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| <math>\underline{1}</math>
 
| <math>\underline{1}</math>
 
| <math>\underline{((} ~ \underline{))}</math>
 
| <math>\underline{((} ~ \underline{))}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ '''Table 17.  Boolean Functions on Two Variables'''
 +
|- style="background:ghostwhite"
 +
| width="14%" | <math>F\!</math>
 +
| width="14%" | <math>F\!</math>
 +
| colspan="4" | <math>F(x, y)\!</math>
 +
| width="24%" | <math>F\!</math>
 +
|- style="background:ghostwhite"
 +
| width="14%" | &nbsp;
 +
| width="14%" | &nbsp;
 +
| width="12%" | <math>F(\underline{1}, \underline{1})</math>
 +
| width="12%" | <math>F(\underline{1}, \underline{0})</math>
 +
| width="12%" | <math>F(\underline{0}, \underline{1})</math>
 +
| width="12%" | <math>F(\underline{0}, \underline{0})</math>
 +
| width="24%" | &nbsp;
 +
|-
 +
| <math>F_{0}^{(2)}\!</math>
 +
| <math>F_{0000}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>(~)</math>
 +
|-
 +
| <math>F_{1}^{(2)}\!</math>
 +
| <math>F_{0001}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>(x)(y)\!</math>
 +
|-
 +
| <math>F_{2}^{(2)}\!</math>
 +
| <math>F_{0010}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>(x) y\!</math>
 +
|-
 +
| <math>F_{3}^{(2)}\!</math>
 +
| <math>F_{0011}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>(x)\!</math>
 +
|-
 +
| <math>F_{4}^{(2)}\!</math>
 +
| <math>F_{0100}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>x (y)\!</math>
 +
|-
 +
| <math>F_{5}^{(2)}\!</math>
 +
| <math>F_{0101}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>(y)\!</math>
 +
|-
 +
| <math>F_{6}^{(2)}\!</math>
 +
| <math>F_{0110}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>(x, y)\!</math>
 +
|-
 +
| <math>F_{7}^{(2)}\!</math>
 +
| <math>F_{0111}^{(2)}\!</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>(x y)\!</math>
 +
|-
 +
| <math>F_{8}^{(2)}\!</math>
 +
| <math>F_{1000}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>x y\!</math>
 +
|-
 +
| <math>F_{9}^{(2)}\!</math>
 +
| <math>F_{1001}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>((x, y))\!</math>
 +
|-
 +
| <math>F_{10}^{(2)}\!</math>
 +
| <math>F_{1010}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>y\!</math>
 +
|-
 +
| <math>F_{11}^{(2)}\!</math>
 +
| <math>F_{1011}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>(x (y))\!</math>
 +
|-
 +
| <math>F_{12}^{(2)}\!</math>
 +
| <math>F_{1100}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{0}</math>
 +
| <math>x\!</math>
 +
|-
 +
| <math>F_{13}^{(2)}\!</math>
 +
| <math>F_{1101}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>\underline{1}</math>
 +
| <math>((x)y)\!</math>
 +
|-
 +
| <math>F_{14}^{(2)}\!</math>
 +
| <math>F_{1110}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{0}</math>
 +
| <math>((x)(y))\!</math>
 +
|-
 +
| <math>F_{15}^{(2)}\!</math>
 +
| <math>F_{1111}^{(2)}\!</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>\underline{1}</math>
 +
| <math>((~))</math>
 
|}
 
|}
  

Revision as of 18:30, 22 January 2009

Grammar Stuff


Table 13. Algorithmic Translation Rules
\(\text{Sentence in PARCE}\!\) \(\xrightarrow{\operatorname{Parse}}\) \(\text{Graph in PARC}\!\)
\(\operatorname{Conc}^0\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Node}^0\)
\(\operatorname{Conc}_{j=1}^k s_j\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)\)
\(\operatorname{Surc}^0\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Lobe}^0\)
\(\operatorname{Surc}_{j=1}^k s_j\) \(\xrightarrow{\operatorname{Parse}}\) \(\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)\)


Table 14.1 Semantic Translation : Functional Form
\(\operatorname{Sentence}\) \(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}\) \(\operatorname{Graph}\) \(\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}\) \(\operatorname{Proposition}\)
\(s_j\!\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(C_j\!\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(q_j\!\)
\(\operatorname{Conc}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Node}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\underline{1}\)
\(\operatorname{Conc}^k_j s_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Node}^k_j C_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Conj}^k_j q_j\)
\(\operatorname{Surc}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Lobe}^0\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\underline{0}\)
\(\operatorname{Surc}^k_j s_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Lobe}^k_j C_j\) \(\xrightarrow{\operatorname{~~~~~~~~~~}}\) \(\operatorname{Surj}^k_j q_j\)


Table 14.2 Semantic Translation : Equational Form
\(\downharpoonleft \operatorname{Sentence} \downharpoonright\) \(\stackrel{\operatorname{Parse}}{=}\) \(\downharpoonleft \operatorname{Graph} \downharpoonright\) \(\stackrel{\operatorname{Denotation}}{=}\) \(\operatorname{Proposition}\)
\(\downharpoonleft s_j \downharpoonright\) \(=\!\) \(\downharpoonleft C_j \downharpoonright\) \(=\!\) \(q_j\!\)
\(\downharpoonleft \operatorname{Conc}^0 \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Node}^0 \downharpoonright\) \(=\!\) \(\underline{1}\)
\(\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright\) \(=\!\) \(\operatorname{Conj}^k_j q_j\)
\(\downharpoonleft \operatorname{Surc}^0 \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Lobe}^0 \downharpoonright\) \(=\!\) \(\underline{0}\)
\(\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright\) \(=\!\) \(\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright\) \(=\!\) \(\operatorname{Surj}^k_j q_j\)


Table Stuff


Table 15. Boolean Functions on Zero Variables
\(F\!\) \(F\!\) \(F()\!\) \(F\!\)
\(\underline{0}\) \(F_0^{(0)}\!\) \(\underline{0}\) \(\underline{(} ~ \underline{)}\)
\(\underline{1}\) \(F_1^{(0)}\!\) \(\underline{1}\) \(\underline{((} ~ \underline{))}\)


Table 16. Boolean Functions on One Variable
\(F\!\) \(F\!\) \(F(x)\!\) \(F\!\)
    \(F(\underline{1})\) \(F(\underline{0})\)  
\(F_0^{(1)}\!\) \(F_{00}^{(1)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{(} ~ \underline{)}\)
\(F_1^{(1)}\!\) \(F_{01}^{(1)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{(} x \underline{)}\)
\(F_2^{(1)}\!\) \(F_{10}^{(1)}\!\) \(\underline{1}\) \(\underline{0}\) \(x\!\)
\(F_3^{(1)}\!\) \(F_{11}^{(1)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{((} ~ \underline{))}\)


Table 17. Boolean Functions on Two Variables
\(F\!\) \(F\!\) \(F(x, y)\!\) \(F\!\)
    \(F(\underline{1}, \underline{1})\) \(F(\underline{1}, \underline{0})\) \(F(\underline{0}, \underline{1})\) \(F(\underline{0}, \underline{0})\)  
\(F_{0}^{(2)}\!\) \(F_{0000}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \((~)\)
\(F_{1}^{(2)}\!\) \(F_{0001}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \((x)(y)\!\)
\(F_{2}^{(2)}\!\) \(F_{0010}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \((x) y\!\)
\(F_{3}^{(2)}\!\) \(F_{0011}^{(2)}\!\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \((x)\!\)
\(F_{4}^{(2)}\!\) \(F_{0100}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(x (y)\!\)
\(F_{5}^{(2)}\!\) \(F_{0101}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \((y)\!\)
\(F_{6}^{(2)}\!\) \(F_{0110}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \((x, y)\!\)
\(F_{7}^{(2)}\!\) \(F_{0111}^{(2)}\!\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \((x y)\!\)
\(F_{8}^{(2)}\!\) \(F_{1000}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(\underline{0}\) \(x y\!\)
\(F_{9}^{(2)}\!\) \(F_{1001}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(\underline{1}\) \(((x, y))\!\)
\(F_{10}^{(2)}\!\) \(F_{1010}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(\underline{0}\) \(y\!\)
\(F_{11}^{(2)}\!\) \(F_{1011}^{(2)}\!\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(\underline{1}\) \((x (y))\!\)
\(F_{12}^{(2)}\!\) \(F_{1100}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(\underline{0}\) \(x\!\)
\(F_{13}^{(2)}\!\) \(F_{1101}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(\underline{1}\) \(((x)y)\!\)
\(F_{14}^{(2)}\!\) \(F_{1110}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(\underline{0}\) \(((x)(y))\!\)
\(F_{15}^{(2)}\!\) \(F_{1111}^{(2)}\!\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(\underline{1}\) \(((~))\)


Table 17.  Boolean Functions on Two Variables
o----------o----------o-------------------------------------------o----------o
| Function | Function |                  F(x, y)                  | Function |
o----------o----------o----------o----------o----------o----------o----------o
|          |          | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% |          |
o----------o----------o----------o----------o----------o----------o----------o
|          |          |          |          |          |          |          |
| F^2_00   | F^2_0000 |   %0%    |   %0%    |   %0%    |   %0%    |    ()    |
|          |          |          |          |          |          |          |
| F^2_01   | F^2_0001 |   %0%    |   %0%    |   %0%    |   %1%    |  (x)(y)  |
|          |          |          |          |          |          |          |
| F^2_02   | F^2_0010 |   %0%    |   %0%    |   %1%    |   %0%    |  (x) y   |
|          |          |          |          |          |          |          |
| F^2_03   | F^2_0011 |   %0%    |   %0%    |   %1%    |   %1%    |  (x)     |
|          |          |          |          |          |          |          |
| F^2_04   | F^2_0100 |   %0%    |   %1%    |   %0%    |   %0%    |   x (y)  |
|          |          |          |          |          |          |          |
| F^2_05   | F^2_0101 |   %0%    |   %1%    |   %0%    |   %1%    |     (y)  |
|          |          |          |          |          |          |          |
| F^2_06   | F^2_0110 |   %0%    |   %1%    |   %1%    |   %0%    |  (x, y)  |
|          |          |          |          |          |          |          |
| F^2_07   | F^2_0111 |   %0%    |   %1%    |   %1%    |   %1%    |  (x  y)  |
|          |          |          |          |          |          |          |
| F^2_08   | F^2_1000 |   %1%    |   %0%    |   %0%    |   %0%    |   x  y   |
|          |          |          |          |          |          |          |
| F^2_09   | F^2_1001 |   %1%    |   %0%    |   %0%    |   %1%    | ((x, y)) |
|          |          |          |          |          |          |          |
| F^2_10   | F^2_1010 |   %1%    |   %0%    |   %1%    |   %0%    |      y   |
|          |          |          |          |          |          |          |
| F^2_11   | F^2_1011 |   %1%    |   %0%    |   %1%    |   %1%    |  (x (y)) |
|          |          |          |          |          |          |          |
| F^2_12   | F^2_1100 |   %1%    |   %1%    |   %0%    |   %0%    |   x      |
|          |          |          |          |          |          |          |
| F^2_13   | F^2_1101 |   %1%    |   %1%    |   %0%    |   %1%    | ((x) y)  |
|          |          |          |          |          |          |          |
| F^2_14   | F^2_1110 |   %1%    |   %1%    |   %1%    |   %0%    | ((x)(y)) |
|          |          |          |          |          |          |          |
| F^2_15   | F^2_1111 |   %1%    |   %1%    |   %1%    |   %1%    |   (())   |
|          |          |          |          |          |          |          |
o----------o----------o----------o----------o----------o----------o----------o


Table 7. Propositional Forms on Two Variables

\(\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}\)

\(~\!\) \(x\colon\!\) \(1~1~0~0\!\) \(~\!\) \(~\!\) \(~\!\)
\(~\!\) \(y\colon\!\) \(1~0~1~0\!\) \(~\!\) \(~\!\) \(~\!\)
\(f_{0}\!\) \(f_{0000}\!\) \(0~0~0~0\!\) \((~)\!\) \(\mbox{false}\!\) \(0\!\)
\(f_{1}\!\) \(f_{0001}\!\) \(0~0~0~1\!\) \((x)(y)\!\) \(\mbox{neither}\ x\ \mbox{nor}\ y\!\) \(\lnot x \land \lnot y\!\)
\(f_{2}\!\) \(f_{0010}\!\) \(0~0~1~0\!\) \((x)\ y\!\) \(y\ \mbox{without}\ x\!\) \(\lnot x \land y\!\)
\(f_{3}\!\) \(f_{0011}\!\) \(0~0~1~1\!\) \((x)\!\) \(\mbox{not}\ x\!\) \(\lnot x\!\)
\(f_{4}\!\) \(f_{0100}\!\) \(0~1~0~0\!\) \(x\ (y)\!\) \(x\ \mbox{without}\ y\!\) \(x \land \lnot y\!\)
\(f_{5}\!\) \(f_{0101}\!\) \(0~1~0~1\!\) \((y)\!\) \(\mbox{not}\ y\!\) \(\lnot y\!\)
\(f_{6}\!\) \(f_{0110}\!\) \(0~1~1~0\!\) \((x, y)\!\) \(x\ \mbox{not equal to}\ y\!\) \(x \ne y\!\)
\(f_{7}\!\) \(f_{0111}\!\) \(0~1~1~1\!\) \((x\ y)\!\) \(\mbox{not both}\ x\ \mbox{and}\ y\!\) \(\lnot x \lor \lnot y\!\)
\(f_{8}\!\) \(f_{1000}\!\) \(1~0~0~0\!\) \(x\ y\!\) \(x\ \mbox{and}\ y\!\) \(x \land y\!\)
\(f_{9}\!\) \(f_{1001}\!\) \(1~0~0~1\!\) \(((x, y))\!\) \(x\ \mbox{equal to}\ y\!\) \(x = y\!\)
\(f_{10}\!\) \(f_{1010}\!\) \(1~0~1~0\!\) \(y\!\) \(y\!\) \(y\!\)
\(f_{11}\!\) \(f_{1011}\!\) \(1~0~1~1\!\) \((x\ (y))\!\) \(\mbox{not}\ x\ \mbox{without}\ y\!\) \(x \Rightarrow y\!\)
\(f_{12}\!\) \(f_{1100}\!\) \(1~1~0~0\!\) \(x\!\) \(x\!\) \(x\!\)
\(f_{13}\!\) \(f_{1101}\!\) \(1~1~0~1\!\) \(((x)\ y)\!\) \(\mbox{not}\ y\ \mbox{without}\ x\!\) \(x \Leftarrow y\!\)
\(f_{14}\!\) \(f_{1110}\!\) \(1~1~1~0\!\) \(((x)(y))\!\) \(x\ \mbox{or}\ y\!\) \(x \lor y\!\)
\(f_{15}\!\) \(f_{1111}\!\) \(1~1~1~1\!\) \(((~))\!\) \(\mbox{true}\!\) \(1\!\)




fixy
u =
v =
1 1 0 0
1 0 1 0
= u
= v
fjuv
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


A
u =
v =
1 1 0 0
1 0 1 0
= u
= v
B
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


u =
v =
1 1 0 0
1 0 1 0
= u
= v
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›


u =
v =
x =
y =
1 1 0 0
1 0 1 0
1 1 1 0
1 0 0 1
= u
= v
= f‹u, v›
= g‹u, v›