# Table Work

It looks like WordPress recently changed the way it handles LaTeX tables, perhaps deprecating the Tabular format in favor of ramping up the Array and Matrix formats. At any rate, I’ll have to play around with reworking my old tables on this page.

# New Versions

Fourier Transforms of Boolean Functions : 1 $\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ \boldsymbol{\chi}_\mathcal{S}(x)} \\[4pt] \hline \mathcal{S} & (1, 1) & (1, 0) & (0, 1) & (0, 0) \\ \hline\hline \varnothing & +1 & +1 & +1 & +1 \\ \{ u \} & -1 & -1 & +1 & +1 \\ \{ v \} & -1 & +1 & -1 & +1 \\ \{ u, v \} & +1 & -1 & -1 & +1 \\ \hline \end{array}$ $\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$ $\begin{array}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(\varnothing) & \hat{f}(\{u\}) & \hat{f}(\{v\}) & \hat{f}(\{u,v\}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1/4 & 1/4 & 1/4 & 1/4 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1/4 & 1/4 &-1/4 &-1/4 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1/4 &-1/4 & 1/4 &-1/4 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 1/4 &-1/4 &-1/4 & 1/4 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 1/2 & 1/2 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1/2 &-1/2 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 1/2 & 0 & 0 &-1/2 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1/2 & 0 & 0 & 1/2 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 1/2 & 0 & 1/2 & 0 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1/2 & 0 &-1/2 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 3/4 & 1/4 & 1/4 &-1/4 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 3/4 & 1/4 &-1/4 & 1/4 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 3/4 &-1/4 & 1/4 & 1/4 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 3/4 &-1/4 &-1/4 &-1/4 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 0 & 0 & 0 \\ \hline \end{array}$

Fourier Transforms of Boolean Functions : 2 $\begin{array}{|c||*{4}{c}|} \multicolumn{5}{c}{\text{Table 2.1. Values of}~ g(x)} \\[4pt] \hline g & f_{8} & f_{4} & f_{2} & f_{1} \\ & \texttt{ } u \texttt{ } v \texttt{ } & \texttt{ } u \texttt{ (} v \texttt{)} & \texttt{(} u \texttt{) } v \texttt{ } & \texttt{(} u \texttt{)(} v \texttt{)} \\ \hline\hline f_{7} & 0 & 1 & 1 & 1 \\ f_{11} & 1 & 0 & 1 & 1 \\ f_{13} & 1 & 1 & 0 & 1 \\ f_{14} & 1 & 1 & 1 & 0 \\ \hline \end{array}$ $\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.2. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline &&u = & 1~1~0~0 &&&&& \\ &&v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ \hline f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$ $\begin{array}{|*{9}{c|}} \multicolumn{9}{c}{\text{Table 2.3. Fourier Coefficients of Bivariate Boolean Functions}} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ L_1 & L_2 && L_3 & L_4 & \hat{f}(f_{7}) & \hat{f}(f_{11}) & \hat{f}(f_{13}) & \hat{f}(f_{14}) \\ ~&~&~&~&~&~&~&~&~\\ \hline && u = & 1~1~0~0 &&&&& \\ && v = & 1~0~1~0 &&&&& \\ \hline f_{0} & f_{0000} && 0~0~0~0 & (~) & 0 & 0 & 0 & 0 \\ \hline f_{1} & f_{0001} && 0~0~0~1 & (u)(v) & 1 & 1 & 1 & 0 \\ f_{2} & f_{0010} && 0~0~1~0 & (u)~v~ & 1 & 1 & 0 & 1 \\ f_{4} & f_{0100} && 0~1~0~0 & ~u~(v) & 1 & 0 & 1 & 1 \\ f_{8} & f_{1000} && 1~0~0~0 & ~u~~v~ & 0 & 1 & 1 & 1 \\ \hline f_{3} & f_{0011} && 0~0~1~1 & (u) & 0 & 0 & 1 & 1 \\ f_{12}& f_{1100} && 1~1~0~0 & u & 1 & 1 & 0 & 0 \\ \hline f_{6} & f_{0110} && 0~1~1~0 & (u,~v) & 0 & 1 & 1 & 0 \\ f_{9} & f_{1001} && 1~0~0~1 &((u,~v))& 1 & 0 & 0 & 1 \\ \hline f_{5} & f_{0101} && 0~1~0~1 & (v) & 0 & 1 & 0 & 1 \\ f_{10}& f_{1010} && 1~0~1~0 & v & 1 & 0 & 1 & 0 \\ \hline f_{7} & f_{0111} && 0~1~1~1 & (u~~v) & 1 & 0 & 0 & 0 \\ f_{11}& f_{1011} && 1~0~1~1 &(~u~(v))& 0 & 1 & 0 & 0 \\ f_{13}& f_{1101} && 1~1~0~1 &((u)~v~)& 0 & 0 & 1 & 0 \\ f_{14}& f_{1110} && 1~1~1~0 &((u)(v))& 0 & 0 & 0 & 1 \\ \hline f_{15}& f_{1111} && 1~1~1~1 & ((~)) & 1 & 1 & 1 & 1 \\ \hline \end{array}$

# Old Versions

Fourier Transforms of Boolean Functions : 1 $\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{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{tabular}$ $\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$&$$L_2$$&&$$L_3$$&$$L_4$$& $$\hat{f}(\varnothing)$$&$$\hat{f}(\{u\})$$&$$\hat{f}(\{v\})$$&$$\hat{f}(\{u,v\})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$ $\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$&$$L_2$$&&$$L_3$$&$$L_4$$& $$\hat{f}(\varnothing)$$&$$\hat{f}(\{u\})$$&$$\hat{f}(\{v\})$$&$$\hat{f}(\{u,v\})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$

Fourier Transforms of Boolean Functions : 2 $\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1. Values of $$g(x)$$} \\[4pt] \hline $$g$$ & $$f_{8}$$ & $$f_{4}$$ & $$f_{2}$$ & $$f_{1}$$ \\ & \texttt{ $$u$$ $$v$$ } & \texttt{ $$u$$ ($$v$$)} & \texttt{($$u$$) $$v$$ } & \texttt{($$u$$)($$v$$)} \\ \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{tabular}$ $\begin{tabular}{|*{9}{c|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$& $$L_2$$&& $$L_3$$& $$L_4$$& $$\hat{f}(f_{7} )$$& $$\hat{f}(f_{11})$$& $$\hat{f}(f_{13})$$& $$\hat{f}(f_{14})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$ $\begin{tabular}{|*{9}{c|}} \multicolumn{9}{c}{Table 2.3. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$& $$L_2$$&& $$L_3$$& $$L_4$$& $$\hat{f}(f_{7} )$$& $$\hat{f}(f_{11})$$& $$\hat{f}(f_{13})$$& $$\hat{f}(f_{14})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$

## Array Format $\begin{array}{c|c} t & x \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ 5 & 1 \\ 6 & 0 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ \ldots & \ldots \end{array}$ $\begin{array}{c|cc} t & x & \mathrm{d}x \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \\ 5 & 1 & 1 \\ 6 & 0 & 1 \\ 7 & 1 & 1 \\ 8 & 0 & 1 \\ 9 & 1 & 1 \\ \ldots & \ldots & \ldots \end{array}$ $\begin{array}{c|cccc} t & x & \mathrm{d}x & \mathrm{d}^2 x & \ldots \\ \hline 0 & 0 & 1 & 0 & \ldots \\ 1 & 1 & 1 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 3 & 1 & 1 & 0 & \ldots \\ 4 & 0 & 1 & 0 & \ldots \\ 5 & 1 & 1 & 0 & \ldots \\ 6 & 0 & 1 & 0 & \ldots \\ 7 & 1 & 1 & 0 & \ldots \\ 8 & 0 & 1 & 0 & \ldots \\ 9 & 1 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}$

## Tabular Format

### Test A $\begin{tabular}{|*{7}{c|}} \multicolumn{7}{c}{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& $$(~)$$& false& $$0$$ \\ $$f_{1}$$& $$f_{0001}$$&& 0 0 0 1& $$(x)(y)$$& neither $$x$$ nor $$y$$& $$\lnot x \land \lnot y$$ \\ $$f_{2}$$& $$f_{0010}$$&& 0 0 1 0& $$(x)~y~$$& $$y$$ without $$x$$& $$\lnot x \land y$$ \\ $$f_{3}$$& $$f_{0011}$$&& 0 0 1 1& $$(x)$$& not $$x$$& $$\lnot x$$ \\ $$f_{4}$$& $$f_{0100}$$&& 0 1 0 0& $$~x~(y)$$& $$x$$ without $$y$$& $$x \land \lnot y$$ \\ $$f_{5}$$& $$f_{0101}$$&& 0 1 0 1& $$(y)$$& not $$y$$& $$\lnot y$$ \\ $$f_{6}$$& $$f_{0110}$$&& 0 1 1 0& $$(x,~y)$$& $$x$$ not equal to $$y$$& $$x \ne y$$ \\ $$f_{7}$$& $$f_{0111}$$&& 0 1 1 1& $$(x~~y)$$& not both $$x$$ and $$y$$& $$\lnot x \lor \lnot y$$ \\ \hline $$f_{8}$$& $$f_{1000}$$&& 1 0 0 0& $$~x~~y~$$& $$x$$ and $$y$$& $$x \land y$$ \\ $$f_{9}$$& $$f_{1001}$$&& 1 0 0 1& $$((x,~y))$$& $$x$$ equal to $$y$$& $$x = y$$ \\ $$f_{10}$$& $$f_{1010}$$&& 1 0 1 0& $$y$$& $$y$$& $$y$$ \\ $$f_{11}$$& $$f_{1011}$$&& 1 0 1 1& $$(~x~(y))$$& not $$x$$ without $$y$$& $$x \Rightarrow y$$ \\ $$f_{12}$$& $$f_{1100}$$&& 1 1 0 0& $$x$$& $$x$$& $$x$$ \\ $$f_{13}$$& $$f_{1101}$$&& 1 1 0 1& $$((x)~y~)$$& not $$y$$ without $$x$$& $$x \Leftarrow y$$ \\ $$f_{14}$$& $$f_{1110}$$&& 1 1 1 0& $$((x)(y))$$& $$x$$ or $$y$$& $$x \lor y$$ \\ $$f_{15}$$& $$f_{1111}$$&& 1 1 1 1& $$((~))$$& true& $$1$$ \\ \hline \end{tabular}$

### Test B $\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$&$$L_2$$&&$$L_3$$&$$L_4$$& $$\hat{f}(\varnothing)$$&$$\hat{f}(\{u\})$$&$$\hat{f}(\{v\})$$&$$\hat{f}(\{u,v\})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$

### Working Example $\begin{tabular}{|*{5}{c|}*{4}{r|}} \multicolumn{9}{c}{Table 2.2. Fourier Coefficients of Boolean Functions on Two Variables} \\[4pt] \hline ~&~&~&~&~&~&~&~&~\\ $$L_1$$&$$L_2$$&&$$L_3$$&$$L_4$$& $$\hat{f}(\varnothing)$$&$$\hat{f}(\{u\})$$&$$\hat{f}(\{v\})$$&$$\hat{f}(\{u,v\})$$ \\ ~&~&~&~&~&~&~&~&~\\ \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{tabular}$

### Tabular Test 1 $\begin{tabular}{lll} Chicago & U.S.A. & 1893 \\ Z\"{u}rich & Switzerland & 1897 \\ Paris & France & 1900 \\ Heidelberg & Germany & 1904 \\ Rome & Italy & 1908 \end{tabular}$

### Tabular Test 2 $\begin{tabular}{|r|r|} \hline $$n$$ & $$n!$$ \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5040 \\ 8 & 40320 \\ 9 & 362880 \\ 10 & 3628800 \\ \hline \end{tabular}$

### Tabular Test 3 $\begin{tabular}{|c|c|*{16}{c}|} \multicolumn{18}{c}{Table 1. Higher Order Propositions $$(n = 1)$$} \\[4pt] \hline $$f$$ & $$f$$ & $$m_{0}$$ & $$m_{1}$$ & $$m_{2}$$ & $$m_{3}$$ & $$m_{4}$$ & $$m_{5}$$ & $$m_{6}$$ & $$m_{7}$$ & $$m_{8}$$ & $$m_{9}$$ & $$m_{10}$$ & $$m_{11}$$ & $$m_{12}$$ & $$m_{13}$$ & $$m_{14}$$ & $$m_{15}$$ \\[4pt] \hline $$f_0$$ & \texttt{()} & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\[4pt] $$f_1$$ & \texttt{(}$$x$$\texttt{)} & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\[4pt] $$f_2$$ & $$x$$ & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\[4pt] $$f_3$$ & \texttt{(())} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\[4pt] \hline \end{tabular}$

### Tabular Test 4 $\begin{tabular}{|*{7}{c|}} \multicolumn{7}{c}{\textbf{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 & $$(~)$$ & false & $$0$$ \\ $$f_{1}$$ & $$f_{0001}$$ && 0 0 0 1 & $$(x)(y)$$ & neither $$x$$ nor $$y$$ & $$\lnot x \land \lnot y$$ \\ $$f_{2}$$ & $$f_{0010}$$ && 0 0 1 0 & $$(x)\ y$$ & $$y$$ without $$x$$ & $$\lnot x \land y$$ \\ $$f_{3}$$ & $$f_{0011}$$ && 0 0 1 1 & $$(x)$$ & not $$x$$ & $$\lnot x$$ \\ $$f_{4}$$ & $$f_{0100}$$ && 0 1 0 0 & $$x\ (y)$$ & $$x$$ without $$y$$ & $$x \land \lnot y$$ \\ $$f_{5}$$ & $$f_{0101}$$ && 0 1 0 1 & $$(y)$$ & not $$y$$ & $$\lnot y$$ \\ $$f_{6}$$ & $$f_{0110}$$ && 0 1 1 0 & $$(x,\ y)$$ & $$x$$ not equal to $$y$$ & $$x \ne y$$ \\ $$f_{7}$$ & $$f_{0111}$$ && 0 1 1 1 & $$(x\ y)$$ & not both $$x$$ and $$y$$ & $$\lnot x \lor \lnot y$$ \\ \hline $$f_{8}$$ & $$f_{1000}$$ && 1 0 0 0 & $$x\ y$$ & $$x$$ and $$y$$ & $$x \land y$$ \\ $$f_{9}$$ & $$f_{1001}$$ && 1 0 0 1 & $$((x,\ y))$$ & $$x$$ equal to $$y$$ & $$x = y$$ \\ $$f_{10}$$ & $$f_{1010}$$ && 1 0 1 0 & $$y$$ & $$y$$ & $$y$$ \\ $$f_{11}$$ & $$f_{1011}$$ && 1 0 1 1 & $$(x\ (y))$$ & not $$x$$ without $$y$$ & $$x \Rightarrow y$$ \\ $$f_{12}$$ & $$f_{1100}$$ && 1 1 0 0 & $$x$$ & $$x$$ & $$x$$ \\ $$f_{13}$$ & $$f_{1101}$$ && 1 1 0 1 & $$((x)\ y)$$ & not $$y$$ without $$x$$ & $$x \Leftarrow y$$ \\ $$f_{14}$$ & $$f_{1110}$$ && 1 1 1 0 & $$((x)(y))$$ & $$x$$ or $$y$$ & $$x \lor y$$ \\ $$f_{15}$$ & $$f_{1111}$$ && 1 1 1 1 & $$((~))$$ & true & $$1$$ \\ \hline \end{tabular}$

### Tabular Test 5 $\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $$\chi_S(x)$$ for $$f : \mathbb{B}^2 \to \mathbb{B}$$} \\[4pt] \hline $$(x_1, x_2) \backslash S$$ & $$\varnothing$$ & $$\{ x_1 \}$$ & $$\{ x_2 \}$$ & $$\{ x_1, x_2 \}$$ \\ \hline\hline $$(0, 0)$$ & $$+1$$ & $$+1$$ & $$+1$$ & $$+1$$ \\ $$(1, 0)$$ & $$+1$$ & $$-1$$ & $$+1$$ & $$-1$$ \\ $$(0, 1)$$ & $$+1$$ & $$+1$$ & $$-1$$ & $$-1$$ \\ $$(1, 1)$$ & $$+1$$ & $$-1$$ & $$-1$$ & $$+1$$ \\ \hline \end{tabular}$ $\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $$\chi_S(x)$$ for $$f : \mathbb{B}^2 \to \mathbb{B}$$} \\[4pt] \hline $$S \backslash (x_1, x_2)$$ & $$(0, 0)$$ & $$(1, 0)$$ & $$(0, 1)$$ & $$(1, 1)$$ \\ \hline\hline $$\varnothing$$ & $$+1$$ & $$+1$$ & $$+1$$ & $$+1$$ \\ $$\{ x_1 \}$$ & $$+1$$ & $$-1$$ & $$+1$$ & $$-1$$ \\ $$\{ x_2 \}$$ & $$+1$$ & $$+1$$ & $$-1$$ & $$-1$$ \\ $$\{ x_1, x_2 \}$$ & $$+1$$ & $$-1$$ & $$-1$$ & $$+1$$ \\ \hline \end{tabular}$ $\begin{tabular}{|c||*{4}{c}|} \multicolumn{5}{c}{Table 2.1 Values of $$\chi_S(x)$$ for $$f : \mathbb{B}^2 \to \mathbb{B}$$} \\[4pt] \hline $$S \backslash (x_1, x_2)$$ & $$(1, 1)$$ & $$(1, 0)$$ & $$(0, 1)$$ & $$(0, 0)$$ \\ \hline\hline $$\varnothing$$ & $$+1$$ & $$+1$$ & $$+1$$ & $$+1$$ \\ $$\{ x_1 \}$$ & $$-1$$ & $$-1$$ & $$+1$$ & $$+1$$ \\ $$\{ x_2 \}$$ & $$-1$$ & $$+1$$ & $$-1$$ & $$+1$$ \\ $$\{ x_1, x_2 \}$$ & $$+1$$ & $$-1$$ & $$-1$$ & $$+1$$ \\ \hline \end{tabular}$

This site uses Akismet to reduce spam. Learn how your comment data is processed.