PlanetPhysics/Differential Propositional Calculus Appendix 1

Note. The following Tables are best viewed in the Page Image mode.

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{tabular}{|c|c|c|c|c|c|c|} \multicolumn{7}{'''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}

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{tabular}{|c|c|c|c|c|c|c|} \multicolumn{7}{'''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}

Table A3. $$\operatorname{E}f$$ Expanded Over Differential Features $$\{ \operatorname{d}x, \operatorname{d}y \}$$
\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{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||}{Fixed Point Total:} & 4 & 4 & 4 & 16 \\ \hline \end{tabular}

Table A4. $$\operatorname{D}f$$ Expanded Over Differential Features $$\{ \operatorname{d}x, \operatorname{d}y \}$$
\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{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}

Table A5. $$\operatorname{E}f$$ Expanded Over Ordinary Features $$\{ x, y \}$$
\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{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}

Table A6. $$\operatorname{D}f$$ Expanded Over Ordinary Features $$\{ x, y \}$$
\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{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}