Tables

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Boolean Functions and Propositional Calculus

For comparison, the Wiki-TeX versions of the tables are here.

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}

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