differential propositional calculus : appendix 1
DifferentialPropositionalCalculusAppendices
2008-06-06 22:05:48
2008-08-03 00:05:28
Application
differential propositional calculus
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\textbf{Note.} The following Tables are best viewed in the Page Image mode.
\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
$\mathcal{L}_1$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\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
$f_{8}$ &
$f_{1000}$ &&
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}