4pt] |
− | f_{13}
| + | \operatorname{d}q |
− | \\[4pt]
| |
− | f_{14}
| |
− | \\[4pt]
| |
− | f_{15}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_{1000}
| + | \operatorname{d}q |
| \\[4pt] | | \\[4pt] |
− | f_{1001}
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{d}q |
| \\[4pt] | | \\[4pt] |
− | f_{1010}
| + | \operatorname{d}q |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_7 |
| \\[4pt] | | \\[4pt] |
− | f_{1011} | + | f_{11} |
| \\[4pt] | | \\[4pt] |
− | f_{1100} | + | f_{13} |
| \\[4pt] | | \\[4pt] |
− | f_{1101} | + | f_{14} |
− | \\[4pt]
| |
− | f_{1110}
| |
− | \\[4pt]
| |
− | f_{1111}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 1~0~0~0
| + | (~p~~q~) |
| \\[4pt] | | \\[4pt] |
− | 1~0~0~1
| + | (~p~(q)) |
| \\[4pt] | | \\[4pt] |
− | 1~0~1~0
| + | ((p)~q~) |
| \\[4pt] | | \\[4pt] |
− | 1~0~1~1
| + | ((p)(q)) |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | 1~1~0~0
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | 1~1~0~1
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | 1~1~1~0
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
− | \\[4pt]
| |
− | 1~1~1~1
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~x~~y~~ | + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | ((x,~y)) | + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | ~~~~~y~~ | + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | ~(x~(y)) | + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | ~~x~~~~~ | + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | ((x)~y)~ | + | ((\operatorname{d}p)(\operatorname{d}q)) |
| \\[4pt] | | \\[4pt] |
− | ((x)(y)) | + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
− | \\[4pt] | |
− | ((~))
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | x ~\text{and}~ y
| + | ~~\operatorname{d}p~~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | x ~\text{equal to}~ y
| + | ~~\operatorname{d}p~(\operatorname{d}q)~ |
| \\[4pt] | | \\[4pt] |
− | y
| + | ~(\operatorname{d}p)~\operatorname{d}q~~ |
| \\[4pt] | | \\[4pt] |
− | \text{not}~ x ~\text{without}~ y | + | ((\operatorname{d}p)(\operatorname{d}q)) |
− | \\[4pt]
| |
− | x
| |
− | \\[4pt]
| |
− | \text{not}~ y ~\text{without}~ x
| |
− | \\[4pt]
| |
− | x ~\text{or}~ y
| |
− | \\[4pt]
| |
− | \text{true}
| |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | x \land y
| |
− | \\[4pt]
| |
− | x = y
| |
− | \\[4pt]
| |
− | y
| |
− | \\[4pt]
| |
− | x \Rightarrow y
| |
− | \\[4pt]
| |
− | x
| |
− | \\[4pt]
| |
− | x \Leftarrow y
| |
− | \\[4pt]
| |
− | x \lor y
| |
− | \\[4pt]
| |
− | 1
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | ===Wiki TeX Tables : XY=== |
− | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> | + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="15%" | | | | width="15%" | |
Line 3,796: |
Line 3,803: |
| | | | | |
| |- | | |- |
− | | <math>f_0\!</math> | + | | <math>f_{0}\!</math> |
| | <math>f_{0000}\!</math> | | | <math>f_{0000}\!</math> |
− | | <math>0~0~0~0</math> | + | | <math>0~0~0~0\!</math> |
− | | <math>(~)</math> | + | | <math>(~)\!</math> |
| | <math>\text{false}\!</math> | | | <math>\text{false}\!</math> |
| | <math>0\!</math> | | | <math>0\!</math> |
| |- | | |- |
− | | | + | | <math>f_{1}\!</math> |
− | <math>\begin{matrix} | + | | <math>f_{0001}\!</math> |
− | f_1
| + | | <math>0~0~0~1\!</math> |
− | \\[4pt] | + | | <math>(x)(y)\!</math> |
− | f_2
| + | | <math>\text{neither}~ x ~\text{nor}~ y\!</math> |
− | \\[4pt] | + | | <math>\lnot x \land \lnot y\!</math> |
− | f_4
| + | |- |
− | \\[4pt] | + | | <math>f_{2}\!</math> |
− | f_8
| + | | <math>f_{0010}\!</math> |
− | \end{matrix}</math> | + | | <math>0~0~1~0\!</math> |
− | | | + | | <math>(x)~y\!</math> |
− | <math>\begin{matrix} | + | | <math>y ~\text{without}~ x\!</math> |
− | f_{0001} | + | | <math>\lnot x \land y\!</math> |
− | \\[4pt] | + | |- |
− | f_{0010} | + | | <math>f_{3}\!</math> |
− | \\[4pt] | + | | <math>f_{0011}\!</math> |
− | f_{0100}
| + | | <math>0~0~1~1\!</math> |
− | \\[4pt] | + | | <math>(x)\!</math> |
− | f_{1000} | + | | <math>\text{not}~ x\!</math> |
− | \end{matrix}</math> | + | | <math>\lnot x\!</math> |
− | | | + | |- |
− | <math>\begin{matrix} | + | | <math>f_{4}\!</math> |
− | 0~0~0~1 | + | | <math>f_{0100}\!</math> |
− | \\[4pt] | + | | <math>0~1~0~0\!</math> |
− | 0~0~1~0
| + | | <math>x~(y)\!</math> |
− | \\[4pt] | + | | <math>x ~\text{without}~ y\!</math> |
− | 0~1~0~0
| + | | <math>x \land \lnot y\!</math> |
− | \\[4pt] | + | |- |
− | 1~0~0~0 | + | | <math>f_{5}\!</math> |
− | \end{matrix}</math> | + | | <math>f_{0101}\!</math> |
− | | | + | | <math>0~1~0~1\!</math> |
− | <math>\begin{matrix} | + | | <math>(y)\!</math> |
− | (x)(y)
| + | | <math>\text{not}~ y\!</math> |
− | \\[4pt] | + | | <math>\lnot y\!</math> |
− | (x)~y~
| + | |- |
− | \\[4pt] | + | | <math>f_{6}\!</math> |
− | ~x~(y)
| + | | <math>f_{0110}\!</math> |
− | \\[4pt] | + | | <math>0~1~1~0\!</math> |
− | ~x~~y~
| + | | <math>(x,~y)\!</math> |
− | \end{matrix}</math> | + | | <math>x ~\text{not equal to}~ y\!</math> |
− | | | + | | <math>x \ne y\!</math> |
− | <math>\begin{matrix} | |
− | \text{neither}~ x ~\text{nor}~ y
| |
− | \\[4pt] | |
− | y ~\text{without}~ x | |
− | \\[4pt]
| |
− | x ~\text{without}~ y
| |
− | \\[4pt] | |
− | x ~\text{and}~ y
| |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | \lnot x \land \lnot y | |
− | \\[4pt] | |
− | \lnot x \land y
| |
− | \\[4pt] | |
− | x \land \lnot y | |
− | \\[4pt] | |
− | x \land y | |
− | \end{matrix}</math> | |
| |- | | |- |
− | | | + | | <math>f_{7}\!</math> |
− | <math>\begin{matrix} | + | | <math>f_{0111}\!</math> |
− | f_3
| + | | <math>0~1~1~1\!</math> |
− | \\[4pt] | + | | <math>(x~y)\!</math> |
− | f_{12} | + | | <math>\text{not both}~ x ~\text{and}~ y\!</math> |
− | \end{matrix}</math> | + | | <math>\lnot x \lor \lnot y\!</math> |
− | | | + | |- |
− | <math>\begin{matrix} | + | | <math>f_{8}\!</math> |
− | f_{0011}
| + | | <math>f_{1000}\!</math> |
− | \\[4pt] | + | | <math>1~0~0~0\!</math> |
− | f_{1100} | + | | <math>x~y\!</math> |
− | \end{matrix}</math> | + | | <math>x ~\text{and}~ y\!</math> |
− | | | + | | <math>x \land y\!</math> |
− | <math>\begin{matrix} | + | |- |
− | 0~0~1~1
| + | | <math>f_{9}\!</math> |
− | \\[4pt] | + | | <math>f_{1001}\!</math> |
− | 1~1~0~0 | + | | <math>1~0~0~1\!</math> |
− | \end{matrix}</math> | + | | <math>((x,~y))\!</math> |
− | | | + | | <math>x ~\text{equal to}~ y\!</math> |
− | <math>\begin{matrix} | + | | <math>x = y\!</math> |
− | (x)
| |
− | \\[4pt] | |
− | ~x~
| |
− | \end{matrix}</math>
| |
− | | | |
− | <math>\begin{matrix} | |
− | \text{not}~ x
| |
− | \\[4pt] | |
− | x | |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | \lnot x | |
− | \\[4pt]
| |
− | x | |
− | \end{matrix}</math> | |
| |- | | |- |
− | | | + | | <math>f_{10}\!</math> |
− | <math>\begin{matrix} | + | | <math>f_{1010}\!</math> |
− | f_6
| + | | <math>1~0~1~0\!</math> |
− | \\[4pt] | + | | <math>y\!</math> |
− | f_9
| + | | <math>y\!</math> |
− | \end{matrix}</math>
| + | | <math>y\!</math> |
− | | | + | |- |
− | <math>\begin{matrix} | + | | <math>f_{11}\!</math> |
− | f_{0110}
| + | | <math>f_{1011}\!</math> |
− | \\[4pt] | + | | <math>1~0~1~1\!</math> |
− | f_{1001} | + | | <math>(x~(y))\!</math> |
− | \end{matrix}</math> | + | | <math>\text{not}~ x ~\text{without}~ y\!</math> |
− | | | + | | <math>x \Rightarrow y\!</math> |
− | <math>\begin{matrix} | + | |- |
− | 0~1~1~0
| + | | <math>f_{12}\!</math> |
− | \\[4pt] | + | | <math>f_{1100}\!</math> |
− | 1~0~0~1 | + | | <math>1~1~0~0\!</math> |
− | \end{matrix}</math> | + | | <math>x\!</math> |
− | | | + | | <math>x\!</math> |
− | <math>\begin{matrix} | + | | <math>x\!</math> |
− | ~(x,~y)~
| |
− | \\[4pt]
| |
− | ((x,~y))
| |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | x ~\text{not equal to}~ y | |
− | \\[4pt] | |
− | x ~\text{equal to}~ y
| |
− | \end{matrix}</math>
| |
− | | | |
− | <math>\begin{matrix} | |
− | x \ne y | |
− | \\[4pt] | |
− | x = y | |
− | \end{matrix}</math> | |
| |- | | |- |
− | | | + | | <math>f_{13}\!</math> |
− | <math>\begin{matrix} | + | | <math>f_{1101}\!</math> |
− | f_5
| + | | <math>1~1~0~1\!</math> |
− | \\[4pt] | + | | <math>((x)~y)\!</math> |
− | f_{10} | + | | <math>\text{not}~ y ~\text{without}~ x\!</math> |
− | \end{matrix}</math> | + | | <math>x \Leftarrow y\!</math> |
− | | | + | |- |
− | <math>\begin{matrix} | + | | <math>f_{14}\!</math> |
− | f_{0101}
| + | | <math>f_{1110}\!</math> |
− | \\[4pt] | + | | <math>1~1~1~0\!</math> |
− | f_{1010} | + | | <math>((x)(y))\!</math> |
− | \end{matrix}</math> | + | | <math>x ~\text{or}~ y\!</math> |
− | | | + | | <math>x \lor y\!</math> |
− | <math>\begin{matrix} | |
− | 0~1~0~1
| |
− | \\[4pt]
| |
− | 1~0~1~0 | |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | (y) | |
− | \\[4pt]
| |
− | ~y~
| |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | \text{not}~ y | |
− | \\[4pt]
| |
− | y
| |
− | \end{matrix}</math> | |
− | | | |
− | <math>\begin{matrix} | |
− | \lnot y
| |
− | \\[4pt] | |
− | y
| |
− | \end{matrix}</math>
| |
| |- | | |- |
− | | | + | | <math>f_{15}\!</math> |
− | <math>\begin{matrix} | + | | <math>f_{1111}\!</math> |
− | f_7
| + | | <math>1~1~1~1\!</math> |
− | \\[4pt] | + | | <math>((~))\!</math> |
− | f_{11}
| + | | <math>\text{true}\!</math> |
− | \\[4pt] | + | | <math>1\!</math> |
− | f_{13}
| + | |} |
− | \\[4pt] | + | |
− | f_{14}
| + | <br> |
− | \end{matrix}</math> | + | |
− | | | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | <math>\begin{matrix} | + | |+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math> |
− | f_{0111}
| + | |- style="background:#f0f0ff" |
− | \\[4pt] | + | | width="15%" | |
− | f_{1011}
| + | <p><math>\mathcal{L}_1</math></p> |
| + | <p><math>\text{Decimal}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_2</math></p> |
| + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>x\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>y\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | f_0 |
| + | \\[4pt] |
| + | f_1 |
| + | \\[4pt] |
| + | f_2 |
| \\[4pt] | | \\[4pt] |
− | f_{1101}
| + | f_3 |
| \\[4pt] | | \\[4pt] |
− | f_{1110}
| + | f_4 |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | 0~1~1~1
| |
| \\[4pt] | | \\[4pt] |
− | 1~0~1~1
| + | f_5 |
| \\[4pt] | | \\[4pt] |
− | 1~1~0~1
| + | f_6 |
| \\[4pt] | | \\[4pt] |
− | 1~1~1~0
| + | f_7 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(x~~y)~
| + | f_{0000} |
| + | \\[4pt] |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0011} |
| + | \\[4pt] |
| + | f_{0100} |
| \\[4pt] | | \\[4pt] |
− | ~(x~(y))
| + | f_{0101} |
| \\[4pt] | | \\[4pt] |
− | ((x)~y)~
| + | f_{0110} |
| \\[4pt] | | \\[4pt] |
− | ((x)(y))
| + | f_{0111} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \text{not both}~ x ~\text{and}~ y | + | 0~0~0~0 |
| + | \\[4pt] |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~0~1~1 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| \\[4pt] | | \\[4pt] |
− | \text{not}~ x ~\text{without}~ y
| + | 0~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | \text{not}~ y ~\text{without}~ x
| + | 0~1~1~0 |
| \\[4pt] | | \\[4pt] |
− | x ~\text{or}~ y
| + | 0~1~1~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \lnot x \lor \lnot y | + | (~) |
| + | \\[4pt] |
| + | (x)(y) |
| + | \\[4pt] |
| + | (x)~y~ |
| + | \\[4pt] |
| + | (x)~~~ |
| + | \\[4pt] |
| + | ~x~(y) |
| \\[4pt] | | \\[4pt] |
− | x \Rightarrow y
| + | ~~~(y) |
| \\[4pt] | | \\[4pt] |
− | x \Leftarrow y | + | (x,~y) |
| \\[4pt] | | \\[4pt] |
− | x \lor y | + | (x~~y) |
| \end{matrix}</math> | | \end{matrix}</math> |
− | |- | + | | |
− | | <math>f_{15}\!</math>
| + | <math>\begin{matrix} |
− | | <math>f_{1111}\!</math>
| + | \text{false} |
− | | <math>1~1~1~1</math>
| + | \\[4pt] |
− | | <math>((~))</math>
| + | \text{neither}~ x ~\text{nor}~ y |
− | | <math>\text{true}\!</math>
| |
− | | <math>1\!</math>
| |
− | |}
| |
− | | |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
| |
− | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="10%" |
| |
− | | width="18%" | <math>f\!</math>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{11} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{10} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{01} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
| |
− | | width="18%" |
| |
− | <p><math>\operatorname{T}_{00} f</math></p>
| |
− | <p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
| |
− | |-
| |
− | | <math>f_0\!</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_1
| |
| \\[4pt] | | \\[4pt] |
− | f_2
| + | y ~\text{without}~ x |
| \\[4pt] | | \\[4pt] |
− | f_4
| + | \text{not}~ x |
| \\[4pt] | | \\[4pt] |
− | f_8
| + | x ~\text{without}~ y |
− | \end{matrix}</math> | |
− | |
| |
− | <math>\begin{matrix}
| |
− | (x)(y)
| |
| \\[4pt] | | \\[4pt] |
− | (x)~y~
| + | \text{not}~ y |
| \\[4pt] | | \\[4pt] |
− | ~x~(y) | + | x ~\text{not equal to}~ y |
| \\[4pt] | | \\[4pt] |
− | ~x~~y~ | + | \text{not both}~ x ~\text{and}~ y |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~x~~y~
| + | 0 |
| \\[4pt] | | \\[4pt] |
− | ~x~(y)
| + | \lnot x \land \lnot y |
| \\[4pt] | | \\[4pt] |
− | (x)~y~
| + | \lnot x \land y |
| \\[4pt] | | \\[4pt] |
− | (x)(y)
| + | \lnot x |
| + | \\[4pt] |
| + | x \land \lnot y |
| + | \\[4pt] |
| + | \lnot y |
| + | \\[4pt] |
| + | x \ne y |
| + | \\[4pt] |
| + | \lnot x \lor \lnot y |
| \end{matrix}</math> | | \end{matrix}</math> |
| + | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~x~(y)
| + | f_8 |
| + | \\[4pt] |
| + | f_9 |
| \\[4pt] | | \\[4pt] |
− | ~x~~y~
| + | f_{10} |
| \\[4pt] | | \\[4pt] |
− | (x)(y)
| + | f_{11} |
| \\[4pt] | | \\[4pt] |
− | (x)~y~
| + | f_{12} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (x)~y~
| |
| \\[4pt] | | \\[4pt] |
− | (x)(y)
| + | f_{13} |
| \\[4pt] | | \\[4pt] |
− | ~x~~y~
| + | f_{14} |
| \\[4pt] | | \\[4pt] |
− | ~x~(y)
| + | f_{15} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (x)(y)
| + | f_{1000} |
| + | \\[4pt] |
| + | f_{1001} |
| + | \\[4pt] |
| + | f_{1010} |
| \\[4pt] | | \\[4pt] |
− | (x)~y~
| + | f_{1011} |
| \\[4pt] | | \\[4pt] |
− | ~x~(y)
| + | f_{1100} |
| \\[4pt] | | \\[4pt] |
− | ~x~~y~
| + | f_{1101} |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_3
| |
| \\[4pt] | | \\[4pt] |
− | f_{12} | + | f_{1110} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (x)
| |
| \\[4pt] | | \\[4pt] |
− | ~x~
| + | f_{1111} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~x~ | + | 1~0~0~0 |
| \\[4pt] | | \\[4pt] |
− | (x)
| + | 1~0~0~1 |
− | \end{matrix}</math> | + | \\[4pt] |
− | |
| + | 1~0~1~0 |
− | <math>\begin{matrix}
| + | \\[4pt] |
− | ~x~ | + | 1~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | (x)
| + | 1~1~0~0 |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (x)
| |
| \\[4pt] | | \\[4pt] |
− | ~x~ | + | 1~1~0~1 |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (x)
| |
| \\[4pt] | | \\[4pt] |
− | ~x~ | + | 1~1~1~0 |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_6
| |
| \\[4pt] | | \\[4pt] |
− | f_9
| + | 1~1~1~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(x,~y)~ | + | ~~x~~y~~ |
| \\[4pt] | | \\[4pt] |
| ((x,~y)) | | ((x,~y)) |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~(x,~y)~
| |
| \\[4pt] | | \\[4pt] |
− | ((x,~y))
| + | ~~~~~y~~ |
− | \end{matrix}</math> | + | \\[4pt] |
− | |
| + | ~(x~(y)) |
− | <math>\begin{matrix}
| + | \\[4pt] |
− | ((x,~y)) | + | ~~x~~~~~ |
| + | \\[4pt] |
| + | ((x)~y)~ |
| + | \\[4pt] |
| + | ((x)(y)) |
| \\[4pt] | | \\[4pt] |
− | ~(x,~y)~
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((x,~y))
| + | x ~\text{and}~ y |
| \\[4pt] | | \\[4pt] |
− | ~(x,~y)~
| + | x ~\text{equal to}~ y |
− | \end{matrix}</math> | |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~(x,~y)~
| |
| \\[4pt] | | \\[4pt] |
− | ((x,~y))
| + | y |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_5
| |
| \\[4pt] | | \\[4pt] |
− | f_{10}
| + | \text{not}~ x ~\text{without}~ y |
− | \end{matrix}</math> | |
− | |
| |
− | <math>\begin{matrix}
| |
− | (y)
| |
| \\[4pt] | | \\[4pt] |
− | ~y~
| + | x |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~y~
| |
| \\[4pt] | | \\[4pt] |
− | (y)
| + | \text{not}~ y ~\text{without}~ x |
− | \end{matrix}</math> | |
− | |
| |
− | <math>\begin{matrix}
| |
− | (y)
| |
| \\[4pt] | | \\[4pt] |
− | ~y~ | + | x ~\text{or}~ y |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~y~ | |
| \\[4pt] | | \\[4pt] |
− | (y)
| + | \text{true} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (y)
| + | x \land y |
| \\[4pt] | | \\[4pt] |
− | ~y~
| + | x = y |
− | \end{matrix}</math>
| |
− | |-
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | f_7
| |
| \\[4pt] | | \\[4pt] |
− | f_{11}
| + | y |
| \\[4pt] | | \\[4pt] |
− | f_{13}
| + | x \Rightarrow y |
| \\[4pt] | | \\[4pt] |
− | f_{14}
| + | x |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (~x~~y~)
| |
| \\[4pt] | | \\[4pt] |
− | (~x~(y))
| + | x \Leftarrow y |
| \\[4pt] | | \\[4pt] |
− | ((x)~y~)
| + | x \lor y |
| \\[4pt] | | \\[4pt] |
− | ((x)(y))
| + | 1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math> |
| + | |- style="background:#f0f0ff" |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_1</math></p> |
| + | <p><math>\text{Decimal}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_2</math></p> |
| + | <p><math>\text{Binary}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_3</math></p> |
| + | <p><math>\text{Vector}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_4</math></p> |
| + | <p><math>\text{Cactus}</math></p> |
| + | | width="25%" | |
| + | <p><math>\mathcal{L}_5</math></p> |
| + | <p><math>\text{English}</math></p> |
| + | | width="15%" | |
| + | <p><math>\mathcal{L}_6</math></p> |
| + | <p><math>\text{Ordinary}</math></p> |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>x\colon\!</math> |
| + | | <math>1~1~0~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- style="background:#f0f0ff" |
| + | | |
| + | | align="right" | <math>y\colon\!</math> |
| + | | <math>1~0~1~0\!</math> |
| + | | |
| + | | |
| + | | |
| + | |- |
| + | | <math>f_0\!</math> |
| + | | <math>f_{0000}\!</math> |
| + | | <math>0~0~0~0</math> |
| + | | <math>(~)</math> |
| + | | <math>\text{false}\!</math> |
| + | | <math>0\!</math> |
| + | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((x)(y))
| + | f_1 |
| \\[4pt] | | \\[4pt] |
− | ((x)~y~)
| + | f_2 |
| \\[4pt] | | \\[4pt] |
− | (~x~(y))
| + | f_4 |
| \\[4pt] | | \\[4pt] |
− | (~x~~y~)
| + | f_8 |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0001} |
| + | \\[4pt] |
| + | f_{0010} |
| + | \\[4pt] |
| + | f_{0100} |
| + | \\[4pt] |
| + | f_{1000} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | 0~0~0~1 |
| + | \\[4pt] |
| + | 0~0~1~0 |
| + | \\[4pt] |
| + | 0~1~0~0 |
| + | \\[4pt] |
| + | 1~0~0~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((x)~y~)
| + | (x)(y) |
| \\[4pt] | | \\[4pt] |
− | ((x)(y))
| + | (x)~y~ |
| \\[4pt] | | \\[4pt] |
− | (~x~~y~)
| + | ~x~(y) |
| \\[4pt] | | \\[4pt] |
− | (~x~(y))
| + | ~x~~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~x~(y))
| + | \text{neither}~ x ~\text{nor}~ y |
| \\[4pt] | | \\[4pt] |
− | (~x~~y~)
| + | y ~\text{without}~ x |
| \\[4pt] | | \\[4pt] |
− | ((x)(y))
| + | x ~\text{without}~ y |
| \\[4pt] | | \\[4pt] |
− | ((x)~y~)
| + | x ~\text{and}~ y |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~x~~y~)
| + | \lnot x \land \lnot y |
| \\[4pt] | | \\[4pt] |
− | (~x~(y))
| + | \lnot x \land y |
| \\[4pt] | | \\[4pt] |
− | ((x)~y~)
| + | x \land \lnot y |
| \\[4pt] | | \\[4pt] |
− | ((x)(y))
| + | x \land y |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
− | | <math>f_{15}\!</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | | <math>((~))</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | colspan="2" | <math>\text{Fixed Point Total}\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>4\!</math>
| |
− | | <math>16\!</math>
| |
− | |}
| |
− |
| |
− | <br>
| |
− |
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
| |
− | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="10%" |
| |
− | | width="18%" | <math>f\!</math>
| |
− | | width="18%" |
| |
− | <math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
| |
− | | width="18%" |
| |
− | <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
| |
− | | width="18%" |
| |
− | <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
| |
− | | width="18%" |
| |
− | <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
| |
− | |-
| |
− | | <math>f_0\!</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
− | |-
| |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_1
| + | f_3 |
| + | \\[4pt] |
| + | f_{12} |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | f_{0011} |
| \\[4pt] | | \\[4pt] |
− | f_2
| + | f_{1100} |
− | \\[4pt]
| |
− | f_4
| |
− | \\[4pt]
| |
− | f_8
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (x)(y)
| + | 0~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | (x)~y~
| + | 1~1~0~0 |
− | \\[4pt]
| |
− | ~x~(y)
| |
− | \\[4pt]
| |
− | ~x~~y~
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((x,~y))
| + | (x) |
| \\[4pt] | | \\[4pt] |
− | ~(x,~y)~ | + | ~x~ |
− | \\[4pt]
| |
− | ~(x,~y)~
| |
− | \\[4pt]
| |
− | ((x,~y))
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (y)
| + | \text{not}~ x |
| \\[4pt] | | \\[4pt] |
− | ~y~
| + | x |
− | \\[4pt]
| |
− | (y)
| |
− | \\[4pt]
| |
− | ~y~
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (x)
| + | \lnot x |
| \\[4pt] | | \\[4pt] |
− | (x)
| + | x |
− | \\[4pt]
| |
− | ~x~
| |
− | \\[4pt]
| |
− | ~x~
| |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_3
| + | f_6 |
| \\[4pt] | | \\[4pt] |
− | f_{12}
| + | f_9 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (x)
| + | f_{0110} |
| \\[4pt] | | \\[4pt] |
− | ~x~
| + | f_{1001} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | 0~1~1~0 |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | 1~0~0~1 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~)) | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ((~)) | + | ((x,~y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | x ~\text{not equal to}~ y |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | x ~\text{equal to}~ y |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | x \ne y |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | x = y |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_6
| + | f_5 |
| \\[4pt] | | \\[4pt] |
− | f_9
| + | f_{10} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(x,~y)~
| + | f_{0101} |
| \\[4pt] | | \\[4pt] |
− | ((x,~y))
| + | f_{1010} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | 0~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | 1~0~1~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~)) | + | (y) |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | ~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((~))
| + | \text{not}~ y |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | y |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | \lnot y |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | y |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_5
| + | f_7 |
| \\[4pt] | | \\[4pt] |
− | f_{10} | + | f_{11} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (y)
| |
| \\[4pt] | | \\[4pt] |
− | ~y~
| + | f_{13} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ((~))
| |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | f_{14} |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~)
| + | f_{0111} |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | f_{1011} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ((~))
| |
| \\[4pt] | | \\[4pt] |
− | ((~))
| + | f_{1101} |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (~)
| |
| \\[4pt] | | \\[4pt] |
− | (~)
| + | f_{1110} |
| \end{matrix}</math> | | \end{matrix}</math> |
− | |-
| |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | f_7
| + | 0~1~1~1 |
| \\[4pt] | | \\[4pt] |
− | f_{11}
| + | 1~0~1~1 |
| \\[4pt] | | \\[4pt] |
− | f_{13}
| + | 1~1~0~1 |
| \\[4pt] | | \\[4pt] |
− | f_{14}
| + | 1~1~1~0 |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
Line 4,535: |
Line 4,432: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((x,~y))
| + | \text{not both}~ x ~\text{and}~ y |
| \\[4pt] | | \\[4pt] |
− | ~(x,~y)~ | + | \text{not}~ x ~\text{without}~ y |
| \\[4pt] | | \\[4pt] |
− | ~(x,~y)~ | + | \text{not}~ y ~\text{without}~ x |
| \\[4pt] | | \\[4pt] |
− | ((x,~y))
| + | x ~\text{or}~ y |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~y~
| + | \lnot x \lor \lnot y |
| \\[4pt] | | \\[4pt] |
− | (y)
| + | x \Rightarrow y |
| \\[4pt] | | \\[4pt] |
− | ~y~
| + | x \Leftarrow y |
| \\[4pt] | | \\[4pt] |
− | (y)
| + | x \lor y |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | ~x~
| |
− | \\[4pt] | |
− | ~x~
| |
− | \\[4pt]
| |
− | (x)
| |
− | \\[4pt]
| |
− | (x)
| |
− | \end{matrix}</math>
| |
− | |
| |
− | <math>\begin{matrix}
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
− | \\[4pt]
| |
− | (~)
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
| + | | <math>f_{1111}\!</math> |
| + | | <math>1~1~1~1</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
− | | <math>(~)</math> | + | | <math>\text{true}\!</math> |
− | | <math>(~)</math> | + | | <math>1\!</math> |
− | | <math>(~)</math>
| |
− | | <math>(~)</math>
| |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> | + | |+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="10%" | | | | width="10%" | |
| | width="18%" | <math>f\!</math> | | | width="18%" | <math>f\!</math> |
− | | width="18%" | <math>\operatorname{E}f|_{xy}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{E}f|_{x(y)}</math> | + | <p><math>\operatorname{T}_{11} f</math></p> |
− | | width="18%" | <math>\operatorname{E}f|_{(x)y}</math> | + | <p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p> |
− | | width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math> | + | | width="18%" | |
− | |- | + | <p><math>\operatorname{T}_{10} f</math></p> |
− | | <math>f_0\!</math> | + | <p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{01} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p> |
| + | | width="18%" | |
| + | <p><math>\operatorname{T}_{00} f</math></p> |
| + | <p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p> |
| + | |- |
| + | | <math>f_0\!</math> |
| | <math>(~)</math> | | | <math>(~)</math> |
| | <math>(~)</math> | | | <math>(~)</math> |
Line 4,623: |
Line 4,508: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}x~~\operatorname{d}y~ | + | ~x~~y~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~(\operatorname{d}y) | + | ~x~(y) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)~\operatorname{d}y~ | + | (x)~y~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)(\operatorname{d}y) | + | (x)(y) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}x~(\operatorname{d}y) | + | ~x~(y) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~~\operatorname{d}y~ | + | ~x~~y~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)(\operatorname{d}y) | + | (x)(y) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)~\operatorname{d}y~ | + | (x)~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x)~\operatorname{d}y~ | + | (x)~y~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)(\operatorname{d}y) | + | (x)(y) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~~\operatorname{d}y~ | + | ~x~~y~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~(\operatorname{d}y) | + | ~x~(y) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x)(\operatorname{d}y) | + | (x)(y) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x)~\operatorname{d}y~ | + | (x)~y~ |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~(\operatorname{d}y) | + | ~x~(y) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~~\operatorname{d}y~ | + | ~x~~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,676: |
Line 4,561: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}x~ | + | ~x~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x) | + | (x) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}x~ | + | ~x~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x) | + | (x) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x) | + | (x) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~ | + | ~x~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x) | + | (x) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}x~ | + | ~x~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,713: |
Line 4,598: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}x,~\operatorname{d}y)~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x,~\operatorname{d}y)) | + | ((x,~y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x,~\operatorname{d}y)) | + | ((x,~y)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x,~\operatorname{d}y)~ | + | ~(x,~y)~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x,~\operatorname{d}y)) | + | ((x,~y)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x,~\operatorname{d}y)~ | + | ~(x,~y)~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}x,~\operatorname{d}y)~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x,~\operatorname{d}y)) | + | ((x,~y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,750: |
Line 4,635: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}y~ | + | ~y~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}y) | + | (y) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}y) | + | (y) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}y~ | + | ~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~\operatorname{d}y~ | + | ~y~ |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}y) | + | (y) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}y) | + | (y) |
| \\[4pt] | | \\[4pt] |
− | ~\operatorname{d}y~ | + | ~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,795: |
Line 4,680: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x)(y)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)~\operatorname{d}y~) | + | ((x)~y~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~(\operatorname{d}y)) | + | (~x~(y)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~~\operatorname{d}y~) | + | (~x~~y~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x)~\operatorname{d}y~) | + | ((x)~y~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x)(y)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~~\operatorname{d}y~) | + | (~x~~y~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~(\operatorname{d}y)) | + | (~x~(y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~\operatorname{d}x~(\operatorname{d}y)) | + | (~x~(y)) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~~\operatorname{d}y~) | + | (~x~~y~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x)(y)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)~\operatorname{d}y~) | + | ((x)~y~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~\operatorname{d}x~~\operatorname{d}y~) | + | (~x~~y~) |
| \\[4pt] | | \\[4pt] |
− | (~\operatorname{d}x~(\operatorname{d}y)) | + | (~x~(y)) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)~\operatorname{d}y~) | + | ((x)~y~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x)(y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,840: |
Line 4,725: |
| | <math>((~))</math> | | | <math>((~))</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
| + | |- style="background:#f0f0ff" |
| + | | colspan="2" | <math>\text{Fixed Point Total}\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>4\!</math> |
| + | | <math>16\!</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> | + | |+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math> |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="10%" | | | | width="10%" | |
| | width="18%" | <math>f\!</math> | | | width="18%" | <math>f\!</math> |
− | | width="18%" | <math>\operatorname{D}f|_{xy}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{D}f|_{x(y)}</math> | + | <math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math> |
− | | width="18%" | <math>\operatorname{D}f|_{(x)y}</math> | + | | width="18%" | |
− | | width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math> | + | <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math> |
| + | | width="18%" | |
| + | <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math> |
| |- | | |- |
| | <math>f_0\!</math> | | | <math>f_0\!</math> |
Line 4,883: |
Line 4,778: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}x~~\operatorname{d}y~~
| + | ((x,~y)) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x,~y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}x~(\operatorname{d}y)~
| + | (y) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | ~y~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (y) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | ~y~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}x)~\operatorname{d}y~~
| + | (x) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (x) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | ~x~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | ~x~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,936: |
Line 4,831: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}x
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}x
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}x
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}x
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}x
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}x
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}x
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}x
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 4,973: |
Line 4,868: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x,~\operatorname{d}y) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x,~\operatorname{d}y) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x,~\operatorname{d}y) | + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x,~\operatorname{d}y) | + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x,~\operatorname{d}y) | + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x,~\operatorname{d}y) | + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (\operatorname{d}x,~\operatorname{d}y) | + | (~) |
| \\[4pt] | | \\[4pt] |
− | (\operatorname{d}x,~\operatorname{d}y) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 5,010: |
Line 4,905: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}y
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}y
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}y
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}y
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}y
| + | ((~)) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}y
| + | ((~)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \operatorname{d}y
| + | (~) |
| \\[4pt] | | \\[4pt] |
− | \operatorname{d}y
| + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
Line 5,045: |
Line 4,940: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | (~x~~y~) | + | ~(x~~y)~ |
| \\[4pt] | | \\[4pt] |
− | (~x~(y)) | + | ~(x~(y)) |
| \\[4pt] | | \\[4pt] |
− | ((x)~y~) | + | ((x)~y)~ |
| \\[4pt] | | \\[4pt] |
| ((x)(y)) | | ((x)(y)) |
Line 5,055: |
Line 4,950: |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | ((x,~y)) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | ~(x,~y)~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~
| + | ((x,~y)) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | ~y~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (y) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | ~y~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~
| + | (y) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | ~x~ |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | ~x~ |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (x) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~
| + | (x) |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | ~~\operatorname{d}x~~\operatorname{d}y~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~~\operatorname{d}x~(\operatorname{d}y)~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ~(\operatorname{d}x)~\operatorname{d}y~~ | + | (~) |
| \\[4pt] | | \\[4pt] |
− | ((\operatorname{d}x)(\operatorname{d}y)) | + | (~) |
| \end{matrix}</math> | | \end{matrix}</math> |
| |- | | |- |
| | <math>f_{15}\!</math> | | | <math>f_{15}\!</math> |
| | <math>((~))</math> | | | <math>((~))</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
− | | <math>((~))</math> | + | | <math>(~)</math> |
| |} | | |} |
| | | |
| <br> | | <br> |
| | | |
− | ===Klein Four-Group V<sub>4</sub>===
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | | + | |+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
− | <br>
| + | |- style="background:#f0f0ff" |
− | | + | | width="10%" | |
− | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" | + | | width="18%" | <math>f\!</math> |
− | |- style="height:50px" | + | | width="18%" | <math>\operatorname{E}f|_{xy}</math> |
− | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math> | + | | width="18%" | <math>\operatorname{E}f|_{x(y)}</math> |
− | | width="22%" style="border-bottom:1px solid black" | | + | | width="18%" | <math>\operatorname{E}f|_{(x)y}</math> |
− | <math>\operatorname{T}_{00}</math> | + | | width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math> |
− | | width="22%" style="border-bottom:1px solid black" | | + | |- |
− | <math>\operatorname{T}_{01}</math> | + | | <math>f_0\!</math> |
− | | width="22%" style="border-bottom:1px solid black" | | + | | <math>(~)</math> |
− | <math>\operatorname{T}_{10}</math> | + | | <math>(~)</math> |
− | | width="22%" style="border-bottom:1px solid black" | | + | | <math>(~)</math> |
− | <math>\operatorname{T}_{11}</math> | + | | <math>(~)</math> |
− | |- style="height:50px" | + | | <math>(~)</math> |
− | | style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
| + | |- |
− | | <math>\operatorname{T}_{00}</math> | + | | |
− | | <math>\operatorname{T}_{01}</math> | + | <math>\begin{matrix} |
− | | <math>\operatorname{T}_{10}</math> | + | f_1 |
− | | <math>\operatorname{T}_{11}</math> | + | \\[4pt] |
− | |- style="height:50px"
| + | f_2 |
− | | style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
| + | \\[4pt] |
− | | <math>\operatorname{T}_{01}</math> | + | f_4 |
− | | <math>\operatorname{T}_{00}</math> | + | \\[4pt] |
− | | <math>\operatorname{T}_{11}</math>
| + | f_8 |
− | | <math>\operatorname{T}_{10}</math>
| + | \end{matrix}</math> |
− | |- style="height:50px"
| + | | |
− | | style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
| + | <math>\begin{matrix} |
− | | <math>\operatorname{T}_{10}</math>
| + | (x)(y) |
− | | <math>\operatorname{T}_{11}</math> | + | \\[4pt] |
− | | <math>\operatorname{T}_{00}</math>
| + | (x)~y~ |
− | | <math>\operatorname{T}_{01}</math>
| + | \\[4pt] |
− | |- style="height:50px"
| + | ~x~(y) |
− | | style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
| + | \\[4pt] |
− | | <math>\operatorname{T}_{11}</math>
| + | ~x~~y~ |
− | | <math>\operatorname{T}_{10}</math>
| + | \end{matrix}</math> |
− | | <math>\operatorname{T}_{01}</math>
| + | | |
− | | <math>\operatorname{T}_{00}</math>
| + | <math>\begin{matrix} |
− | |}
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
− | | + | \\[4pt] |
− | <br>
| + | ~\operatorname{d}x~(\operatorname{d}y) |
− | | + | \\[4pt] |
− | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
| + | (\operatorname{d}x)~\operatorname{d}y~ |
− | |- style="height:50px"
| + | \\[4pt] |
− | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| + | (\operatorname{d}x)(\operatorname{d}y) |
− | | width="22%" style="border-bottom:1px solid black" |
| + | \end{matrix}</math> |
− | <math>\operatorname{e}</math>
| + | | |
− | | width="22%" style="border-bottom:1px solid black" |
| + | <math>\begin{matrix} |
− | <math>\operatorname{f}</math>
| + | ~\operatorname{d}x~(\operatorname{d}y) |
− | | width="22%" style="border-bottom:1px solid black" |
| + | \\[4pt] |
− | <math>\operatorname{g}</math>
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
− | | width="22%" style="border-bottom:1px solid black" |
| + | \\[4pt] |
− | <math>\operatorname{h}</math> | + | (\operatorname{d}x)(\operatorname{d}y) |
− | |- style="height:50px"
| + | \\[4pt] |
− | | style="border-right:1px solid black" | <math>\operatorname{e}</math>
| + | (\operatorname{d}x)~\operatorname{d}y~ |
− | | <math>\operatorname{e}</math>
| + | \end{matrix}</math> |
− | | <math>\operatorname{f}</math>
| + | | |
− | | <math>\operatorname{g}</math>
| + | <math>\begin{matrix} |
− | | <math>\operatorname{h}</math>
| + | (\operatorname{d}x)~\operatorname{d}y~ |
− | |- style="height:50px"
| + | \\[4pt] |
− | | style="border-right:1px solid black" | <math>\operatorname{f}</math>
| + | (\operatorname{d}x)(\operatorname{d}y) |
− | | <math>\operatorname{f}</math>
| + | \\[4pt] |
− | | <math>\operatorname{e}</math>
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
− | | <math>\operatorname{h}</math>
| + | \\[4pt] |
− | | <math>\operatorname{g}</math>
| + | ~\operatorname{d}x~(\operatorname{d}y) |
− | |- style="height:50px" | + | \end{matrix}</math> |
− | | style="border-right:1px solid black" | <math>\operatorname{g}</math>
| + | | |
− | | <math>\operatorname{g}</math>
| + | <math>\begin{matrix} |
− | | <math>\operatorname{h}</math>
| + | (\operatorname{d}x)(\operatorname{d}y) |
− | | <math>\operatorname{e}</math>
| + | \\[4pt] |
− | | <math>\operatorname{f}</math>
| + | (\operatorname{d}x)~\operatorname{d}y~ |
− | |- style="height:50px"
| + | \\[4pt] |
− | | style="border-right:1px solid black" | <math>\operatorname{h}</math>
| + | ~\operatorname{d}x~(\operatorname{d}y) |
− | | <math>\operatorname{h}</math>
| + | \\[4pt] |
− | | <math>\operatorname{g}</math>
| + | ~\operatorname{d}x~~\operatorname{d}y~ |
− | | <math>\operatorname{f}</math>
| + | \end{matrix}</math> |
− | | <math>\operatorname{e}</math>
| |
− | |} | |
− | | |
− | <br> | |
− | | |
− | ===Symmetric Group S<sub>3</sub>===
| |
− | | |
− | <br>
| |
− | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%" | |
− | |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="16%" | <math>\operatorname{e}</math>
| |
− | | width="16%" | <math>\operatorname{f}</math>
| |
− | | width="16%" | <math>\operatorname{g}</math>
| |
− | | width="16%" | <math>\operatorname{h}</math>
| |
− | | width="16%" | <math>\operatorname{i}</math>
| |
− | | width="16%" | <math>\operatorname{j}</math>
| |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C}
| + | f_3 |
− | \\[3pt]
| + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow
| + | f_{12} |
− | \\[6pt] | |
− | \mathrm{A} & \mathrm{B} & \mathrm{C}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C}
| + | (x) |
− | \\[3pt] | + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow
| + | ~x~ |
− | \\[6pt]
| |
− | \mathrm{C} & \mathrm{A} & \mathrm{B}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C} | + | ~\operatorname{d}x~ |
− | \\[3pt] | + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow | + | (\operatorname{d}x) |
− | \\[6pt]
| |
− | \mathrm{B} & \mathrm{C} & \mathrm{A}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C} | + | ~\operatorname{d}x~ |
− | \\[3pt] | + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow | + | (\operatorname{d}x) |
− | \\[6pt]
| |
− | \mathrm{A} & \mathrm{C} & \mathrm{B}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C} | + | (\operatorname{d}x) |
− | \\[3pt] | + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow | + | ~\operatorname{d}x~ |
− | \\[6pt]
| |
− | \mathrm{C} & \mathrm{B} & \mathrm{A}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | \mathrm{A} & \mathrm{B} & \mathrm{C} | + | (\operatorname{d}x) |
− | \\[3pt] | + | \\[4pt] |
− | \downarrow & \downarrow & \downarrow | + | ~\operatorname{d}x~ |
− | \\[6pt]
| |
− | \mathrm{B} & \mathrm{A} & \mathrm{C}
| |
| \end{matrix}</math> | | \end{matrix}</math> |
− | |}
| |
− |
| |
− | <br>
| |
− |
| |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
| |
− | |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
| |
− | |- style="background:#f0f0ff"
| |
− | | width="16%" | <math>\operatorname{e}</math>
| |
− | | width="16%" | <math>\operatorname{f}</math>
| |
− | | width="16%" | <math>\operatorname{g}</math>
| |
− | | width="16%" | <math>\operatorname{h}</math>
| |
− | | width="16%" | <math>\operatorname{i}</math>
| |
− | | width="16%" | <math>\operatorname{j}</math>
| |
| |- | | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 1 & 0 & 0
| + | f_6 |
− | \\ | + | \\[4pt] |
− | 0 & 1 & 0
| + | f_9 |
− | \\
| |
− | 0 & 0 & 1
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0 & 0 & 1
| + | ~(x,~y)~ |
− | \\ | + | \\[4pt] |
− | 1 & 0 & 0
| + | ((x,~y)) |
− | \\
| |
− | 0 & 1 & 0
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0 & 1 & 0
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
− | \\ | + | \\[4pt] |
− | 0 & 0 & 1
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
− | \\ | + | \end{matrix}</math> |
− | 1 & 0 & 0
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
| + | \\[4pt] |
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 1 & 0 & 0
| + | ~(\operatorname{d}x,~\operatorname{d}y)~ |
− | \\ | + | \\[4pt] |
− | 0 & 0 & 1
| + | ((\operatorname{d}x,~\operatorname{d}y)) |
− | \\ | |
− | 0 & 1 & 0
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| + | |- |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0 & 0 & 1
| + | f_5 |
− | \\ | + | \\[4pt] |
− | 0 & 1 & 0
| + | f_{10} |
− | \\
| |
− | 1 & 0 & 0
| |
| \end{matrix}</math> | | \end{matrix}</math> |
| | | | | |
| <math>\begin{matrix} | | <math>\begin{matrix} |
− | 0 & 1 & 0
| + | (y) |
− | \\ | + | \\[4pt] |
− | 1 & 0 & 0
| + | ~y~ |
− | \\
| |
− | 0 & 0 & 1
| |
| \end{matrix}</math> | | \end{matrix}</math> |
− | |} | + | | |
− | | + | <math>\begin{matrix} |
− | <br> | + | ~\operatorname{d}y~ |
− | | + | \\[4pt] |
− | <pre> | + | (\operatorname{d}y) |
− | Symmetric Group S_3
| + | \end{matrix}</math> |
− | o-------------------------------------------------o
| + | | |
− | | |
| + | <math>\begin{matrix} |
− | | ^ |
| + | (\operatorname{d}y) |
− | | e / \ e |
| + | \\[4pt] |
− | | / \ |
| + | ~\operatorname{d}y~ |
− | | / e \ |
| + | \end{matrix}</math> |
− | | f / \ / \ f |
| + | | |
− | | / \ / \ |
| + | <math>\begin{matrix} |
− | | / f \ f \ |
| + | ~\operatorname{d}y~ |
− | | g / \ / \ / \ g | | + | \\[4pt] |
− | | / \ / \ / \ |
| + | (\operatorname{d}y) |
− | | / g \ g \ g \ |
| + | \end{matrix}</math> |
− | | h / \ / \ / \ / \ h |
| + | | |
− | | / \ / \ / \ / \ |
| + | <math>\begin{matrix} |
− | | / h \ e \ e \ h \ |
| + | (\operatorname{d}y) |
− | | i / \ / \ / \ / \ / \ i |
| + | \\[4pt] |
− | | / \ / \ / \ / \ / \ |
| + | ~\operatorname{d}y~ |
− | | / i \ i \ f \ j \ i \ |
| + | \end{matrix}</math> |
− | | j / \ / \ / \ / \ / \ / \ j |
| + | |- |
− | | / \ / \ / \ / \ / \ / \ |
| + | | |
− | | ( j \ j \ j \ i \ h \ j ) |
| + | <math>\begin{matrix} |
− | | \ / \ / \ / \ / \ / \ / |
| + | f_7 |
− | | \ / \ / \ / \ / \ / \ / |
| + | \\[4pt] |
− | | \ h \ h \ e \ j \ i / |
| + | f_{11} |
− | | \ / \ / \ / \ / \ / |
| + | \\[4pt] |
− | | \ / \ / \ / \ / \ / |
| + | f_{13} |
− | | \ i \ g \ f \ h / |
| + | \\[4pt] |
− | | \ / \ / \ / \ / |
| + | f_{14} |
− | | \ / \ / \ / \ / |
| + | \end{matrix}</math> |
− | | \ f \ e \ g / |
| + | | |
− | | \ / \ / \ / |
| + | <math>\begin{matrix} |
− | | \ / \ / \ / |
| + | (~x~~y~) |
− | | \ g \ f / |
| + | \\[4pt] |
− | | \ / \ / |
| + | (~x~(y)) |
− | | \ / \ / |
| + | \\[4pt] |
− | | \ e / |
| + | ((x)~y~) |
− | | \ / |
| + | \\[4pt] |
− | | \ / |
| + | ((x)(y)) |
− | | v |
| + | \end{matrix}</math> |
− | | |
| + | | |
− | o-------------------------------------------------o
| + | <math>\begin{matrix} |
− | </pre>
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | | + | \\[4pt] |
− | <br>
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
− | | + | \\[4pt] |
− | ===TeX Tables===
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
− | | + | \\[4pt] |
− | <pre> | + | (~\operatorname{d}x~~\operatorname{d}y~) |
− | \tableofcontents | + | \end{matrix}</math> |
− | | + | | |
− | \subsection{Table A1. Propositional Forms on Two Variables}
| + | <math>\begin{matrix} |
− | | + | ((\operatorname{d}x)~\operatorname{d}y~) |
− | Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.
| + | \\[4pt] |
− | | + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | \begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
| + | \\[4pt] |
− | \multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
− | \hline
| + | \\[4pt] |
− | $\mathcal{L}_1$ &
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
− | $\mathcal{L}_2$ &&
| + | \end{matrix}</math> |
− | $\mathcal{L}_3$ &
| + | | |
− | $\mathcal{L}_4$ &
| + | <math>\begin{matrix} |
− | $\mathcal{L}_5$ &
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
− | $\mathcal{L}_6$ \\
| + | \\[4pt] |
− | \hline
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
− | & & $x =$ & 1 1 0 0 & & & \\
| + | \\[4pt] |
− | & & $y =$ & 1 0 1 0 & & & \\
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | \hline
| + | \\[4pt] |
− | $f_{0}$ &
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
− | $f_{0000}$ &&
| + | \end{matrix}</math> |
− | 0 0 0 0 &
| + | | |
− | $(~)$ &
| + | <math>\begin{matrix} |
− | $\operatorname{false}$ &
| + | (~\operatorname{d}x~~\operatorname{d}y~) |
− | $0$ \\
| + | \\[4pt] |
− | $f_{1}$ &
| + | (~\operatorname{d}x~(\operatorname{d}y)) |
− | $f_{0001}$ &&
| + | \\[4pt] |
− | 0 0 0 1 &
| + | ((\operatorname{d}x)~\operatorname{d}y~) |
− | $(x)(y)$ &
| + | \\[4pt] |
− | $\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | $\lnot x \land \lnot y$ \\
| + | \end{matrix}</math> |
− | $f_{2}$ &
| + | |- |
− | $f_{0010}$ &&
| + | | <math>f_{15}\!</math> |
− | 0 0 1 0 &
| + | | <math>((~))</math> |
− | $(x)\ y$ &
| + | | <math>((~))</math> |
− | $y\ \operatorname{without}\ x$ &
| + | | <math>((~))</math> |
− | $\lnot x \land y$ \\
| + | | <math>((~))</math> |
− | $f_{3}$ &
| + | | <math>((~))</math> |
− | $f_{0011}$ &&
| + | |} |
− | 0 0 1 1 &
| + | |
− | $(x)$ &
| + | <br> |
− | $\operatorname{not}\ x$ &
| + | |
− | $\lnot x$ \\
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | $f_{4}$ &
| + | |+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math> |
− | $f_{0100}$ &&
| + | |- style="background:#f0f0ff" |
− | 0 1 0 0 &
| + | | width="10%" | |
− | $x\ (y)$ &
| + | | width="18%" | <math>f\!</math> |
− | $x\ \operatorname{without}\ y$ &
| + | | width="18%" | <math>\operatorname{D}f|_{xy}</math> |
− | $x \land \lnot y$ \\
| + | | width="18%" | <math>\operatorname{D}f|_{x(y)}</math> |
− | $f_{5}$ &
| + | | width="18%" | <math>\operatorname{D}f|_{(x)y}</math> |
− | $f_{0101}$ &&
| + | | width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math> |
− | 0 1 0 1 &
| + | |- |
− | $(y)$ &
| + | | <math>f_0\!</math> |
− | $\operatorname{not}\ y$ &
| + | | <math>(~)</math> |
− | $\lnot y$ \\
| + | | <math>(~)</math> |
− | $f_{6}$ &
| + | | <math>(~)</math> |
− | $f_{0110}$ &&
| + | | <math>(~)</math> |
− | 0 1 1 0 &
| + | | <math>(~)</math> |
− | $(x,\ y)$ &
| + | |- |
− | $x\ \operatorname{not~equal~to}\ y$ &
| + | | |
− | $x \ne y$ \\
| + | <math>\begin{matrix} |
− | $f_{7}$ &
| + | f_1 |
− | $f_{0111}$ &&
| + | \\[4pt] |
− | 0 1 1 1 &
| + | f_2 |
− | $(x\ y)$ &
| + | \\[4pt] |
− | $\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
| + | f_4 |
− | $\lnot x \lor \lnot y$ \\
| + | \\[4pt] |
− | \hline | + | f_8 |
− | $f_{8}$ &
| + | \end{matrix}</math> |
− | $f_{1000}$ &&
| + | | |
− | 1 0 0 0 &
| + | <math>\begin{matrix} |
− | $x\ y$ &
| + | (x)(y) |
− | $x\ \operatorname{and}\ y$ &
| + | \\[4pt] |
− | $x \land y$ \\
| + | (x)~y~ |
− | $f_{9}$ &
| + | \\[4pt] |
− | $f_{1001}$ &&
| + | ~x~(y) |
− | 1 0 0 1 &
| + | \\[4pt] |
− | $((x,\ y))$ &
| + | ~x~~y~ |
− | $x\ \operatorname{equal~to}\ y$ &
| + | \end{matrix}</math> |
− | $x = y$ \\
| + | | |
− | $f_{10}$ &
| + | <math>\begin{matrix} |
− | $f_{1010}$ &&
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | 1 0 1 0 &
| + | \\[4pt] |
− | $y$ &
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | $y$ &
| + | \\[4pt] |
− | $y$ \\
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | $f_{11}$ &
| + | \\[4pt] |
− | $f_{1011}$ &&
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | 1 0 1 1 &
| + | \end{matrix}</math> |
− | $(x\ (y))$ &
| + | | |
− | $\operatorname{not}\ x\ \operatorname{without}\ y$ &
| + | <math>\begin{matrix} |
− | $x \Rightarrow y$ \\
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | $f_{12}$ &
| + | \\[4pt] |
− | $f_{1100}$ &&
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | 1 1 0 0 &
| + | \\[4pt] |
− | $x$ &
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | $x$ &
| + | \\[4pt] |
− | $x$ \\
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | $f_{13}$ &
| + | \end{matrix}</math> |
− | $f_{1101}$ &&
| + | | |
− | 1 1 0 1 &
| + | <math>\begin{matrix} |
− | $((x)\ y)$ &
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | $\operatorname{not}\ y\ \operatorname{without}\ x$ &
| + | \\[4pt] |
− | $x \Leftarrow y$ \\
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | $f_{14}$ &
| + | \\[4pt] |
− | $f_{1110}$ &&
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | 1 1 1 0 &
| + | \\[4pt] |
− | $((x)(y))$ &
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | $x\ \operatorname{or}\ y$ &
| + | \end{matrix}</math> |
− | $x \lor y$ \\
| + | | |
− | $f_{15}$ &
| + | <math>\begin{matrix} |
− | $f_{1111}$ &&
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | 1 1 1 1 &
| + | \\[4pt] |
− | $((~))$ &
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | $\operatorname{true}$ &
| + | \\[4pt] |
− | $1$ \\
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | \hline | + | \\[4pt] |
− | \end{tabular}\end{quote}
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | | + | \end{matrix}</math> |
− | \subsection{Table A2. Propositional Forms on Two Variables} | + | |- |
− | | + | | |
− | Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
| + | <math>\begin{matrix} |
− | | + | f_3 |
− | \begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} | + | \\[4pt] |
− | \multicolumn{7}{c}{\textbf{Table A2. Propositional Forms on Two Variables}} \\ | + | f_{12} |
− | \hline | + | \end{matrix}</math> |
− | $\mathcal{L}_1$ &
| + | | |
− | $\mathcal{L}_2$ &&
| + | <math>\begin{matrix} |
− | $\mathcal{L}_3$ &
| + | (x) |
− | $\mathcal{L}_4$ &
| + | \\[4pt] |
− | $\mathcal{L}_5$ &
| + | ~x~ |
− | $\mathcal{L}_6$ \\
| + | \end{matrix}</math> |
− | \hline
| + | | |
− | & & $x =$ & 1 1 0 0 & & & \\
| + | <math>\begin{matrix} |
− | & & $y =$ & 1 0 1 0 & & & \\
| + | \operatorname{d}x |
− | \hline | + | \\[4pt] |
− | $f_{0}$ &
| + | \operatorname{d}x |
− | $f_{0000}$ &&
| + | \end{matrix}</math> |
− | 0 0 0 0 &
| + | | |
− | $(~)$ &
| + | <math>\begin{matrix} |
− | $\operatorname{false}$ &
| + | \operatorname{d}x |
− | $0$ \\
| + | \\[4pt] |
− | \hline | + | \operatorname{d}x |
− | $f_{1}$ &
| + | \end{matrix}</math> |
− | $f_{0001}$ &&
| + | | |
− | 0 0 0 1 &
| + | <math>\begin{matrix} |
− | $(x)(y)$ &
| + | \operatorname{d}x |
− | $\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
| + | \\[4pt] |
− | $\lnot x \land \lnot y$ \\
| + | \operatorname{d}x |
− | $f_{2}$ &
| + | \end{matrix}</math> |
− | $f_{0010}$ &&
| + | | |
− | 0 0 1 0 &
| + | <math>\begin{matrix} |
− | $(x)\ y$ &
| + | \operatorname{d}x |
− | $y\ \operatorname{without}\ x$ &
| + | \\[4pt] |
− | $\lnot x \land y$ \\
| + | \operatorname{d}x |
− | $f_{4}$ &
| + | \end{matrix}</math> |
− | $f_{0100}$ &&
| + | |- |
− | 0 1 0 0 &
| + | | |
− | $x\ (y)$ &
| + | <math>\begin{matrix} |
− | $x\ \operatorname{without}\ y$ &
| + | f_6 |
− | $x \land \lnot y$ \\
| + | \\[4pt] |
− | $f_{8}$ &
| + | f_9 |
− | $f_{1000}$ &&
| + | \end{matrix}</math> |
− | 1 0 0 0 &
| + | | |
− | $x\ y$ &
| + | <math>\begin{matrix} |
− | $x\ \operatorname{and}\ y$ &
| + | ~(x,~y)~ |
− | $x \land y$ \\
| + | \\[4pt] |
− | \hline | + | ((x,~y)) |
− | $f_{3}$ &
| + | \end{matrix}</math> |
− | $f_{0011}$ &&
| + | | |
− | 0 0 1 1 &
| + | <math>\begin{matrix} |
− | $(x)$ &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $\operatorname{not}\ x$ &
| + | \\[4pt] |
− | $\lnot x$ \\
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $f_{12}$ &
| + | \end{matrix}</math> |
− | $f_{1100}$ &&
| + | | |
− | 1 1 0 0 &
| + | <math>\begin{matrix} |
− | $x$ &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $x$ &
| + | \\[4pt] |
− | $x$ \\
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | \hline | + | \end{matrix}</math> |
− | $f_{6}$ &
| + | | |
− | $f_{0110}$ &&
| + | <math>\begin{matrix} |
− | 0 1 1 0 &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $(x,\ y)$ &
| + | \\[4pt] |
− | $x\ \operatorname{not~equal~to}\ y$ &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $x \ne y$ \\
| + | \end{matrix}</math> |
− | $f_{9}$ &
| + | | |
− | $f_{1001}$ &&
| + | <math>\begin{matrix} |
− | 1 0 0 1 &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $((x,\ y))$ &
| + | \\[4pt] |
− | $x\ \operatorname{equal~to}\ y$ &
| + | (\operatorname{d}x,~\operatorname{d}y) |
− | $x = y$ \\
| + | \end{matrix}</math> |
− | \hline | + | |- |
− | $f_{5}$ &
| + | | |
− | $f_{0101}$ &&
| + | <math>\begin{matrix} |
− | 0 1 0 1 &
| + | f_5 |
− | $(y)$ &
| + | \\[4pt] |
− | $\operatorname{not}\ y$ &
| + | f_{10} |
− | $\lnot y$ \\
| + | \end{matrix}</math> |
− | $f_{10}$ &
| + | | |
− | $f_{1010}$ &&
| + | <math>\begin{matrix} |
− | 1 0 1 0 &
| + | (y) |
− | $y$ &
| + | \\[4pt] |
− | $y$ &
| + | ~y~ |
− | $y$ \\
| + | \end{matrix}</math> |
− | \hline | + | | |
− | $f_{7}$ &
| + | <math>\begin{matrix} |
− | $f_{0111}$ &&
| + | \operatorname{d}y |
− | 0 1 1 1 &
| + | \\[4pt] |
− | $(x\ y)$ &
| + | \operatorname{d}y |
− | $\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
| + | \end{matrix}</math> |
− | $\lnot x \lor \lnot y$ \\
| + | | |
− | $f_{11}$ &
| + | <math>\begin{matrix} |
− | $f_{1011}$ &&
| + | \operatorname{d}y |
− | 1 0 1 1 &
| + | \\[4pt] |
− | $(x\ (y))$ &
| + | \operatorname{d}y |
− | $\operatorname{not}\ x\ \operatorname{without}\ y$ &
| + | \end{matrix}</math> |
− | $x \Rightarrow y$ \\
| + | | |
− | $f_{13}$ &
| + | <math>\begin{matrix} |
− | $f_{1101}$ &&
| + | \operatorname{d}y |
− | 1 1 0 1 &
| + | \\[4pt] |
− | $((x)\ y)$ &
| + | \operatorname{d}y |
− | $\operatorname{not}\ y\ \operatorname{without}\ x$ &
| + | \end{matrix}</math> |
− | $x \Leftarrow y$ \\
| + | | |
− | $f_{14}$ &
| + | <math>\begin{matrix} |
− | $f_{1110}$ &&
| + | \operatorname{d}y |
− | 1 1 1 0 &
| + | \\[4pt] |
− | $((x)(y))$ &
| + | \operatorname{d}y |
− | $x\ \operatorname{or}\ y$ &
| + | \end{matrix}</math> |
− | $x \lor y$ \\
| + | |- |
− | \hline | + | | |
− | $f_{15}$ &
| + | <math>\begin{matrix} |
− | $f_{1111}$ &&
| + | f_7 |
− | 1 1 1 1 &
| + | \\[4pt] |
− | $((~))$ &
| + | f_{11} |
− | $\operatorname{true}$ &
| + | \\[4pt] |
− | $1$ \\
| + | f_{13} |
− | \hline
| + | \\[4pt] |
− | \end{tabular}\end{quote}
| + | f_{14} |
− | | + | \end{matrix}</math> |
− | \subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} | + | | |
− | | + | <math>\begin{matrix} |
− | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} | + | (~x~~y~) |
− | \multicolumn{6}{c}{\textbf{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\ | + | \\[4pt] |
− | \hline
| + | (~x~(y)) |
− | & &
| + | \\[4pt] |
− | $\operatorname{T}_{11}$ &
| + | ((x)~y~) |
− | $\operatorname{T}_{10}$ &
| + | \\[4pt] |
− | $\operatorname{T}_{01}$ &
| + | ((x)(y)) |
− | $\operatorname{T}_{00}$ \\
| + | \end{matrix}</math> |
− | & $f$ &
| + | | |
− | $\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
| + | <math>\begin{matrix} |
− | $\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | $\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
| + | \\[4pt] |
− | $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | \hline | + | \\[4pt] |
− | $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | \hline
| + | \\[4pt] |
− | $f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | $f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\
| + | \end{matrix}</math> |
− | $f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\
| + | | |
− | $f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\
| + | <math>\begin{matrix} |
− | \hline
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | $f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\
| + | \\[4pt] |
− | $f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | \hline
| + | \\[4pt] |
− | $f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | $f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\
| + | \\[4pt] |
− | \hline
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | $f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\
| + | \end{matrix}</math> |
− | $f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\
| + | | |
− | \hline
| + | <math>\begin{matrix} |
− | $f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
− | $f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\
| + | \\[4pt] |
− | $f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
− | $f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\
| + | \\[4pt] |
− | \hline
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
− | $f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\
| + | \\[4pt] |
− | \hline
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
− | \multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\
| + | \end{matrix}</math> |
− | \hline
| + | | |
− | \end{tabular}\end{quote}
| + | <math>\begin{matrix} |
| + | ~~\operatorname{d}x~~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ~~\operatorname{d}x~(\operatorname{d}y)~ |
| + | \\[4pt] |
| + | ~(\operatorname{d}x)~\operatorname{d}y~~ |
| + | \\[4pt] |
| + | ((\operatorname{d}x)(\operatorname{d}y)) |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>f_{15}\!</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | | <math>((~))</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ===Klein Four-Group V<sub>4</sub>=== |
| | | |
− | \subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
| + | <br> |
| | | |
− | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | \multicolumn{6}{c}{\textbf{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\ | + | |- style="height:50px" |
− | \hline
| + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
− | & $f$ &
| + | | width="22%" style="border-bottom:1px solid black" | |
− | $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
| + | <math>\operatorname{T}_{00}</math> |
− | $\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
| + | | width="22%" style="border-bottom:1px solid black" | |
− | $\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
| + | <math>\operatorname{T}_{01}</math> |
− | $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
| + | | width="22%" style="border-bottom:1px solid black" | |
− | \hline | + | <math>\operatorname{T}_{10}</math> |
− | $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
| + | | width="22%" style="border-bottom:1px solid black" | |
− | \hline
| + | <math>\operatorname{T}_{11}</math> |
− | $f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\
| + | |- style="height:50px" |
− | $f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math> |
− | $f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\
| + | | <math>\operatorname{T}_{00}</math> |
− | $f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\
| + | | <math>\operatorname{T}_{01}</math> |
− | \hline | + | | <math>\operatorname{T}_{10}</math> |
− | $f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\
| + | | <math>\operatorname{T}_{11}</math> |
− | $f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\
| + | |- style="height:50px" |
− | \hline | + | | style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math> |
− | $f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\
| + | | <math>\operatorname{T}_{01}</math> |
− | $f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\
| + | | <math>\operatorname{T}_{00}</math> |
− | \hline | + | | <math>\operatorname{T}_{11}</math> |
− | $f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\
| + | | <math>\operatorname{T}_{10}</math> |
− | $f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\
| + | |- style="height:50px" |
− | \hline | + | | style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math> |
− | $f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\
| + | | <math>\operatorname{T}_{10}</math> |
− | $f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\
| + | | <math>\operatorname{T}_{11}</math> |
− | $f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\
| + | | <math>\operatorname{T}_{00}</math> |
− | $f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\
| + | | <math>\operatorname{T}_{01}</math> |
− | \hline | + | |- style="height:50px" |
− | $f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
| + | | style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math> |
− | \hline | + | | <math>\operatorname{T}_{11}</math> |
− | \end{tabular}\end{quote}
| + | | <math>\operatorname{T}_{10}</math> |
| + | | <math>\operatorname{T}_{01}</math> |
| + | | <math>\operatorname{T}_{00}</math> |
| + | |} |
| | | |
− | \subsection{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}
| + | <br> |
| | | |
− | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
− | \multicolumn{6}{c}{\textbf{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\
| + | |- style="height:50px" |
− | \hline
| + | | width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
− | & $f$ &
| + | | width="22%" style="border-bottom:1px solid black" | |
− | $\operatorname{E}f|_{x\ y}$ &
| + | <math>\operatorname{e}</math> |
− | $\operatorname{E}f|_{x (y)}$ &
| + | | width="22%" style="border-bottom:1px solid black" | |
− | $\operatorname{E}f|_{(x) y}$ &
| + | <math>\operatorname{f}</math> |
− | $\operatorname{E}f|_{(x)(y)}$ \\
| + | | width="22%" style="border-bottom:1px solid black" | |
− | \hline | + | <math>\operatorname{g}</math> |
− | $f_{0}$ &
| + | | width="22%" style="border-bottom:1px solid black" | |
− | $(~)$ &
| + | <math>\operatorname{h}</math> |
− | $(~)$ &
| + | |- style="height:50px" |
− | $(~)$ &
| + | | style="border-right:1px solid black" | <math>\operatorname{e}</math> |
− | $(~)$ &
| + | | <math>\operatorname{e}</math> |
− | $(~)$ \\
| + | | <math>\operatorname{f}</math> |
− | \hline
| + | | <math>\operatorname{g}</math> |
− | $f_{1}$ &
| + | | <math>\operatorname{h}</math> |
− | $(x)(y)$ &
| + | |- style="height:50px" |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | | <math>\operatorname{f}</math> |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | | <math>\operatorname{e}</math> |
− | $(\operatorname{d}x)(\operatorname{d}y)$ \\
| + | | <math>\operatorname{h}</math> |
− | $f_{2}$ &
| + | | <math>\operatorname{g}</math> |
− | $(x)\ y$ &
| + | |- style="height:50px" |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | <math>\operatorname{g}</math> |
− | $(\operatorname{d}x)(\operatorname{d}y)$ &
| + | | <math>\operatorname{h}</math> |
− | $(\operatorname{d}x)\ \operatorname{d}y$ \\
| + | | <math>\operatorname{e}</math> |
− | $f_{4}$ &
| + | | <math>\operatorname{f}</math> |
− | $x\ (y)$ &
| + | |- style="height:50px" |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
− | $(\operatorname{d}x)(\operatorname{d}y)$ &
| + | | <math>\operatorname{h}</math> |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | <math>\operatorname{g}</math> |
− | $\operatorname{d}x\ (\operatorname{d}y)$ \\
| + | | <math>\operatorname{f}</math> |
− | $f_{8}$ &
| + | | <math>\operatorname{e}</math> |
− | $x\ y$ &
| + | |} |
− | $(\operatorname{d}x)(\operatorname{d}y)$ &
| + | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | <br> |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | |
− | $\operatorname{d}x\ \operatorname{d}y$ \\
| + | ===Symmetric Group S<sub>3</sub>=== |
− | \hline | + | |
− | $f_{3}$ &
| + | <br> |
− | $(x)$ &
| + | |
− | $\operatorname{d}x$ &
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | $\operatorname{d}x$ &
| + | |+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math> |
− | $(\operatorname{d}x)$ &
| + | |- style="background:#f0f0ff" |
− | $(\operatorname{d}x)$ \\
| + | | width="16%" | <math>\operatorname{e}</math> |
− | $f_{12}$ &
| + | | width="16%" | <math>\operatorname{f}</math> |
− | $x$ &
| + | | width="16%" | <math>\operatorname{g}</math> |
− | $(\operatorname{d}x)$ &
| + | | width="16%" | <math>\operatorname{h}</math> |
− | $(\operatorname{d}x)$ &
| + | | width="16%" | <math>\operatorname{i}</math> |
− | $\operatorname{d}x$ &
| + | | width="16%" | <math>\operatorname{j}</math> |
− | $\operatorname{d}x$ \\
| + | |- |
− | \hline | + | | |
− | $f_{6}$ &
| + | <math>\begin{matrix} |
− | $(x,\ y)$ &
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | \\[3pt] |
− | $((\operatorname{d}x,\ \operatorname{d}y))$ &
| + | \downarrow & \downarrow & \downarrow |
− | $((\operatorname{d}x,\ \operatorname{d}y))$ &
| + | \\[6pt] |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ \\
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $f_{9}$ &
| + | \end{matrix}</math> |
− | $((x,\ y))$ &
| + | | |
− | $((\operatorname{d}x,\ \operatorname{d}y))$ &
| + | <math>\begin{matrix} |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | \\[3pt] |
− | $((\operatorname{d}x,\ \operatorname{d}y))$ \\
| + | \downarrow & \downarrow & \downarrow |
− | \hline | + | \\[6pt] |
− | $f_{5}$ &
| + | \mathrm{C} & \mathrm{A} & \mathrm{B} |
− | $(y)$ &
| + | \end{matrix}</math> |
− | $\operatorname{d}y$ &
| + | | |
− | $(\operatorname{d}y)$ &
| + | <math>\begin{matrix} |
− | $\operatorname{d}y$ &
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $(\operatorname{d}y)$ \\
| + | \\[3pt] |
− | $f_{10}$ &
| + | \downarrow & \downarrow & \downarrow |
− | $y$ &
| + | \\[6pt] |
− | $(\operatorname{d}y)$ &
| + | \mathrm{B} & \mathrm{C} & \mathrm{A} |
− | $\operatorname{d}y$ &
| + | \end{matrix}</math> |
− | $(\operatorname{d}y)$ &
| + | | |
− | $\operatorname{d}y$ \\
| + | <math>\begin{matrix} |
− | \hline | + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $f_{7}$ &
| + | \\[3pt] |
− | $(x\ y)$ &
| + | \downarrow & \downarrow & \downarrow |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | \\[6pt] |
− | $((\operatorname{d}x)\ \operatorname{d}y)$ &
| + | \mathrm{A} & \mathrm{C} & \mathrm{B} |
− | $(\operatorname{d}x\ (\operatorname{d}y))$ &
| + | \end{matrix}</math> |
− | $(\operatorname{d}x\ \operatorname{d}y)$ \\
| + | | |
− | $f_{11}$ &
| + | <math>\begin{matrix} |
− | $(x\ (y))$ &
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $((\operatorname{d}x)\ \operatorname{d}y)$ &
| + | \\[3pt] |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | \downarrow & \downarrow & \downarrow |
− | $(\operatorname{d}x\ \operatorname{d}y)$ &
| + | \\[6pt] |
− | $(\operatorname{d}x\ (\operatorname{d}y))$ \\
| + | \mathrm{C} & \mathrm{B} & \mathrm{A} |
− | $f_{13}$ &
| + | \end{matrix}</math> |
− | $((x)\ y)$ &
| + | | |
− | $(\operatorname{d}x\ (\operatorname{d}y))$ &
| + | <math>\begin{matrix} |
− | $(\operatorname{d}x\ \operatorname{d}y)$ &
| + | \mathrm{A} & \mathrm{B} & \mathrm{C} |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | \\[3pt] |
− | $((\operatorname{d}x)\ \operatorname{d}y)$ \\
| + | \downarrow & \downarrow & \downarrow |
− | $f_{14}$ &
| + | \\[6pt] |
− | $((x)(y))$ &
| + | \mathrm{B} & \mathrm{A} & \mathrm{C} |
− | $(\operatorname{d}x\ \operatorname{d}y)$ &
| + | \end{matrix}</math> |
− | $(\operatorname{d}x\ (\operatorname{d}y))$ &
| + | |} |
− | $((\operatorname{d}x)\ \operatorname{d}y)$ &
| |
− | $((\operatorname{d}x)(\operatorname{d}y))$ \\
| |
− | \hline | |
− | $f_{15}$ &
| |
− | $((~))$ &
| |
− | $((~))$ &
| |
− | $((~))$ &
| |
− | $((~))$ &
| |
− | $((~))$ \\
| |
− | \hline
| |
− | \end{tabular}\end{quote}
| |
| | | |
− | \subsection{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}
| + | <br> |
| | | |
− | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
− | \multicolumn{6}{c}{\textbf{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\ | + | |+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math> |
− | \hline
| + | |- style="background:#f0f0ff" |
− | & $f$ &
| + | | width="16%" | <math>\operatorname{e}</math> |
− | $\operatorname{D}f|_{x\ y}$ &
| + | | width="16%" | <math>\operatorname{f}</math> |
− | $\operatorname{D}f|_{x (y)}$ &
| + | | width="16%" | <math>\operatorname{g}</math> |
− | $\operatorname{D}f|_{(x) y}$ &
| + | | width="16%" | <math>\operatorname{h}</math> |
− | $\operatorname{D}f|_{(x)(y)}$ \\
| + | | width="16%" | <math>\operatorname{i}</math> |
− | \hline | + | | width="16%" | <math>\operatorname{j}</math> |
− | $f_{0}$ &
| + | |- |
− | $(~)$ &
| + | | |
− | $(~)$ &
| + | <math>\begin{matrix} |
− | $(~)$ &
| + | 1 & 0 & 0 |
− | $(~)$ &
| + | \\ |
− | $(~)$ \\
| + | 0 & 1 & 0 |
− | \hline
| + | \\ |
− | $f_{1}$ &
| + | 0 & 0 & 1 |
− | $(x)(y)$ &
| + | \end{matrix}</math> |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | <math>\begin{matrix} |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | 0 & 0 & 1 |
− | $((\operatorname{d}x)(\operatorname{d}y))$ \\
| + | \\ |
− | $f_{2}$ &
| + | 1 & 0 & 0 |
− | $(x)\ y$ &
| + | \\ |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | 0 & 1 & 0 |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | \end{matrix}</math> |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ \\
| + | <math>\begin{matrix} |
− | $f_{4}$ &
| + | 0 & 1 & 0 |
− | $x\ (y)$ &
| + | \\ |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | 0 & 0 & 1 |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | \\ |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | 1 & 0 & 0 |
− | $\operatorname{d}x\ (\operatorname{d}y)$ \\
| + | \end{matrix}</math> |
− | $f_{8}$ &
| + | | |
− | $x\ y$ &
| + | <math>\begin{matrix} |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | 1 & 0 & 0 |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | \\ |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | 0 & 0 & 1 |
− | $\operatorname{d}x\ \operatorname{d}y$ \\
| + | \\ |
− | \hline
| + | 0 & 1 & 0 |
− | $f_{3}$ &
| + | \end{matrix}</math> |
− | $(x)$ &
| + | | |
− | $\operatorname{d}x$ &
| + | <math>\begin{matrix} |
− | $\operatorname{d}x$ &
| + | 0 & 0 & 1 |
− | $\operatorname{d}x$ &
| + | \\ |
− | $\operatorname{d}x$ \\
| + | 0 & 1 & 0 |
− | $f_{12}$ &
| + | \\ |
− | $x$ &
| + | 1 & 0 & 0 |
− | $\operatorname{d}x$ &
| + | \end{matrix}</math> |
− | $\operatorname{d}x$ &
| + | | |
− | $\operatorname{d}x$ &
| + | <math>\begin{matrix} |
− | $\operatorname{d}x$ \\
| + | 0 & 1 & 0 |
− | \hline | + | \\ |
− | $f_{6}$ &
| + | 1 & 0 & 0 |
− | $(x,\ y)$ &
| + | \\ |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | 0 & 0 & 1 |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | \end{matrix}</math> |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | |} |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ \\
| + | |
− | $f_{9}$ &
| + | <br> |
− | $((x,\ y))$ &
| + | |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | <pre> |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | Symmetric Group S_3 |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ &
| + | o-------------------------------------------------o |
− | $(\operatorname{d}x,\ \operatorname{d}y)$ \\
| + | | | |
− | \hline | + | | ^ | |
− | $f_{5}$ &
| + | | e / \ e | |
− | $(y)$ &
| + | | / \ | |
− | $\operatorname{d}y$ &
| + | | / e \ | |
− | $\operatorname{d}y$ &
| + | | f / \ / \ f | |
− | $\operatorname{d}y$ &
| + | | / \ / \ | |
− | $\operatorname{d}y$ \\
| + | | / f \ f \ | |
− | $f_{10}$ &
| + | | g / \ / \ / \ g | |
− | $y$ &
| + | | / \ / \ / \ | |
− | $\operatorname{d}y$ &
| + | | / g \ g \ g \ | |
− | $\operatorname{d}y$ &
| + | | h / \ / \ / \ / \ h | |
− | $\operatorname{d}y$ &
| + | | / \ / \ / \ / \ | |
− | $\operatorname{d}y$ \\
| + | | / h \ e \ e \ h \ | |
− | \hline | + | | i / \ / \ / \ / \ / \ i | |
− | $f_{7}$ &
| + | | / \ / \ / \ / \ / \ | |
− | $(x\ y)$ &
| + | | / i \ i \ f \ j \ i \ | |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | | j / \ / \ / \ / \ / \ / \ j | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | | / \ / \ / \ / \ / \ / \ | |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | | ( j \ j \ j \ i \ h \ j ) | |
− | $\operatorname{d}x\ \operatorname{d}y$ \\
| + | | \ / \ / \ / \ / \ / \ / | |
− | $f_{11}$ &
| + | | \ / \ / \ / \ / \ / \ / | |
− | $(x\ (y))$ &
| + | | \ h \ h \ e \ j \ i / | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | | \ / \ / \ / \ / \ / | |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | | \ / \ / \ / \ / \ / | |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | \ i \ g \ f \ h / | |
− | $\operatorname{d}x\ (\operatorname{d}y)$ \\
| + | | \ / \ / \ / \ / | |
− | $f_{13}$ &
| + | | \ / \ / \ / \ / | |
− | $((x)\ y)$ &
| + | | \ f \ e \ g / | |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | | \ / \ / \ / | |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | \ / \ / \ / | |
− | $((\operatorname{d}x)(\operatorname{d}y))$ &
| + | | \ g \ f / | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ \\
| + | | \ / \ / | |
− | $f_{14}$ &
| + | | \ / \ / | |
− | $((x)(y))$ &
| + | | \ e / | |
− | $\operatorname{d}x\ \operatorname{d}y$ &
| + | | \ / | |
− | $\operatorname{d}x\ (\operatorname{d}y)$ &
| + | | \ / | |
− | $(\operatorname{d}x)\ \operatorname{d}y$ &
| + | | v | |
− | $((\operatorname{d}x)(\operatorname{d}y))$ \\
| + | | | |
| + | o-------------------------------------------------o |
| + | </pre> |
| + | |
| + | <br> |
| + | |
| + | ===TeX Tables=== |
| + | |
| + | <pre> |
| + | \tableofcontents |
| + | |
| + | \subsection{Table A1. Propositional Forms on Two Variables} |
| + | |
| + | Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems. |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} |
| + | \multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\ |
| \hline | | \hline |
− | $f_{15}$ & | + | $\mathcal{L}_1$ & |
− | $((~))$ & | + | $\mathcal{L}_2$ && |
− | $(~)$ & | + | $\mathcal{L}_3$ & |
− | $(~)$ & | + | $\mathcal{L}_4$ & |
− | $(~)$ & | + | $\mathcal{L}_5$ & |
− | $(~)$ \\ | + | $\mathcal{L}_6$ \\ |
| + | \hline |
| + | & & $x =$ & 1 1 0 0 & & & \\ |
| + | & & $y =$ & 1 0 1 0 & & & \\ |
| + | \hline |
| + | $f_{0}$ & |
| + | $f_{0000}$ && |
| + | 0 0 0 0 & |
| + | $(~)$ & |
| + | $\operatorname{false}$ & |
| + | $0$ \\ |
| + | $f_{1}$ & |
| + | $f_{0001}$ && |
| + | 0 0 0 1 & |
| + | $(x)(y)$ & |
| + | $\operatorname{neither}\ x\ \operatorname{nor}\ y$ & |
| + | $\lnot x \land \lnot y$ \\ |
| + | $f_{2}$ & |
| + | $f_{0010}$ && |
| + | 0 0 1 0 & |
| + | $(x)\ y$ & |
| + | $y\ \operatorname{without}\ x$ & |
| + | $\lnot x \land y$ \\ |
| + | $f_{3}$ & |
| + | $f_{0011}$ && |
| + | 0 0 1 1 & |
| + | $(x)$ & |
| + | $\operatorname{not}\ x$ & |
| + | $\lnot x$ \\ |
| + | $f_{4}$ & |
| + | $f_{0100}$ && |
| + | 0 1 0 0 & |
| + | $x\ (y)$ & |
| + | $x\ \operatorname{without}\ y$ & |
| + | $x \land \lnot y$ \\ |
| + | $f_{5}$ & |
| + | $f_{0101}$ && |
| + | 0 1 0 1 & |
| + | $(y)$ & |
| + | $\operatorname{not}\ y$ & |
| + | $\lnot y$ \\ |
| + | $f_{6}$ & |
| + | $f_{0110}$ && |
| + | 0 1 1 0 & |
| + | $(x,\ y)$ & |
| + | $x\ \operatorname{not~equal~to}\ y$ & |
| + | $x \ne y$ \\ |
| + | $f_{7}$ & |
| + | $f_{0111}$ && |
| + | 0 1 1 1 & |
| + | $(x\ y)$ & |
| + | $\operatorname{not~both}\ x\ \operatorname{and}\ y$ & |
| + | $\lnot x \lor \lnot y$ \\ |
| \hline | | \hline |
− | \end{tabular}\end{quote} | + | $f_{8}$ & |
| + | $f_{1000}$ && |
| + | 1 0 0 0 & |
| + | $x\ y$ & |
| + | $x\ \operatorname{and}\ y$ & |
| + | $x \land y$ \\ |
| + | $f_{9}$ & |
| + | $f_{1001}$ && |
| + | 1 0 0 1 & |
| + | $((x,\ y))$ & |
| + | $x\ \operatorname{equal~to}\ y$ & |
| + | $x = y$ \\ |
| + | $f_{10}$ & |
| + | $f_{1010}$ && |
| + | 1 0 1 0 & |
| + | $y$ & |
| + | $y$ & |
| + | $y$ \\ |
| + | $f_{11}$ & |
| + | $f_{1011}$ && |
| + | 1 0 1 1 & |
| + | $(x\ (y))$ & |
| + | $\operatorname{not}\ x\ \operatorname{without}\ y$ & |
| + | $x \Rightarrow y$ \\ |
| + | $f_{12}$ & |
| + | $f_{1100}$ && |
| + | 1 1 0 0 & |
| + | $x$ & |
| + | $x$ & |
| + | $x$ \\ |
| + | $f_{13}$ & |
| + | $f_{1101}$ && |
| + | 1 1 0 1 & |
| + | $((x)\ y)$ & |
| + | $\operatorname{not}\ y\ \operatorname{without}\ x$ & |
| + | $x \Leftarrow y$ \\ |
| + | $f_{14}$ & |
| + | $f_{1110}$ && |
| + | 1 1 1 0 & |
| + | $((x)(y))$ & |
| + | $x\ \operatorname{or}\ y$ & |
| + | $x \lor y$ \\ |
| + | $f_{15}$ & |
| + | $f_{1111}$ && |
| + | 1 1 1 1 & |
| + | $((~))$ & |
| + | $\operatorname{true}$ & |
| + | $1$ \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| + | |
| + | \subsection{Table A2. Propositional Forms on Two Variables} |
| + | |
| + | Table A2 lists the sixteen Boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes. |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} |
| + | \multicolumn{7}{c}{\textbf{Table A2. Propositional Forms on Two Variables}} \\ |
| + | \hline |
| + | $\mathcal{L}_1$ & |
| + | $\mathcal{L}_2$ && |
| + | $\mathcal{L}_3$ & |
| + | $\mathcal{L}_4$ & |
| + | $\mathcal{L}_5$ & |
| + | $\mathcal{L}_6$ \\ |
| + | \hline |
| + | & & $x =$ & 1 1 0 0 & & & \\ |
| + | & & $y =$ & 1 0 1 0 & & & \\ |
| + | \hline |
| + | $f_{0}$ & |
| + | $f_{0000}$ && |
| + | 0 0 0 0 & |
| + | $(~)$ & |
| + | $\operatorname{false}$ & |
| + | $0$ \\ |
| + | \hline |
| + | $f_{1}$ & |
| + | $f_{0001}$ && |
| + | 0 0 0 1 & |
| + | $(x)(y)$ & |
| + | $\operatorname{neither}\ x\ \operatorname{nor}\ y$ & |
| + | $\lnot x \land \lnot y$ \\ |
| + | $f_{2}$ & |
| + | $f_{0010}$ && |
| + | 0 0 1 0 & |
| + | $(x)\ y$ & |
| + | $y\ \operatorname{without}\ x$ & |
| + | $\lnot x \land y$ \\ |
| + | $f_{4}$ & |
| + | $f_{0100}$ && |
| + | 0 1 0 0 & |
| + | $x\ (y)$ & |
| + | $x\ \operatorname{without}\ y$ & |
| + | $x \land \lnot y$ \\ |
| + | $f_{8}$ & |
| + | $f_{1000}$ && |
| + | 1 0 0 0 & |
| + | $x\ y$ & |
| + | $x\ \operatorname{and}\ y$ & |
| + | $x \land y$ \\ |
| + | \hline |
| + | $f_{3}$ & |
| + | $f_{0011}$ && |
| + | 0 0 1 1 & |
| + | $(x)$ & |
| + | $\operatorname{not}\ x$ & |
| + | $\lnot x$ \\ |
| + | $f_{12}$ & |
| + | $f_{1100}$ && |
| + | 1 1 0 0 & |
| + | $x$ & |
| + | $x$ & |
| + | $x$ \\ |
| + | \hline |
| + | $f_{6}$ & |
| + | $f_{0110}$ && |
| + | 0 1 1 0 & |
| + | $(x,\ y)$ & |
| + | $x\ \operatorname{not~equal~to}\ y$ & |
| + | $x \ne y$ \\ |
| + | $f_{9}$ & |
| + | $f_{1001}$ && |
| + | 1 0 0 1 & |
| + | $((x,\ y))$ & |
| + | $x\ \operatorname{equal~to}\ y$ & |
| + | $x = y$ \\ |
| + | \hline |
| + | $f_{5}$ & |
| + | $f_{0101}$ && |
| + | 0 1 0 1 & |
| + | $(y)$ & |
| + | $\operatorname{not}\ y$ & |
| + | $\lnot y$ \\ |
| + | $f_{10}$ & |
| + | $f_{1010}$ && |
| + | 1 0 1 0 & |
| + | $y$ & |
| + | $y$ & |
| + | $y$ \\ |
| + | \hline |
| + | $f_{7}$ & |
| + | $f_{0111}$ && |
| + | 0 1 1 1 & |
| + | $(x\ y)$ & |
| + | $\operatorname{not~both}\ x\ \operatorname{and}\ y$ & |
| + | $\lnot x \lor \lnot y$ \\ |
| + | $f_{11}$ & |
| + | $f_{1011}$ && |
| + | 1 0 1 1 & |
| + | $(x\ (y))$ & |
| + | $\operatorname{not}\ x\ \operatorname{without}\ y$ & |
| + | $x \Rightarrow y$ \\ |
| + | $f_{13}$ & |
| + | $f_{1101}$ && |
| + | 1 1 0 1 & |
| + | $((x)\ y)$ & |
| + | $\operatorname{not}\ y\ \operatorname{without}\ x$ & |
| + | $x \Leftarrow y$ \\ |
| + | $f_{14}$ & |
| + | $f_{1110}$ && |
| + | 1 1 1 0 & |
| + | $((x)(y))$ & |
| + | $x\ \operatorname{or}\ y$ & |
| + | $x \lor y$ \\ |
| + | \hline |
| + | $f_{15}$ & |
| + | $f_{1111}$ && |
| + | 1 1 1 1 & |
| + | $((~))$ & |
| + | $\operatorname{true}$ & |
| + | $1$ \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| + | |
| + | \subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| + | \multicolumn{6}{c}{\textbf{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\ |
| + | \hline |
| + | & & |
| + | $\operatorname{T}_{11}$ & |
| + | $\operatorname{T}_{10}$ & |
| + | $\operatorname{T}_{01}$ & |
| + | $\operatorname{T}_{00}$ \\ |
| + | & $f$ & |
| + | $\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
| + | $\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
| + | $\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
| + | $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
| + | \hline |
| + | $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| + | \hline |
| + | $f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\ |
| + | $f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\ |
| + | $f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\ |
| + | $f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\ |
| + | \hline |
| + | $f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\ |
| + | $f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\ |
| + | \hline |
| + | $f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\ |
| + | $f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\ |
| + | \hline |
| + | $f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\ |
| + | $f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\ |
| + | \hline |
| + | $f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\ |
| + | $f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\ |
| + | $f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\ |
| + | $f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\ |
| + | \hline |
| + | $f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\ |
| + | \hline |
| + | \multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| + | |
| + | \subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| + | \multicolumn{6}{c}{\textbf{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\ |
| + | \hline |
| + | & $f$ & |
| + | $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
| + | $\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
| + | $\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
| + | $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
| + | \hline |
| + | $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| + | \hline |
| + | $f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
| + | $f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
| + | $f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
| + | $f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
| + | \hline |
| + | $f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
| + | $f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
| + | \hline |
| + | $f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
| + | $f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
| + | \hline |
| + | $f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
| + | $f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
| + | \hline |
| + | $f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
| + | $f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
| + | $f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
| + | $f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
| + | \hline |
| + | $f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| + | |
| + | \subsection{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$} |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| + | \multicolumn{6}{c}{\textbf{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\ |
| + | \hline |
| + | & $f$ & |
| + | $\operatorname{E}f|_{x\ y}$ & |
| + | $\operatorname{E}f|_{x (y)}$ & |
| + | $\operatorname{E}f|_{(x) y}$ & |
| + | $\operatorname{E}f|_{(x)(y)}$ \\ |
| + | \hline |
| + | $f_{0}$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ \\ |
| + | \hline |
| + | $f_{1}$ & |
| + | $(x)(y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $(\operatorname{d}x)(\operatorname{d}y)$ \\ |
| + | $f_{2}$ & |
| + | $(x)\ y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $(\operatorname{d}x)(\operatorname{d}y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ \\ |
| + | $f_{4}$ & |
| + | $x\ (y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $(\operatorname{d}x)(\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ \\ |
| + | $f_{8}$ & |
| + | $x\ y$ & |
| + | $(\operatorname{d}x)(\operatorname{d}y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ \\ |
| + | \hline |
| + | $f_{3}$ & |
| + | $(x)$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ & |
| + | $(\operatorname{d}x)$ & |
| + | $(\operatorname{d}x)$ \\ |
| + | $f_{12}$ & |
| + | $x$ & |
| + | $(\operatorname{d}x)$ & |
| + | $(\operatorname{d}x)$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ \\ |
| + | \hline |
| + | $f_{6}$ & |
| + | $(x,\ y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $((\operatorname{d}x,\ \operatorname{d}y))$ & |
| + | $((\operatorname{d}x,\ \operatorname{d}y))$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ \\ |
| + | $f_{9}$ & |
| + | $((x,\ y))$ & |
| + | $((\operatorname{d}x,\ \operatorname{d}y))$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $((\operatorname{d}x,\ \operatorname{d}y))$ \\ |
| + | \hline |
| + | $f_{5}$ & |
| + | $(y)$ & |
| + | $\operatorname{d}y$ & |
| + | $(\operatorname{d}y)$ & |
| + | $\operatorname{d}y$ & |
| + | $(\operatorname{d}y)$ \\ |
| + | $f_{10}$ & |
| + | $y$ & |
| + | $(\operatorname{d}y)$ & |
| + | $\operatorname{d}y$ & |
| + | $(\operatorname{d}y)$ & |
| + | $\operatorname{d}y$ \\ |
| + | \hline |
| + | $f_{7}$ & |
| + | $(x\ y)$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $((\operatorname{d}x)\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x\ (\operatorname{d}y))$ & |
| + | $(\operatorname{d}x\ \operatorname{d}y)$ \\ |
| + | $f_{11}$ & |
| + | $(x\ (y))$ & |
| + | $((\operatorname{d}x)\ \operatorname{d}y)$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $(\operatorname{d}x\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x\ (\operatorname{d}y))$ \\ |
| + | $f_{13}$ & |
| + | $((x)\ y)$ & |
| + | $(\operatorname{d}x\ (\operatorname{d}y))$ & |
| + | $(\operatorname{d}x\ \operatorname{d}y)$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $((\operatorname{d}x)\ \operatorname{d}y)$ \\ |
| + | $f_{14}$ & |
| + | $((x)(y))$ & |
| + | $(\operatorname{d}x\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x\ (\operatorname{d}y))$ & |
| + | $((\operatorname{d}x)\ \operatorname{d}y)$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ \\ |
| + | \hline |
| + | $f_{15}$ & |
| + | $((~))$ & |
| + | $((~))$ & |
| + | $((~))$ & |
| + | $((~))$ & |
| + | $((~))$ \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| + | |
| + | \subsection{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$} |
| + | |
| + | \begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| + | \multicolumn{6}{c}{\textbf{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\ |
| + | \hline |
| + | & $f$ & |
| + | $\operatorname{D}f|_{x\ y}$ & |
| + | $\operatorname{D}f|_{x (y)}$ & |
| + | $\operatorname{D}f|_{(x) y}$ & |
| + | $\operatorname{D}f|_{(x)(y)}$ \\ |
| + | \hline |
| + | $f_{0}$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ \\ |
| + | \hline |
| + | $f_{1}$ & |
| + | $(x)(y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ \\ |
| + | $f_{2}$ & |
| + | $(x)\ y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ \\ |
| + | $f_{4}$ & |
| + | $x\ (y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ \\ |
| + | $f_{8}$ & |
| + | $x\ y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ \\ |
| + | \hline |
| + | $f_{3}$ & |
| + | $(x)$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ \\ |
| + | $f_{12}$ & |
| + | $x$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ & |
| + | $\operatorname{d}x$ \\ |
| + | \hline |
| + | $f_{6}$ & |
| + | $(x,\ y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ \\ |
| + | $f_{9}$ & |
| + | $((x,\ y))$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ & |
| + | $(\operatorname{d}x,\ \operatorname{d}y)$ \\ |
| + | \hline |
| + | $f_{5}$ & |
| + | $(y)$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ \\ |
| + | $f_{10}$ & |
| + | $y$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ & |
| + | $\operatorname{d}y$ \\ |
| + | \hline |
| + | $f_{7}$ & |
| + | $(x\ y)$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ \\ |
| + | $f_{11}$ & |
| + | $(x\ (y))$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ \\ |
| + | $f_{13}$ & |
| + | $((x)\ y)$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ \\ |
| + | $f_{14}$ & |
| + | $((x)(y))$ & |
| + | $\operatorname{d}x\ \operatorname{d}y$ & |
| + | $\operatorname{d}x\ (\operatorname{d}y)$ & |
| + | $(\operatorname{d}x)\ \operatorname{d}y$ & |
| + | $((\operatorname{d}x)(\operatorname{d}y))$ \\ |
| + | \hline |
| + | $f_{15}$ & |
| + | $((~))$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ & |
| + | $(~)$ \\ |
| + | \hline |
| + | \end{tabular}\end{quote} |
| </pre> | | </pre> |
| + | |
| + | ==Group Operation Tables== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
| + | |+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math> |
| + | | style="border-bottom:1px solid black" | <math>x_0\!</math> |
| + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
| + | | style="border-bottom:1px solid black" | <math>x_j\!</math> |
| + | | style="border-bottom:1px solid black" | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>x_0 * x_0\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>x_0 * x_j\!</math> |
| + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>x_i * x_0\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>x_i * x_j\!</math> |
| + | | <math>\cdots\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
| + | |+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_0 * x_j),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_i * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_i * x_j),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%" |
| + | |+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_0\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_j * x_0),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>x_i\!</math> |
| + | | <math>\{\!</math> |
| + | | <math>(x_0 ~,~ x_0 * x_i),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>(x_j ~,~ x_j * x_i),\!</math> |
| + | | <math>\cdots\!</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | width="12%" style="border-right:1px solid black" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="22%" | <math>\cdots\!</math> |
| + | | width="18%" | <math>\cdots\!</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{h}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{e}</math> |
| + | | <math>\operatorname{f}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{h}</math> |
| + | | <math>\operatorname{g}</math> |
| + | | <math>\operatorname{f}</math> |
| + | | <math>\operatorname{e}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math> |
| + | | width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math> |
| + | | width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math> |
| + | | width="16%" | <math>(\operatorname{h}, \operatorname{h})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{f}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{e}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{g})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{g}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{e}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{f})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{e}, \operatorname{h}),</math> |
| + | | <math>(\operatorname{f}, \operatorname{g}),</math> |
| + | | <math>(\operatorname{g}, \operatorname{f}),</math> |
| + | | <math>(\operatorname{h}, \operatorname{e})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{f}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{g}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{h}</math> |
| + | | <math>\{\!</math> |
| + | | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math> |
| + | | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{b}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{c}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{a}</math> |
| + | | <math>\operatorname{b}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math> |
| + | | width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math> |
| + | | width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math> |
| + | | width="16%" | <math>(\operatorname{c}, \operatorname{c})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{a}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{a}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{b}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{1})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{b}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{b}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{a})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{c}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{1}, \operatorname{c}),</math> |
| + | | <math>(\operatorname{a}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{b}, \operatorname{a}),</math> |
| + | | <math>(\operatorname{c}, \operatorname{b})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math> |
| + | | width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{2}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{3}</math> |
| + | | <math>\operatorname{0}</math> |
| + | | <math>\operatorname{1}</math> |
| + | | <math>\operatorname{2}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%" |
| + | |+ <math>\text{Table 35.2}~~\text{Regular Representation of the Group}~Z_4(+)</math> |
| + | |- style="height:50px" |
| + | | style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math> |
| + | | colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math> |
| + | |- style="height:50px" |
| + | | width="20%" style="border-right:1px solid black" | <math>\operatorname{0}</math> |
| + | | width="4%" | <math>\{\!</math> |
| + | | width="16%" | <math>(\operatorname{0}, \operatorname{0}),</math> |
| + | | width="20%" | <math>(\operatorname{1}, \operatorname{1}),</math> |
| + | | width="20%" | <math>(\operatorname{2}, \operatorname{2}),</math> |
| + | | width="16%" | <math>(\operatorname{3}, \operatorname{3})</math> |
| + | | width="4%" | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{1}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{2}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{0})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{2}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{2}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{0}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{1})</math> |
| + | | <math>\}\!</math> |
| + | |- style="height:50px" |
| + | | style="border-right:1px solid black" | <math>\operatorname{3}</math> |
| + | | <math>\{\!</math> |
| + | | <math>(\operatorname{0}, \operatorname{3}),</math> |
| + | | <math>(\operatorname{1}, \operatorname{0}),</math> |
| + | | <math>(\operatorname{2}, \operatorname{1}),</math> |
| + | | <math>(\operatorname{3}, \operatorname{2})</math> |
| + | | <math>\}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ==Higher Order Propositions== |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"><math>x:</math></td> |
| + | <td style="border-bottom:2px solid black"><math>1 ~ 0</math></td> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{0}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{5}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{6}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{9}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>m_{15}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0 ~ 0</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0 ~ 1</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} x \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>1 ~ 0</math></td> |
| + | <td style="border-right:2px solid black"><math>x</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>1 ~ 1</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 2.} ~~ \text{Interpretive Categories for Higher Order Propositions} ~ (n = 1)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black">Measure</td> |
| + | <td style="border-bottom:2px solid black">Happening</td> |
| + | <td style="border-bottom:2px solid black">Exactness</td> |
| + | <td style="border-bottom:2px solid black">Existence</td> |
| + | <td style="border-bottom:2px solid black">Linearity</td> |
| + | <td style="border-bottom:2px solid black">Uniformity</td> |
| + | <td style="border-bottom:2px solid black">Information</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{0}</math></td> |
| + | <td>Nothing happens</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{1}</math></td> |
| + | <td> </td> |
| + | <td>Just false</td> |
| + | <td>Nothing exists</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{2}</math></td> |
| + | <td> </td> |
| + | <td>Just not <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{3}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Nothing is <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{4}</math></td> |
| + | <td> </td> |
| + | <td>Just <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{5}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Everything is <math>x</math></td> |
| + | <td><math>f</math> is linear</td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{6}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>f</math> is not uniform</td> |
| + | <td><math>f</math> is informed</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{7}</math></td> |
| + | <td> </td> |
| + | <td>Not just true</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{8}</math></td> |
| + | <td> </td> |
| + | <td>Just true</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{9}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>f</math> is uniform</td> |
| + | <td><math>f</math> is not informed</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{10}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Something is not <math>x</math></td> |
| + | <td><math>f</math> is not linear</td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{11}</math></td> |
| + | <td> </td> |
| + | <td>Not just <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{12}</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td>Something is <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{13}</math></td> |
| + | <td> </td> |
| + | <td>Not just not <math>x</math></td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{14}</math></td> |
| + | <td> </td> |
| + | <td>Not just false</td> |
| + | <td>Something exists</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-right:2px solid black"><math>m_{15}</math></td> |
| + | <td>Anything happens</td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td> </td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="background:white; color:black; text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 3.} ~~ \text{Higher Order Propositions} ~ (n = 2)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"><math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{0}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{1}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{2}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{3}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{4}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{5}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{6}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{7}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{8}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{9}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{10}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{11}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{12}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{13}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{14}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{15}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{16}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{17}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{18}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{19}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{20}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{21}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{22}{m}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\underset{23}{m}</math></td> |
| + | </tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td> |
| + | <td>0</td><td>0</td><td>0</td><td>0</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 4.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \alpha_{i} f = \Upsilon (f_{i}, f) = \Upsilon (f_{i} \Rightarrow f)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{15}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{9}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{6}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{5}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\alpha_{0}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="1" cellspacing="0" style="text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 5.} ~~ \text{Qualifiers of the Implication Ordering:} ~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:2px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td style="border-bottom:2px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | |
| + | <td style="border-bottom:2px solid black; border-right:2px solid black"><math>f</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{0}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{1}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{2}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{3}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{4}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{5}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{6}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{7}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{8}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{9}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{10}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{11}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{12}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{13}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{14}</math></td> |
| + | <td style="border-bottom:2px solid black"><math>\beta_{15}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:2px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:2px solid black"><math>v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:2px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:2px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%" |
| + | |+ <math>\text{Table 7.} ~~ \text{Syllogistic Premisses as Higher Order Indicator Functions}</math> |
| + | | |
| + | <math>\begin{array}{clcl} |
| + | \mathrm{A} |
| + | & \mathrm{Universal~Affirmative} |
| + | & \mathrm{All} ~ u ~ \mathrm{is} ~ v |
| + | & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 0 |
| + | \\ |
| + | \mathrm{E} |
| + | & \mathrm{Universal~Negative} |
| + | & \mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | & \mathrm{Indicator~of} ~ u \cdot v = 0 |
| + | \\ |
| + | \mathrm{I} |
| + | & \mathrm{Particular~Affirmative} |
| + | & \mathrm{Some} ~ u ~ \mathrm{is} ~ v |
| + | & \mathrm{Indicator~of} ~ u \cdot v = 1 |
| + | \\ |
| + | \mathrm{O} |
| + | & \mathrm{Particular~Negative} |
| + | & \mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | & \mathrm{Indicator~of} ~ u \texttt{(} v \texttt{)} = 1 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 8.} ~~ \text{Simple Qualifiers of Propositions (Version 1)}</math></font></caption> |
| + | |
| + | <tr> |
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td width="6%" style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
| + | <math>f</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{11} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{10} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{01} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{00} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{00} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{01} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{10} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{11} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{0}</math></td> |
| + | <td><math>0000</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{8}</math></td> |
| + | <td><math>1000</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{9}</math></td> |
| + | <td><math>1001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{10}</math></td> |
| + | <td><math>1010</math></td> |
| + | <td style="border-right:1px solid black"><math>v</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{12}</math></td> |
| + | <td><math>1100</math></td> |
| + | <td style="border-right:1px solid black"><math>u</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{14}</math></td> |
| + | <td><math>1110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 9.} ~~ \text{Simple Qualifiers of Propositions (Version 2)}</math></font></caption> |
| + | |
| + | <tr> |
| + | <td width="4%" style="border-bottom:1px solid black" align="right"> |
| + | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math></td> |
| + | <td width="6%" style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix}1100\\1010\end{matrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black; border-right:1px solid black"> |
| + | <math>f</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{11} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{10} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{01} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \texttt{(} \ell_{00} \texttt{)} |
| + | \\ |
| + | \mathrm{No} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{00} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{01} |
| + | \\ |
| + | \mathrm{Some} ~ \texttt{(} u \texttt{)} |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{10} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ \texttt{(} v \texttt{)} |
| + | \end{smallmatrix}</math></td> |
| + | <td width="10%" style="border-bottom:1px solid black"> |
| + | <math>\begin{smallmatrix} |
| + | \ell_{11} |
| + | \\ |
| + | \mathrm{Some} ~ u |
| + | \\ |
| + | \mathrm{is} ~ v |
| + | \end{smallmatrix}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{0}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>0000</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{(~)}</math></td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{1}</math></td> |
| + | <td><math>0001</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{2}</math></td> |
| + | <td><math>0010</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u\texttt{)} ~ v</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{4}</math></td> |
| + | <td><math>0100</math></td> |
| + | <td style="border-right:1px solid black"><math>u ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{8}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1000</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u ~ v</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{3}</math></td> |
| + | <td><math>0011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{12}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1100</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>u</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{6}</math></td> |
| + | <td><math>0110</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u \texttt{,} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{9}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1001</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{,} v \texttt{))}</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{5}</math></td> |
| + | <td><math>0101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{10}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1010</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>v</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{7}</math></td> |
| + | <td><math>0111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ v \texttt{)}</math></td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{11}</math></td> |
| + | <td><math>1011</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{13}</math></td> |
| + | <td><math>1101</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>f_{14}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>1110</math></td> |
| + | <td style="border-bottom:1px solid black; border-right:1px solid black"><math>\texttt{((} u \texttt{)(} v \texttt{))}</math></td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:white; color:black">0</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td> |
| + | <td style="border-bottom:1px solid black; background:black; color:white">1</td></tr> |
| + | |
| + | <tr> |
| + | <td><math>f_{15}</math></td> |
| + | <td><math>1111</math></td> |
| + | <td style="border-right:1px solid black"><math>\texttt{((~))}</math></td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:white; color:black">0</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td> |
| + | <td style="background:black; color:white">1</td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| + | |
| + | <table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%"> |
| + | |
| + | <caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Mnemonic}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Category}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Classical~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Alternate~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Symmetric~Form}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Operator}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{E} |
| + | \\ |
| + | \mathrm{Exclusive} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Universal} |
| + | \\ |
| + | \mathrm{Negative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix} |
| + | \mathrm{A} |
| + | \\ |
| + | \mathrm{Absolute} |
| + | \end{matrix}</math></td> |
| + | <td style="border-bottom:1px solid black"> |
| + | <math>\begin{matrix} |
| + | \mathrm{Universal} |
| + | \\ |
| + | \mathrm{Affirmative} |
| + | \end{matrix}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td> |
| + | <td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td> |
| + | <td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td> </td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td><math>\ell_{00}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"> </td> |
| + | <td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td> |
| + | <td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{O} |
| + | \\ |
| + | \mathrm{Obtrusive} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Particular} |
| + | \\ |
| + | \mathrm{Negative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td> |
| + | <td><math>\ell_{10}</math></td></tr> |
| + | |
| + | <tr> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{I} |
| + | \\ |
| + | \mathrm{Indefinite} |
| + | \end{matrix}</math></td> |
| + | <td><math>\begin{matrix} |
| + | \mathrm{Particular} |
| + | \\ |
| + | \mathrm{Affirmative} |
| + | \end{matrix}</math></td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td> </td> |
| + | <td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td> |
| + | <td><math>\ell_{11}</math></td></tr> |
| + | |
| + | </table> |
| + | |
| + | <br> |
| | | |
| ==Inquiry Driven Systems== | | ==Inquiry Driven Systems== |
Line 7,789: |
Line 10,595: |
| |- | | |- |
| | | | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%" |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="33%" | <math>\text{Object}\!</math> | | | width="33%" | <math>\text{Object}\!</math> |
Line 7,819: |
Line 10,625: |
| |- | | |- |
| | | | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:100%" |
| |- style="background:#f0f0ff" | | |- style="background:#f0f0ff" |
| | width="33%" | <math>\text{Object}\!</math> | | | width="33%" | <math>\text{Object}\!</math> |