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Directory:Jon Awbrey/Papers/Cactus Language (view source)

Revision as of 01:50, 30 May 2009

4,629 bytes added ,  01:50, 30 May 2009
→‎Note 6: + Standard Tables A1 & A2
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To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  For future reference, I will set here a few Tables that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> and on each of these functions, allowing us to view the results in several different ways.
 
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  For future reference, I will set here a few Tables that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> and on each of these functions, allowing us to view the results in several different ways.
   −
By way of initial orientation, Table&nbsp;0 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
+
Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
−
|+ <math>\text{Table 0.}~~\text{Propositional Forms on Two Variables}</math>
+
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
 
| style="width:15%" |
 
| style="width:15%" |
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| &nbsp;
 
| &nbsp;
 
| &nbsp;
 
| &nbsp;
−
|-
   
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
 
| &nbsp;
 
| &nbsp;
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| <math>1~1~1~1\!</math>
 
| <math>1~1~1~1\!</math>
 
| <math>((~))\!</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 A2.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| style="width:25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| style="width:15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0</math>
 +
| <math>(~)</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
(x)(y)
 +
\\[4pt]
 +
(x)~y~
 +
\\[4pt]
 +
~x~(y)
 +
\\[4pt]
 +
~x~~y~
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
(x)
 +
\\[4pt]
 +
~x~
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ x
 +
\\[4pt]
 +
x
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\lnot x
 +
\\[4pt]
 +
x
 +
\end{matrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
~(x,~y)~
 +
\\[4pt]
 +
((x,~y))
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
x ~\text{not equal to}~ y
 +
\\[4pt]
 +
x ~\text{equal to}~ y
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
x \ne y
 +
\\[4pt]
 +
x = y
 +
\end{matrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
(y)
 +
\\[4pt]
 +
~y~
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ y
 +
\\[4pt]
 +
y
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\lnot y
 +
\\[4pt]
 +
y
 +
\end{matrix}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<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>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
~(x~~y)~
 +
\\[4pt]
 +
~(x~(y))
 +
\\[4pt]
 +
((x)~y)~
 +
\\[4pt]
 +
((x)(y))
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ x ~\text{and}~ y
 +
\\[4pt]
 +
\text{not}~ x ~\text{without}~ y
 +
\\[4pt]
 +
\text{not}~ y ~\text{without}~ x
 +
\\[4pt]
 +
x ~\text{or}~ y
 +
\end{matrix}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<math>\begin{matrix}
 +
\lnot x \lor \lnot y
 +
\\[4pt]
 +
x \Rightarrow y
 +
\\[4pt]
 +
x \Leftarrow y
 +
\\[4pt]
 +
x \lor y
 +
\end{matrix}</math>
 +
|}
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1</math>
 +
| <math>((~))</math>
 
| <math>\text{true}\!</math>
 
| <math>\text{true}\!</math>
 
| <math>1\!</math>
 
| <math>1\!</math>
Jon Awbrey
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