| Version 8 |
Version 7 |
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| \textbf{Note.} The following Tables are best viewed in the Page Image mode. |
\textbf{Note.} The following Tables are best viewed in the Page Image mode. |
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| \tableofcontents |
\tableofcontents |
|
|
| \subsection{Table A1. Propositional Forms on Two Variables} |
\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. |
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|} |
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} |
| \multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\ |
\multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\ |
| \hline |
\hline |
| $\mathcal{L}_1$ & $\mathcal{L}_2$ && |
$\mathcal{L}_1$ & $\mathcal{L}_2$ && |
| $\mathcal{L}_3$ & $\mathcal{L}_4$ & |
$\mathcal{L}_3$ & $\mathcal{L}_4$ & |
| $\mathcal{L}_5$ & $\mathcal{L}_6$ \\ |
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\ |
| \hline |
\hline |
| & & $x =$ & 1 1 0 0 & & & \\ |
& & $x =$ & 1 1 0 0 & & & \\ |
| & & $y =$ & 1 0 1 0 & & & \\ |
& & $y =$ & 1 0 1 0 & & & \\ |
| \hline |
\hline |
| $f_{0}$ & $f_{0000}$ & & 0 0 0 0 & $(~)$ & false & $0$ \\ |
$f_{0}$ & $f_{0000}$ & & 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 $ \\ |
$f_{1}$ & $f_{0001}$ & & 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 $x$ & $\lnot x \land y$ \\
|
$f_{2}$ & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$ & $y$ and not $x$ & $\lnot x \land y$ \\
|
| $f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\ |
$f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\ |
|
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ without $y$ & $x \land \lnot y$ \\
|
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ and not $y$ & $x \land \lnot y$ \\
|
| $f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\ |
$f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\ |
| $f_{6}$ & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$ & $x$ not equal to $y$ & $x \ne y$ \\ |
$f_{6}$ & $f_{0110}$ & & 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 $y$ & $\lnot x \lor \lnot y$ \\ |
$f_{7}$ & $f_{0111}$ & & 0 1 1 1 & $(x\ y)$ & not both $x$ and $y$ & $\lnot x \lor \lnot y$ \\ |
| \hline |
\hline |
| $f_{8}$ & $f_{1000}$ & & 1 0 0 0 & $x\ y$ & $x$ and $y$ & $x \land y$ \\ |
$f_{8}$ & $f_{1000}$ & & 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 $y$ & $x = y$ \\ |
$f_{9}$ & $f_{1001}$ & & 1 0 0 1 & $((x,\ y))$ & $x$ equal to $y$ & $x = y$ \\ |
| $f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\ |
$f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\ |
| $f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\ |
$f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\ |
| $f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\ |
$f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\ |
| $f_{13}$ & $f_{1101}$ & & 1 1 0 1 & $((x)\ y)$ & not $y$ without $x$ & $x \Leftarrow y$ \\ |
$f_{13}$ & $f_{1101}$ & & 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 $y$ & $x \lor y$ \\ |
$f_{14}$ & $f_{1110}$ & & 1 1 1 0 & $((x)(y))$ & $x$ or $y$ & $x \lor y$ \\ |
| $f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\ |
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\ |
| \hline |
\hline |
| \end{tabular}\end{quote} |
\end{tabular}\end{quote} |
|
|
| \subsection{Table A2. Propositional Forms on Two Variables} |
\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. |
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|} |
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} |
| \multicolumn{7}{c}{Table A2. Propositional Forms on Two Variables} \\ |
\multicolumn{7}{c}{Table A2. Propositional Forms on Two Variables} \\ |
| \hline |
\hline |
| $\mathcal{L}_1$ & $\mathcal{L}_2$ && |
$\mathcal{L}_1$ & $\mathcal{L}_2$ && |
| $\mathcal{L}_3$ & $\mathcal{L}_4$ & |
$\mathcal{L}_3$ & $\mathcal{L}_4$ & |
| $\mathcal{L}_5$ & $\mathcal{L}_6$ \\ |
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\ |
| \hline |
\hline |
| & & $x =$ & 1 1 0 0 & & & \\ |
& & $x =$ & 1 1 0 0 & & & \\ |
| & & $y =$ & 1 0 1 0 & & & \\ |
& & $y =$ & 1 0 1 0 & & & \\ |
| \hline |
\hline |
| $f_{0}$ & $f_{0000}$ & & 0 0 0 0 & $(~)$ & false & $0$ \\ |
$f_{0}$ & $f_{0000}$ & & 0 0 0 0 & $(~)$ & false & $0$ \\ |
| \hline |
\hline |
| $f_{1}$ & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$ & neither $x$ nor $y$ & $\lnot x \land \lnot y $ \\ |
$f_{1}$ & $f_{0001}$ & & 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 $x$ & $\lnot x \land y$ \\
|
$f_{2}$ & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$ & $y$ and not $x$ & $\lnot x \land y$ \\
|
|
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ without $y$ & $x \land \lnot y$ \\
|
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ and not $y$ & $x \land \lnot y$ \\
|
| $f_{8}$ & $f_{1000}$ & & 1 0 0 0 & $x\ y$ & $x$ and $y$ & $x \land y$ \\ |
$f_{8}$ & $f_{1000}$ & & 1 0 0 0 & $x\ y$ & $x$ and $y$ & $x \land y$ \\ |
| \hline |
\hline |
| $f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\ |
$f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\ |
| $f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\ |
$f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\ |
| \hline |
\hline |
| $f_{6}$ & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$ & $x$ not equal to $y$ & $x \ne y$ \\ |
$f_{6}$ & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$ & $x$ not equal to $y$ & $x \ne y$ \\ |
| $f_{9}$ & $f_{1001}$ & & 1 0 0 1 & $((x,\ y))$ & $x$ equal to $y$ & $x = y$ \\ |
$f_{9}$ & $f_{1001}$ & & 1 0 0 1 & $((x,\ y))$ & $x$ equal to $y$ & $x = y$ \\ |
| \hline |
\hline |
| $f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\ |
$f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\ |
| $f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\ |
$f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\ |
| \hline |
\hline |
| $f_{7}$ & $f_{0111}$ & & 0 1 1 1 & $(x\ y)$ & not both $x$ and $y$ & $\lnot x \lor \lnot y$ \\ |
$f_{7}$ & $f_{0111}$ & & 0 1 1 1 & $(x\ y)$ & not both $x$ and $y$ & $\lnot x \lor \lnot y$ \\ |
| $f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\ |
$f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\ |
| $f_{13}$ & $f_{1101}$ & & 1 1 0 1 & $((x)\ y)$ & not $y$ without $x$ & $x \Leftarrow y$ \\ |
$f_{13}$ & $f_{1101}$ & & 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 $y$ & $x \lor y$ \\ |
$f_{14}$ & $f_{1110}$ & & 1 1 1 0 & $((x)(y))$ & $x$ or $y$ & $x \lor y$ \\ |
| \hline |
\hline |
| $f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\ |
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\ |
| \hline |
\hline |
| \end{tabular}\end{quote} |
\end{tabular}\end{quote} |
|
|
| \subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} |
\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|} |
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| \multicolumn{6}{c}{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} \\ |
\multicolumn{6}{c}{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} \\ |
| \hline |
\hline |
| & & |
& & |
| $\operatorname{T}_{11}$ & |
$\operatorname{T}_{11}$ & |
| $\operatorname{T}_{10}$ & |
$\operatorname{T}_{10}$ & |
| $\operatorname{T}_{01}$ & |
$\operatorname{T}_{01}$ & |
| $\operatorname{T}_{00}$ \\ |
$\operatorname{T}_{00}$ \\ |
| & $f$ & |
& $f$ & |
| $\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
| $\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
| $\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
| $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
| \hline |
\hline |
| $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\ |
$f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\ |
| $f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\ |
$f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\ |
| $f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\ |
$f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\ |
| $f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\ |
$f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\ |
| \hline |
\hline |
| $f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\ |
$f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\ |
| $f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\ |
$f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\ |
| \hline |
\hline |
| $f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\ |
$f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\ |
| $f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\ |
$f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\ |
| \hline |
\hline |
| $f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\ |
$f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\ |
| $f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\ |
$f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\ |
| \hline |
\hline |
| $f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\ |
$f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\ |
| $f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\ |
$f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\ |
| $f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\ |
$f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\ |
| $f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\ |
$f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\ |
| \hline |
\hline |
| $f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\ |
$f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\ |
| \hline |
\hline |
| \multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\ |
\multicolumn{2}{|c||}{\PMlinkname{Fixed Point}{FixedPoint} Total:} & 4 & 4 & 4 & 16 \\ |
| \hline |
\hline |
| \end{tabular}\end{quote} |
\end{tabular}\end{quote} |
|
|
| \subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} |
\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|} |
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|} |
| \multicolumn{6}{c}{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} \\ |
\multicolumn{6}{c}{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} \\ |
| \hline |
\hline |
| & $f$ & |
& $f$ & |
| $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
$\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ & |
| $\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
$\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ & |
| $\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
$\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ & |
| $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
$\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ |
| \hline |
\hline |
| $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
$f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
| $f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
$f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
| $f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
$f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
| $f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
$f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
$f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
| $f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
$f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
$f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
| $f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
$f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
$f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
| $f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
$f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
$f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ |
| $f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
$f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ |
| $f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
$f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ |
| $f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
$f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ |
| \hline |
\hline |
| $f_{15}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
$f_{15}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ |
| \hline |
\hline |
| \end{tabular}\end{quote} |
\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}{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} |
|
