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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
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, 03:50, 2 March 2009→Note 8: sub full table for link
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The underlined parenthetical expressions on the right are the cactus forms for the boolean functions that correspond to inclusive disjunction and logical equivalence, respectively. By way of a reminder, consult Table 1 on the page at this location:
The underlined parenthetical expressions on the right are the cactus forms for the boolean functions that correspond to inclusive disjunction and logical equivalence, respectively. Table 1 summarizes the basic elements of the cactus notation for propositional logic.
:* [http://stderr.org/pipermail/inquiry/2003-May/000478.html DLOG D1]
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
|+ <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
|- style="background:whitesmoke"
| <math>\text{Expression}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
|-
| <math>~</math>
| <math>\operatorname{True}</math>
| <math>1\!</math>
|-
| <math>(~)</math>
| <math>\operatorname{False}</math>
| <math>0\!</math>
|-
| <math>x\!</math>
| <math>x\!</math>
| <math>x\!</math>
|-
| <math>(x)\!</math>
| <math>\operatorname{Not}\ x</math>
|
<math>\begin{matrix}
x' \\
\tilde{x} \\
\lnot x \\
\end{matrix}</math>
|-
| <math>x\ y\ z</math>
| <math>x\ \operatorname{and}\ y\ \operatorname{and}\ z</math>
| <math>x \land y \land z</math>
|-
| <math>((x)(y)(z))\!</math>
| <math>x\ \operatorname{or}\ y\ \operatorname{or}\ z</math>
| <math>x \lor y \lor z</math>
|-
| <math>(x\ (y))\!</math>
|
<math>\begin{matrix}
x\ \operatorname{implies}\ y \\
\operatorname{If}\ x\ \operatorname{then}\ y \\
\end{matrix}</math>
| <math>x \Rightarrow y\!</math>
|-
| <math>(x, y)\!</math>
|
<math>\begin{matrix}
x\ \operatorname{not~equal~to}\ y \\
x\ \operatorname{exclusive~or}\ y \\
\end{matrix}</math>
|
<math>\begin{matrix}
x \neq y \\
x + y \\
\end{matrix}</math>
|-
| <math>((x, y))\!</math>
|
<math>\begin{matrix}
x\ \operatorname{is~equal~to}\ y \\
x\ \operatorname{if~and~only~if}\ y \\
\end{matrix}</math>
|
<math>\begin{matrix}
x = y \\
x \Leftrightarrow y \\
\end{matrix}</math>
|-
| <math>(x, y, z)\!</math>
|
<math>\begin{matrix}
\operatorname{Just~one~of} \\
x, y, z \\
\operatorname{is~false}. \\
\end{matrix}</math>
|
<math>\begin{matrix}
x'y~z~ & \lor \\
x~y'z~ & \lor \\
x~y~z' & \\
\end{matrix}</math>
|-
| <math>((x),(y),(z))\!</math>
|
<math>\begin{matrix}
\operatorname{Just~one~of} \\
x, y, z \\
\operatorname{is~true}. \\
& \\
\operatorname{Partition~all} \\
\operatorname{into}\ x, y, z. \\
\end{matrix}</math>
|
<math>\begin{matrix}
x~y'z' & \lor \\
x'y~z' & \lor \\
x'y'z~ & \\
\end{matrix}</math>
|-
|
<math>\begin{matrix}
((x, y), z) \\
& \\
(x, (y, z)) \\
\end{matrix}</math>
|
<math>\begin{matrix}
\operatorname{Oddly~many~of} \\
x, y, z \\
\operatorname{are~true}. \\
\end{matrix}</math>
|
<p><math>x + y + z\!</math></p>
<br>
<p><math>\begin{matrix}
x~y~z~ & \lor \\
x~y'z' & \lor \\
x'y~z' & \lor \\
x'y'z~ & \\
\end{matrix}</math></p>
|-
| <math>(w, (x),(y),(z))\!</math>
|
<math>\begin{matrix}
\operatorname{Partition}\ w \\
\operatorname{into}\ x, y, z. \\
& \\
\operatorname{Genus}\ w\ \operatorname{comprises} \\
\operatorname{species}\ x, y, z. \\
\end{matrix}</math>
|
<math>\begin{matrix}
w'x'y'z' & \lor \\
w~x~y'z' & \lor \\
w~x'y~z' & \lor \\
w~x'y'z~ & \\
\end{matrix}</math>
|}
<br>
<pre>
<pre>