Tables

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Boolean Functions and Propositional Calculus

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Differential Logic and Dynamic Systems • Appendices

Table A1. Propositional Forms on Two Variables

$\begin{array}{|*{7}{c|}} \multicolumn{7}{c}{\text{Table A1. Propositional Forms on Two Variables}} \\ \hline L_1 & L_2 && L_3 & L_4 & L_5 & L_6 \\ \hline && x= & 1~1~0~0 &&& \\ && y= & 1~0~1~0 &&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & \text{false} & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (x)(y) & \text{neither}~ x ~\text{nor}~ y & \lnot x \land \lnot y \\ f_{2} & f_{0010} && 0~0~1~0 & (x)~y~ & y ~\text{without}~ x & \lnot x \land y \\ f_{3} & f_{0011} && 0~0~1~1 & (x) & \text{not}~ x & \lnot x \\ f_{4} & f_{0100} && 0~1~0~0 & ~x~(y) & x ~\text{without}~ y & x \land \lnot y \\ f_{5} & f_{0101} && 0~1~0~1 & (y) & \text{not}~ y & \lnot y \\ f_{6} & f_{0110} && 0~1~1~0 & (x,~y) & x ~\text{not equal to}~ y & x \ne y \\ f_{7} & f_{0111} && 0~1~1~1 & (x~~y) & \text{not both}~ x ~\text{and}~ y & \lnot x \lor \lnot y \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~x~~y~ & x ~\text{and}~ y & x \land y \\ f_{9} & f_{1001} && 1~0~0~1 &((x,~y))& x ~\text{equal to}~ y & x = y \\ f_{10}& f_{1010} && 1~0~1~0 & y & y & y \\ f_{11}& f_{1011} && 1~0~1~1 &(~x~(y))& \text{not}~ x ~\text{without}~ y & x \Rightarrow y \\ f_{12}& f_{1100} && 1~1~0~0 & x & x & x \\ f_{13}& f_{1101} && 1~1~0~1 &((x)~y~)& \text{not}~ y ~\text{without}~ x & x \Leftarrow y \\ f_{14}& f_{1110} && 1~1~1~0 &((x)(y))& x ~\text{or}~ y & x \lor y \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & \text{true} & 1 \\ \hline \end{array}$

Table A2. Propositional Forms on Two Variables

$\begin{array}{|*{7}{c|}} \multicolumn{7}{c}{\text{Table A2. Propositional Forms on Two Variables}} \\ \hline L_1 & L_2 && L_3 & L_4 & L_5 & L_6 \\ \hline && x = & 1~1~0~0 &&& \\ && y = & 1~0~1~ 0&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & \text{false} & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (x)(y) & \text{neither}~ x ~\text{nor}~ y & \lnot x \land \lnot y \\ f_{2} & f_{0010} && 0~0~1~0 & (x)~y~ & y ~\text{without}~ x & \lnot x \land y \\ f_{4} & f_{0100} && 0~1~0~0 & ~x~(y) & x ~\text{without}~ y & x \land \lnot y \\ f_{8} & f_{1000} && 1~0~0~0 & ~x~~y~ & x ~\text{and}~ y & x \land y \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (x) & \text{not}~ x & \lnot x \\ f_{12}& f_{1100} && 1~1~0~0 & x & x & x \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (x,~y) & x ~\text{not equal to}~ y & x \ne y \\ f_{9} & f_{1001} && 1~0~0~1 &((x,~y))& x ~\text{equal to}~ y & x = y \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (y) & \text{not}~ y & \lnot y \\ f_{10}& f_{1010} && 1~0~1~0 & y & y & y \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (x~~y) & \text{not both}~ x ~\text{and}~ y & \lnot x \lor \lnot y \\ f_{11}& f_{1011} && 1~0~1~1 &(~x~(y))& \text{not}~ x ~\text{without}~ y & x \Rightarrow y \\ f_{13}& f_{1101} && 1~1~0~1 &((x)~y~)& \text{not}~ y ~\text{without}~ x & x \Leftarrow y \\ f_{14}& f_{1110} && 1~1~1~0 &((x)(y))& x ~\text{or}~ y & x \lor y \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & \text{true} & 1 \\ \hline \end{array}$

Table A3. Ef Expanded Over Differential Features

$\begin{array}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\text{Table A3.}~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{\mathrm{d}x, \mathrm{d}y\}} \\ \hline & f & \mathrm{T}_{11}f & \mathrm{T}_{10}f & \mathrm{T}_{01}f & \mathrm{T}_{00}f \\ && \mathrm{E}f|_{ \mathrm{d}x \; \mathrm{d}y } & \mathrm{E}f|_{ \mathrm{d}x \;(\mathrm{d}y)} & \mathrm{E}f|_{(\mathrm{d}x)\; \mathrm{d}y } & \mathrm{E}f|_{(\mathrm{d}x) (\mathrm{d}y)} \\ \hline\hline f_{0} & 0 & 0 & 0 & 0 & 0 \\ \hline f_{1} & (x)(y) & ~x~~y~ & ~x~(y) & (x)~y~ & (x)(y) \\ f_{2} & (x)~y~ & ~x~(y) & ~x~~y~ & (x)(y) & (x)~y~ \\ f_{4} & ~x~(y) & (x)~y~ & (x)(y) & ~x~~y~ & ~x~(y) \\ f_{8} & ~x~~y~ & (x)(y) & (x)~y~ & ~x~(y) & ~x~~y~ \\ \hline f_{3} & (x) & x & x & (x) & (x) \\ f_{12}& x & (x) & (x) & x & x \\ \hline f_{6} & (x,y) & (x,y) & ((x,y)) & ((x,y)) & (x,y) \\ f_{9} & ((x,y)) & ((x,y)) & (x,y) & (x,y) & ((x,y)) \\ \hline f_{5} & (y) & y & (y) & y & (y) \\ f_{10}& y & (y) & y & (y) & y \\ \hline f_{7} & (~x~~y~) & ((x)(y)) & ((x)~y~) & (~x~(y)) & (~x~~y~) \\ f_{11}& (~x~(y)) & ((x)~y~) & ((x)(y)) & (~x~~y~) & (~x~(y)) \\ f_{13}& ((x)~y~) & (~x~(y)) & (~x~~y~) & ((x)(y)) & ((x)~y~) \\ f_{14}& ((x)(y)) & (~x~~y~) & (~x~(y)) & ((x)~y~) & ((x)(y)) \\ \hline f_{15}& 1 & 1 & 1 & 1 & 1 \\ \hline\hline \multicolumn{2}{|c||}{\text{Fixed Point Total}} & 4 & 4 & 4 & 16 \\ \hline \end{array}$

Table A4. Df Expanded Over Differential Features

$\begin{array}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\text{Table A4.}~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{\mathrm{d}x, \mathrm{d}y\}} \\ \hline \text{~~~~~~} & \text{~~~~~~} f \text{~~~~~~} & \text{~} \mathrm{D}f|_{ \mathrm{d}x \; \mathrm{d}y } \text{~} & \text{~} \mathrm{D}f|_{ \mathrm{d}x \;(\mathrm{d}y)} \text{~} & \text{~} \mathrm{D}f|_{(\mathrm{d}x)\; \mathrm{d}y } \text{~} & \text{~} \mathrm{D}f|_{(\mathrm{d}x) (\mathrm{d}y)} \text{~} \\ \hline\hline f_{0} & 0 & 0 & 0 & 0 & 0 \\ \hline f_{1} & (x)(y) & ((x,y)) & (y) & (x) & 0 \\ f_{2} & (x)~y~ & (x,y) & y & (x) & 0 \\ f_{4} & ~x~(y) & (x,y) & (y) & x & 0 \\ f_{8} & ~x~~y~ & ((x,y)) & y & x & 0 \\ \hline f_{3} & (x) & 1 & 1 & 0 & 0 \\ f_{12}& x & 1 & 1 & 0 & 0 \\ \hline f_{6} & (x,y) & 0 & 1 & 1 & 0 \\ f_{9} & ((x,y)) & 0 & 1 & 1 & 0 \\ \hline f_{5} & (y) & 1 & 0 & 1 & 0 \\ f_{10}& y & 1 & 0 & 1 & 0 \\ \hline f_{7} & (~x~~y~) & ((x,y)) & y & x & 0 \\ f_{11}& (~x~(y)) & (x,y) & (y) & x & 0 \\ f_{13}& ((x)~y~) & (x,y) & y & (x) & 0 \\ f_{14}& ((x)(y)) & ((x,y)) & (y) & (x) & 0 \\ \hline f_{15}& 1 & 0 & 0 & 0 & 0 \\ \hline \end{array}$

Table A5. Ef Expanded Over Ordinary Features

$\begin{array}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\text{Table A5.}~ \text{E}f ~\text{Expanded Over Ordinary Features}~ \{x, y\}} \\ \hline & f & \text{E}f|_{ x \; y } & \text{E}f|_{ x \;(y)} & \text{E}f|_{(x)\; y } & \text{E}f|_{(x) (y)} \\ \hline\hline f_{0} & 0 & 0 & 0 & 0 & 0 \\ \hline f_{1} & (x)(y) & ~\text{d}x~~\text{d}y~ & ~\text{d}x~(\text{d}y) & (\text{d}x)~\text{d}y~ & (\text{d}x)(\text{d}y) \\ f_{2} & (x)~y~ & ~\text{d}x~(\text{d}y) & ~\text{d}x~~\text{d}y~ & (\text{d}x)(\text{d}y) & (\text{d}x)~\text{d}y~ \\ f_{4} & ~x~(y) & (\text{d}x)~\text{d}y~ & (\text{d}x)(\text{d}y) & ~\text{d}x~~\text{d}y~ & ~\text{d}x~(\text{d}y) \\ f_{8} & ~x~~y~ & (\text{d}x)(\text{d}y) & (\text{d}x)~\text{d}y~ & ~\text{d}x~(\text{d}y) & ~\text{d}x~~\text{d}y~ \\ \hline f_{3} & (x) & \text{d}x & \text{d}x & (\text{d}x) & (\text{d}x) \\ f_{12} & x & (\text{d}x) & (\text{d}x) & \text{d}x & \text{d}x \\ \hline f_{6} & (x,y) & (\text{d}x, \text{d}y) & ((\text{d}x, \text{d}y)) & ((\text{d}x, \text{d}y)) & (\text{d}x, \text{d}y) \\ f_{9} & ((x,y)) & ((\text{d}x, \text{d}y)) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) & ((\text{d}x, \text{d}y)) \\ \hline f_{5} & (y) & \text{d}y & (\text{d}y) & \text{d}y & (\text{d}y) \\ f_{10} & y & (\text{d}y) & \text{d}y & (\text{d}y) & \text{d}y \\ \hline f_{7} & (~x~~y~) & ((\text{d}x)(\text{d}y)) & ((\text{d}x)~\text{d}y~) & (~\text{d}x~(\text{d}y)) & (~\text{d}x~~\text{d}y~) \\ f_{11} & (~x~(y)) & ((\text{d}x)~\text{d}y~) & ((\text{d}x)(\text{d}y)) & (~\text{d}x~~\text{d}y~) & (~\text{d}x~(\text{d}y)) \\ f_{13} & ((x)~y~) & (~\text{d}x~(\text{d}y)) & (~\text{d}x~~\text{d}y~) & ((\text{d}x)(\text{d}y)) & ((\text{d}x)~\text{d}y~) \\ f_{14} & ((x)(y)) & (~\text{d}x~~\text{d}y~) & (~\text{d}x~(\text{d}y)) & ((\text{d}x)~\text{d}y~) & ((\text{d}x)(\text{d}y)) \\ \hline f_{15} & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array}$

Table A6. Df Expanded Over Ordinary Features

$\begin{array}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\text{Table A6.}~ \text{D}f ~\text{Expanded Over Ordinary Features}~ \{x, y\}} \\ \hline & f & \text{D}f|_{ x \; y } & \text{D}f|_{ x \;(y)} & \text{D}f|_{(x)\; y } & \text{D}f|_{(x) (y)} \\ \hline\hline f_{0} & 0 & 0 & 0 & 0 & 0 \\ \hline f_{1} & (x)(y) & ~~\text{d}x~~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ & ~(\text{d}x)~\text{d}y~~ & ((\text{d}x)(\text{d}y)) \\ f_{2} & (x)~y~ & ~~\text{d}x~(\text{d}y)~ & ~~\text{d}x~~\text{d}y~~ & ((\text{d}x)(\text{d}y)) & ~(\text{d}x)~\text{d}y~~ \\ f_{4} & ~x~(y) & ~(\text{d}x)~\text{d}y~~ & ((\text{d}x)(\text{d}y)) & ~~\text{d}x~~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ \\ f_{8} & ~x~~y~ & ((\text{d}x)(\text{d}y)) & ~(\text{d}x)~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ & ~~\text{d}x~~\text{d}y~~ \\ \hline f_{3} & (x) & \text{d}x & \text{d}x & \text{d}x & \text{d}x \\ f_{12} & x & \text{d}x & \text{d}x & \text{d}x & \text{d}x \\ \hline f_{6} & (x,y) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) \\ f_{9} & ((x,y)) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) & (\text{d}x, \text{d}y) \\ \hline f_{5} & (y) & \text{d}y & \text{d}y & \text{d}y & \text{d}y \\ f_{10} & y & \text{d}y & \text{d}y & \text{d}y & \text{d}y \\ \hline f_{7} & (~x~~y~) & ((\text{d}x)(\text{d}y)) & ~(\text{d}x)~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ & ~~\text{d}x~~\text{d}y~~ \\ f_{11} & (~x~(y)) & ~(\text{d}x)~\text{d}y~~ & ((\text{d}x)(\text{d}y)) & ~~\text{d}x~~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ \\ f_{13} & ((x)~y~) & ~~\text{d}x~(\text{d}y)~ & ~~\text{d}x~~\text{d}y~~ & ((\text{d}x)(\text{d}y)) & ~(\text{d}x)~\text{d}y~~ \\ f_{14} & ((x)(y)) & ~~\text{d}x~~\text{d}y~~ & ~~\text{d}x~(\text{d}y)~ & ~(\text{d}x)~\text{d}y~~ & ((\text{d}x)(\text{d}y)) \\ \hline f_{15} & 1 & 0 & 0 & 0 & 0 \\ \hline \end{array}$

Fourier Transforms of Boolean Functions

Integer Coefficients

$\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ \boldsymbol{\chi}_\mathcal{S}(x)} \\[4pt] \hline \mathcal{S} & (1, 1) & (1, 0) & (0, 1) & (0, 0) \\ \hline\hline \varnothing & +1 & +1 & +1 & +1 \\ \{ u \} & -1 & -1 & +1 & +1 \\ \{ v \} & -1 & +1 & -1 & +1 \\ \{ u, v \} & +1 & -1 & -1 & +1 \\ \hline \end{array}$

$\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$

$\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$

Boolean Coefficients

$\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ g(x)} \\[4pt] \hline g & f_{8} & f_{4} & f_{2} & f_{1} \\ & \texttt{ } u \texttt{ } v \texttt{ } & \texttt{ } u \texttt{ (} v \texttt{)} & \texttt{(} u \texttt{) } v \texttt{ } & \texttt{(} u \texttt{)(} v \texttt{)} \\ \hline\hline f_{7} & 0 & 1 & 1 & 1 \\ f_{11} & 1 & 0 & 1 & 1 \\ f_{13} & 1 & 1 & 0 & 1 \\ f_{14} & 1 & 1 & 1 & 0 \\ \hline \end{array}$

$\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline &&u = & 1~1~0~0 &&&&& \\ &&v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$

$\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables}}\\[4pt] \hline \text{~~~~~~~~} & \text{~~~~~~~~} & & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~} & \text{~~~~~~~~~} \\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$

Dyadic Relations • Divisibility

$\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 1. Elementary Relatives for the ``Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1\!\!:\!\!1&1\!:\!2&1\!:\!3&1\!:\!4&1\!:\!5&1\!:\!6&1\!:\!7&1\!:\!8&1\!:\!9&1\!:\!10&\dots \\ 2&&2\!:\!2&&2\!:\!4&&2\!:\!6&&2\!:\!8&&2\!:\!10&\dots \\ 3&&&3\!:\!3&&&3\!:\!6&&&3\!:\!9&&\dots \\ 4&&&&4\!:\!4&&&&4\!:\!8&&&\dots \\ 5&&&&&5\!:\!5&&&&&5\!:\!10&\dots \\ 6&&&&&&6\!:\!6&&&&&\dots \\ 7&&&&&&&7\!:\!7&&&&\dots \\ 8&&&&&&&&8\!:\!8&&&\dots \\ 9&&&&&&&&&9\!:\!9&&\dots \\ 10&&&&&&&&&&10\!:\!10&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}$

$\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 2. Logical Matrix for the ``Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1&1&1&1&1&1&1&1&1&1&\dots \\ 2& &1& &1& &1& &1& &1&\dots \\ 3& & &1& & &1& & &1& &\dots \\ 4& & & &1& & & &1& & &\dots \\ 5& & & & &1& & & & &1&\dots \\ 6& & & & & &1& & & & &\dots \\ 7& & & & & & &1& & & &\dots \\ 8& & & & & & & &1& & &\dots \\ 9& & & & & & & & &1& &\dots \\ 10&& & & & & & & & &1&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}$