Table A1 lists equivalent expressions for the Boolean functions of two variables in a number of different notational systems.

Table A1. Propositional Forms on Two Variables
$\mathcal{L}_{1}$	$\mathcal{L}_{2}$		$\mathcal{L}_{3}$	$\mathcal{L}_{4}$	$\mathcal{L}_{5}$	$\mathcal{L}_{6}$
		$x=$	1 1 0 0
		$y=$	1 0 1 0
$f_{{0}}$	$f_{{0000}}$		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	$(x,\ y)$	$x\ \operatorname{not~{}equal~{}to}\ y$	$x\neq y$
$f_{{7}}$	$f_{{0111}}$		0 1 1 1	$(x\ y)$	$\operatorname{not~{}both}\ x\ \operatorname{and}\ y$	$\lnot x\lor\lnot y$
$f_{{8}}$	$f_{{1000}}$		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	$((~{}))$	$\operatorname{true}$	$1$

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. Propositional Forms on Two Variables
$\mathcal{L}_{1}$	$\mathcal{L}_{2}$		$\mathcal{L}_{3}$	$\mathcal{L}_{4}$	$\mathcal{L}_{5}$	$\mathcal{L}_{6}$
		$x=$	1 1 0 0
		$y=$	1 0 1 0
$f_{{0}}$	$f_{{0000}}$		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	$(x)\ y$	$y\ \operatorname{without}\ x$	$\lnot x\land y$
$f_{{4}}$	$f_{{0100}}$		0 1 0 0	$x\ (y)$	$x\ \operatorname{without}\ y$	$x\land\lnot y$
$f_{{8}}$	$f_{{1000}}$		1 0 0 0	$x\ y$	$x\ \operatorname{and}\ y$	$x\land y$
$f_{{3}}$	$f_{{0011}}$		0 0 1 1	$(x)$	$\operatorname{not}\ x$	$\lnot x$
$f_{{12}}$	$f_{{1100}}$		1 1 0 0	$x$	$x$	$x$
$f_{{6}}$	$f_{{0110}}$		0 1 1 0	$(x,\ y)$	$x\ \operatorname{not~{}equal~{}to}\ y$	$x\neq y$
$f_{{9}}$	$f_{{1001}}$		1 0 0 1	$((x,\ y))$	$x\ \operatorname{equal~{}to}\ y$	$x=y$
$f_{{5}}$	$f_{{0101}}$		0 1 0 1	$(y)$	$\operatorname{not}\ y$	$\lnot y$
$f_{{10}}$	$f_{{1010}}$		1 0 1 0	$y$	$y$	$y$
$f_{{7}}$	$f_{{0111}}$		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	$((x)(y))$	$x\ \operatorname{or}\ y$	$x\lor y$
$f_{{15}}$	$f_{{1111}}$		1 1 1 1	$((~{}))$	$\operatorname{true}$	$1$

Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$
		$\operatorname{T}_{{11}}$	$\operatorname{T}_{{10}}$	$\operatorname{T}_{{01}}$	$\operatorname{T}_{{00}}$
	$f$	$\operatorname{E}f|_{{\operatorname{d}x\ \operatorname{d}y}}$	$\operatorname{E}f|_{{\operatorname{d}x(\operatorname{d}y)}}$	$\operatorname{E}f|_{{(\operatorname{d}x)\operatorname{d}y}}$	$\operatorname{E}f|_{{(\operatorname{d}x)(\operatorname{d}y)}}$
$f_{{0}}$	$(~{})$	$(~{})$	$(~{})$	$(~{})$	$(~{})$
$f_{{1}}$	$(x)(y)$	$x\ y$	$x\ (y)$	$(x)\ y$	$(x)(y)$
$f_{{2}}$	$(x)\ y$	$x\ (y)$	$x\ y$	$(x)(y)$	$(x)\ y$
$f_{{4}}$	$x\ (y)$	$(x)\ y$	$(x)(y)$	$x\ y$	$x\ (y)$
$f_{{8}}$	$x\ y$	$(x)(y)$	$(x)\ y$	$x\ (y)$	$x\ y$
$f_{{3}}$	$(x)$	$x$	$x$	$(x)$	$(x)$
$f_{{12}}$	$x$	$(x)$	$(x)$	$x$	$x$
$f_{{6}}$	$(x,\ y)$	$(x,\ y)$	$((x,\ y))$	$((x,\ y))$	$(x,\ y)$
$f_{{9}}$	$((x,\ y))$	$((x,\ y))$	$(x,\ y)$	$(x,\ y)$	$((x,\ y))$
$f_{{5}}$	$(y)$	$y$	$(y)$	$y$	$(y)$
$f_{{10}}$	$y$	$(y)$	$y$	$(y)$	$y$
$f_{{7}}$	$(x\ y)$	$((x)(y))$	$((x)\ y)$	$(x\ (y))$	$(x\ y)$
$f_{{11}}$	$(x\ (y))$	$((x)\ y)$	$((x)(y))$	$(x\ y)$	$(x\ (y))$
$f_{{13}}$	$((x)\ y)$	$(x\ (y))$	$(x\ y)$	$((x)(y))$	$((x)\ y)$
$f_{{14}}$	$((x)(y))$	$(x\ y)$	$(x\ (y))$	$((x)\ y)$	$((x)(y))$
$f_{{15}}$	$((~{}))$	$((~{}))$	$((~{}))$	$((~{}))$	$((~{}))$
Fixed Point Total:		4	4	4	16

Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{\operatorname{d}x,\operatorname{d}y\}$
	$f$	$\operatorname{D}f|_{{\operatorname{d}x\ \operatorname{d}y}}$	$\operatorname{D}f|_{{\operatorname{d}x(\operatorname{d}y)}}$	$\operatorname{D}f|_{{(\operatorname{d}x)\operatorname{d}y}}$	$\operatorname{D}f|_{{(\operatorname{d}x)(\operatorname{d}y)}}$
$f_{{0}}$	$(~{})$	$(~{})$	$(~{})$	$(~{})$	$(~{})$
$f_{{1}}$	$(x)(y)$	$((x,\ y))$	$(y)$	$(x)$	$(~{})$
$f_{{2}}$	$(x)\ y$	$(x,\ y)$	$y$	$(x)$	$(~{})$
$f_{{4}}$	$x\ (y)$	$(x,\ y)$	$(y)$	$x$	$(~{})$
$f_{{8}}$	$x\ y$	$((x,\ y))$	$y$	$x$	$(~{})$
$f_{{3}}$	$(x)$	$((~{}))$	$((~{}))$	$(~{})$	$(~{})$
$f_{{12}}$	$x$	$((~{}))$	$((~{}))$	$(~{})$	$(~{})$
$f_{{6}}$	$(x,\ y)$	$(~{})$	$((~{}))$	$((~{}))$	$(~{})$
$f_{{9}}$	$((x,\ y))$	$(~{})$	$((~{}))$	$((~{}))$	$(~{})$
$f_{{5}}$	$(y)$	$((~{}))$	$(~{})$	$((~{}))$	$(~{})$
$f_{{10}}$	$y$	$((~{}))$	$(~{})$	$((~{}))$	$(~{})$
$f_{{7}}$	$(x\ y)$	$((x,\ y))$	$y$	$x$	$(~{})$
$f_{{11}}$	$(x\ (y))$	$(x,\ y)$	$(y)$	$x$	$(~{})$
$f_{{13}}$	$((x)\ y)$	$(x,\ y)$	$y$	$(x)$	$(~{})$
$f_{{14}}$	$((x)(y))$	$((x,\ y))$	$(y)$	$(x)$	$(~{})$
$f_{{15}}$	$((~{}))$	$(~{})$	$(~{})$	$(~{})$	$(~{})$

Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{x,y\}$
	$f$	$\operatorname{E}f|_{{x\ y}}$	$\operatorname{E}f|_{{x(y)}}$	$\operatorname{E}f|_{{(x)y}}$	$\operatorname{E}f|_{{(x)(y)}}$
$f_{{0}}$	$(~{})$	$(~{})$	$(~{})$	$(~{})$	$(~{})$
$f_{{1}}$	$(x)(y)$	$\operatorname{d}x\ \operatorname{d}y$	$\operatorname{d}x\ (\operatorname{d}y)$	$(\operatorname{d}x)\ \operatorname{d}y$	$(\operatorname{d}x)(\operatorname{d}y)$
$f_{{2}}$	$(x)\ y$	$\operatorname{d}x\ (\operatorname{d}y)$	$\operatorname{d}x\ \operatorname{d}y$	$(\operatorname{d}x)(\operatorname{d}y)$	$(\operatorname{d}x)\ \operatorname{d}y$
$f_{{4}}$	$x\ (y)$	$(\operatorname{d}x)\ \operatorname{d}y$	$(\operatorname{d}x)(\operatorname{d}y)$	$\operatorname{d}x\ \operatorname{d}y$	$\operatorname{d}x\ (\operatorname{d}y)$
$f_{{8}}$	$x\ y$	$(\operatorname{d}x)(\operatorname{d}y)$	$(\operatorname{d}x)\ \operatorname{d}y$	$\operatorname{d}x\ (\operatorname{d}y)$	$\operatorname{d}x\ \operatorname{d}y$
$f_{{3}}$	$(x)$	$\operatorname{d}x$	$\operatorname{d}x$	$(\operatorname{d}x)$	$(\operatorname{d}x)$
$f_{{12}}$	$x$	$(\operatorname{d}x)$	$(\operatorname{d}x)$	$\operatorname{d}x$	$\operatorname{d}x$
$f_{{6}}$	$(x,\ y)$	$(\operatorname{d}x,\ \operatorname{d}y)$	$((\operatorname{d}x,\ \operatorname{d}y))$	$((\operatorname{d}x,\ \operatorname{d}y))$	$(\operatorname{d}x,\ \operatorname{d}y)$
$f_{{9}}$	$((x,\ y))$	$((\operatorname{d}x,\ \operatorname{d}y))$	$(\operatorname{d}x,\ \operatorname{d}y)$	$(\operatorname{d}x,\ \operatorname{d}y)$	$((\operatorname{d}x,\ \operatorname{d}y))$
$f_{{5}}$	$(y)$	$\operatorname{d}y$	$(\operatorname{d}y)$	$\operatorname{d}y$	$(\operatorname{d}y)$
$f_{{10}}$	$y$	$(\operatorname{d}y)$	$\operatorname{d}y$	$(\operatorname{d}y)$	$\operatorname{d}y$
$f_{{7}}$	$(x\ y)$	$((\operatorname{d}x)(\operatorname{d}y))$	$((\operatorname{d}x)\ \operatorname{d}y)$	$(\operatorname{d}x\ (\operatorname{d}y))$	$(\operatorname{d}x\ \operatorname{d}y)$
$f_{{11}}$	$(x\ (y))$	$((\operatorname{d}x)\ \operatorname{d}y)$	$((\operatorname{d}x)(\operatorname{d}y))$	$(\operatorname{d}x\ \operatorname{d}y)$	$(\operatorname{d}x\ (\operatorname{d}y))$
$f_{{13}}$	$((x)\ y)$	$(\operatorname{d}x\ (\operatorname{d}y))$	$(\operatorname{d}x\ \operatorname{d}y)$	$((\operatorname{d}x)(\operatorname{d}y))$	$((\operatorname{d}x)\ \operatorname{d}y)$
$f_{{14}}$	$((x)(y))$	$(\operatorname{d}x\ \operatorname{d}y)$	$(\operatorname{d}x\ (\operatorname{d}y))$	$((\operatorname{d}x)\ \operatorname{d}y)$	$((\operatorname{d}x)(\operatorname{d}y))$
$f_{{15}}$	$((~{}))$	$((~{}))$	$((~{}))$	$((~{}))$	$((~{}))$