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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 19:00, 9 September 2013

166 bytes added ,  19:00, 9 September 2013
→‎Appendix A
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<br>
 
<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
−
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
+
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math>
−
|- style="background:#f0f0ff"
+
|- style="background:ghostwhite"
−
| width="15%" |
+
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
−
<p><math>\mathcal{L}_1</math></p>
+
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
−
<p><math>\text{Decimal}</math></p>
+
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
−
| width="15%" |
+
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
−
<p><math>\mathcal{L}_2</math></p>
+
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
−
<p><math>\text{Binary}</math></p>
+
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
−
| width="15%" |
+
|- style="background:ghostwhite"
−
<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"
   
| &nbsp;
 
| &nbsp;
 
| align="right" | <math>x\colon\!</math>
 
| align="right" | <math>x\colon\!</math>
 
| <math>1~1~0~0\!</math>
 
| <math>1~1~0~0\!</math>
−
| &nbsp;
+
| &nbsp; || &nbsp; || &nbsp;
−
| &nbsp;
+
|- style="background:ghostwhite"
−
| &nbsp;
  −
|- style="background:#f0f0ff"
   
| &nbsp;
 
| &nbsp;
 
| align="right" | <math>y\colon\!</math>
 
| align="right" | <math>y\colon\!</math>
 
| <math>1~0~1~0\!</math>
 
| <math>1~0~1~0\!</math>
−
| &nbsp;
+
| &nbsp; || &nbsp; || &nbsp;
−
| &nbsp;
  −
| &nbsp;
   
|-
 
|-
 
| <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>\texttt{(~)}\!</math>
 
| <math>\text{false}\!</math>
 
| <math>\text{false}\!</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
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|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_1
+
f_{1}\\f_{2}\\f_{4}\\f_{8}
−
\\[4pt]
+
\end{matrix}</math>
−
f_2
  −
\\[4pt]
  −
f_4
  −
\\[4pt]
  −
f_8
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0001}
+
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{0010}
  −
\\[4pt]
  −
f_{0100}
  −
\\[4pt]
  −
f_{1000}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~0~0~1
+
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
−
\\[4pt]
+
\end{matrix}</math>
−
0~0~1~0
  −
\\[4pt]
  −
0~1~0~0
  −
\\[4pt]
  −
1~0~0~0
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(x)(y)
+
\texttt{(} x \texttt{)(} y \texttt{)}
−
\\[4pt]
+
\\
−
(x)~y~
+
\texttt{(} x \texttt{)~} y \texttt{~}
−
\\[4pt]
+
\\
−
~x~(y)
+
\texttt{~} x \texttt{~(} y \texttt{)}
−
\\[4pt]
+
\\
−
~x~~y~
+
\texttt{~} x \texttt{~~} y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\text{neither}~ x ~\text{nor}~ y
 
\text{neither}~ x ~\text{nor}~ y
−
\\[4pt]
+
\\
 
y ~\text{without}~ x
 
y ~\text{without}~ x
−
\\[4pt]
+
\\
 
x ~\text{without}~ y
 
x ~\text{without}~ y
−
\\[4pt]
+
\\
 
x ~\text{and}~ y
 
x ~\text{and}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\lnot x \land \lnot y
 
\lnot x \land \lnot y
−
\\[4pt]
+
\\
 
\lnot x \land y
 
\lnot x \land y
−
\\[4pt]
+
\\
 
x \land \lnot y
 
x \land \lnot y
−
\\[4pt]
+
\\
 
x \land y
 
x \land y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_3
+
f_{3}\\f_{12}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{12}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0011}
+
f_{0011}\\f_{1100}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{1100}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~0~1~1
+
0~0~1~1\\1~1~0~0
−
\\[4pt]
+
\end{matrix}</math>
−
1~1~0~0
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(x)
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
~x~
+
\texttt{~} x \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\text{not}~ x
 
\text{not}~ x
−
\\[4pt]
+
\\
 
x
 
x
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\lnot x
 
\lnot x
−
\\[4pt]
+
\\
 
x
 
x
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_6
+
f_{6}\\f_{9}
−
\\[4pt]
+
\end{matrix}</math>
−
f_9
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0110}
+
f_{0110}\\f_{1001}
−
\\[4pt]
  −
f_{1001}
   
\end{matrix}\!</math>
 
\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~1~1~0
+
0~1~1~0\\1~0~0~1
−
\\[4pt]
+
\end{matrix}</math>
−
1~0~0~1
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\\[4pt]
+
\\
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
x ~\text{not equal to}~ y
 
x ~\text{not equal to}~ y
−
\\[4pt]
+
\\
 
x ~\text{equal to}~ y
 
x ~\text{equal to}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
x \ne y
 
x \ne y
−
\\[4pt]
+
\\
 
x = y
 
x = y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_5
+
f_{5}\\f_{10}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{10}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0101}
+
f_{0101}\\f_{1010}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{1010}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~1~0~1
+
0~1~0~1\\1~0~1~0
−
\\[4pt]
+
\end{matrix}</math>
−
1~0~1~0
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(y)
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~y~
+
\texttt{~} y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\text{not}~ y
 
\text{not}~ y
−
\\[4pt]
+
\\
 
y
 
y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\lnot y
 
\lnot y
−
\\[4pt]
+
\\
 
y
 
y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_7
+
f_{7}\\f_{11}\\f_{13}\\f_{14}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{11}
  −
\\[4pt]
  −
f_{13}
  −
\\[4pt]
  −
f_{14}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0111}
+
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{1011}
  −
\\[4pt]
  −
f_{1101}
  −
\\[4pt]
  −
f_{1110}
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~1~1~1
+
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
−
\\[4pt]
+
\end{matrix}</math>
−
1~0~1~1
  −
\\[4pt]
  −
1~1~0~1
  −
\\[4pt]
  −
1~1~1~0
  −
\end{matrix}\!</math>
   
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(x~~y)~
+
\texttt{~(} x \texttt{~~} y \texttt{)~}
−
\\[4pt]
+
\\
−
~(x~(y))
+
\texttt{~(} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
((x)~y)~
+
\texttt{((} x \texttt{)~} y \texttt{)~}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\text{not both}~ x ~\text{and}~ y
 
\text{not both}~ x ~\text{and}~ y
−
\\[4pt]
+
\\
 
\text{not}~ x ~\text{without}~ y
 
\text{not}~ x ~\text{without}~ y
−
\\[4pt]
+
\\
 
\text{not}~ y ~\text{without}~ x
 
\text{not}~ y ~\text{without}~ x
−
\\[4pt]
+
\\
 
x ~\text{or}~ y
 
x ~\text{or}~ y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\lnot x \lor \lnot y
 
\lnot x \lor \lnot y
−
\\[4pt]
+
\\
 
x \Rightarrow y
 
x \Rightarrow y
−
\\[4pt]
+
\\
 
x \Leftarrow y
 
x \Leftarrow y
−
\\[4pt]
+
\\
 
x \lor y
 
x \lor y
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
 
| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
 
| <math>f_{1111}\!</math>
 
| <math>f_{1111}\!</math>
 
| <math>1~1~1~1\!</math>
 
| <math>1~1~1~1\!</math>
−
| <math>((~))</math>
+
| <math>\texttt{((~))}\!</math>
 
| <math>\text{true}\!</math>
 
| <math>\text{true}\!</math>
 
| <math>1\!</math>
 
| <math>1\!</math>
Jon Awbrey
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