User:Jon Awbrey/SANDBOX

Logic of Relatives

Table 3. Relational Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | L # X | Y | | o---------o---------o---------o---------o | M # | Y | Z | o---------o---------o---------o---------o | L o M # X | | Z | o---------o---------o---------o---------o

$\text{Table 3. Relational Composition}\!$

	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$
$L\!$	$X\!$	$Y\!$
$M\!$		$Y\!$	$Z\!$
$L \circ M$	$X\!$		$Z\!$

Table 9. Composite of Triadic and Dyadic Relations o---------o---------o---------o---------o---------o | # !1! | !1! | !1! | !1! | o=========o=========o=========o=========o=========o | G # T | U | | V | o---------o---------o---------o---------o---------o | L # | U | W | | o---------o---------o---------o---------o---------o | G o L # T | | W | V | o---------o---------o---------o---------o---------o

$\text{Table 9. Composite of Triadic and Dyadic Relations}\!$

	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$
$G\!$	$T\!$	$U\!$		$V\!$
$L\!$		$U\!$	$W\!$
$G \circ L$	$T\!$		$W\!$	$V\!$

Table 13. Another Brand of Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | G # X | Y | Z | o---------o---------o---------o---------o | T # | Y | Z | o---------o---------o---------o---------o | G o T # X | | Z | o---------o---------o---------o---------o

$\text{Table 13. Another Brand of Composition}\!$

	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$
$G\!$	$X\!$	$Y\!$	$Z\!$
$T\!$		$Y\!$	$Z\!$
$G \circ T$	$X\!$		$Z\!$

Table 15. Conjunction Via Composition o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | L, # X | X | Y | o---------o---------o---------o---------o | S # | X | Y | o---------o---------o---------o---------o | L , S # X | | Y | o---------o---------o---------o---------o

$\text{Table 15. Conjunction Via Composition}\!$

	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$
$L,\!$	$X\!$	$X\!$	$Y\!$
$S\!$		$X\!$	$Y\!$
$L,\!S$	$X\!$		$Y\!$

Table 18. Relational Composition P o Q o---------o---------o---------o---------o | # !1! | !1! | !1! | o=========o=========o=========o=========o | P # X | Y | | o---------o---------o---------o---------o | Q # | Y | Z | o---------o---------o---------o---------o | P o Q # X | | Z | o---------o---------o---------o---------o

$\text{Table 18. Relational Composition}~ P \circ Q$

	$\mathit{1}\!$	$\mathit{1}\!$	$\mathit{1}\!$
$P\!$	$X\!$	$Y\!$
$Q\!$		$Y\!$	$Z\!$
$P \circ Q$	$X\!$		$Z\!$

Table 20. Arrow: J(L(u, v)) = K(Ju, Jv) o---------o---------o---------o---------o | # J | J | J | o=========o=========o=========o=========o | K # X | X | X | o---------o---------o---------o---------o | L # Y | Y | Y | o---------o---------o---------o---------o

$\text{Table 20. Arrow Equation:}~~ J(L(u, v)) = K(Ju, Jv)$

	$J\!$	$J\!$	$J\!$
$K\!$	$X\!$	$X\!$	$X\!$
$L\!$	$Y\!$	$Y\!$	$Y\!$

Grammar Stuff

Table 13. Algorithmic Translation Rules

$\text{Sentence in PARCE}\!$ $\xrightarrow{\operatorname{Parse}}$ $\text{Graph in PARC}\!$

$\operatorname{Conc}^0$ $\xrightarrow{\operatorname{Parse}}$ $\operatorname{Node}^0$ $\operatorname{Conc}_{j=1}^k s_j$ $\xrightarrow{\operatorname{Parse}}$ $\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)$

$\operatorname{Surc}^0$ $\xrightarrow{\operatorname{Parse}}$ $\operatorname{Lobe}^0$ $\operatorname{Surc}_{j=1}^k s_j$ $\xrightarrow{\operatorname{Parse}}$ $\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)$

Table 14.1 Semantic Translation : Functional Form

$\operatorname{Sentence}$ $\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}$ $\operatorname{Graph}$ $\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}$ $\operatorname{Proposition}$

$s_j\!$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $C_j\!$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $q_j\!$

$\operatorname{Conc}^0$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Node}^0$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\underline{1}$ $\operatorname{Conc}^k_j s_j$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Node}^k_j C_j$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Conj}^k_j q_j$

$\operatorname{Surc}^0$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Lobe}^0$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\underline{0}$ $\operatorname{Surc}^k_j s_j$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Lobe}^k_j C_j$ $\xrightarrow{\operatorname{~~~~~~~~~~}}$ $\operatorname{Surj}^k_j q_j$

Table 14.2 Semantic Translation : Equational Form

$\downharpoonleft \operatorname{Sentence} \downharpoonright$ $\stackrel{\operatorname{Parse}}{=}$ $\downharpoonleft \operatorname{Graph} \downharpoonright$ $\stackrel{\operatorname{Denotation}}{=}$ $\operatorname{Proposition}$

$\downharpoonleft s_j \downharpoonright$ $=\!$ $\downharpoonleft C_j \downharpoonright$ $=\!$ $q_j\!$

$\downharpoonleft \operatorname{Conc}^0 \downharpoonright$ $=\!$ $\downharpoonleft \operatorname{Node}^0 \downharpoonright$ $=\!$ $\underline{1}$ $\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright$ $=\!$ $\downharpoonleft \operatorname{Node}^k_j C_j \downharpoonright$ $=\!$ $\operatorname{Conj}^k_j q_j$

$\downharpoonleft \operatorname{Surc}^0 \downharpoonright$ $=\!$ $\downharpoonleft \operatorname{Lobe}^0 \downharpoonright$ $=\!$ $\underline{0}$ $\downharpoonleft \operatorname{Surc}^k_j s_j \downharpoonright$ $=\!$ $\downharpoonleft \operatorname{Lobe}^k_j C_j \downharpoonright$ $=\!$ $\operatorname{Surj}^k_j q_j$

Table Stuff

Table 15. Boolean Functions on Zero Variables

$F\!$	$F\!$	$F()\!$	$F\!$
$\underline{0}$	$F_0^{(0)}\!$	$\underline{0}$	$(~)$
$\underline{1}$	$F_1^{(0)}\!$	$\underline{1}$	$((~))$

Table 16. Boolean Functions on One Variable

$F\!$	$F\!$	$F(x)\!$		$F\!$
		$F(\underline{1})$	$F(\underline{0})$
$F_0^{(1)}\!$	$F_{00}^{(1)}\!$	$\underline{0}$	$\underline{0}$	$(~)$
$F_1^{(1)}\!$	$F_{01}^{(1)}\!$	$\underline{0}$	$\underline{1}$	$(x)\!$
$F_2^{(1)}\!$	$F_{10}^{(1)}\!$	$\underline{1}$	$\underline{0}$	$x\!$
$F_3^{(1)}\!$	$F_{11}^{(1)}\!$	$\underline{1}$	$\underline{1}$	$((~))$

Table 17. Boolean Functions on Two Variables

$F\!$	$F\!$	$F(x, y)\!$				$F\!$
		$F(\underline{1}, \underline{1})$	$F(\underline{1}, \underline{0})$	$F(\underline{0}, \underline{1})$	$F(\underline{0}, \underline{0})$
$F_{0}^{(2)}\!$	$F_{0000}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$(~)$
$F_{1}^{(2)}\!$	$F_{0001}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$(x)(y)\!$
$F_{2}^{(2)}\!$	$F_{0010}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$(x) y\!$
$F_{3}^{(2)}\!$	$F_{0011}^{(2)}\!$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$(x)\!$
$F_{4}^{(2)}\!$	$F_{0100}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$x (y)\!$
$F_{5}^{(2)}\!$	$F_{0101}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$(y)\!$
$F_{6}^{(2)}\!$	$F_{0110}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$(x, y)\!$
$F_{7}^{(2)}\!$	$F_{0111}^{(2)}\!$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$(x y)\!$
$F_{8}^{(2)}\!$	$F_{1000}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$\underline{0}$	$x y\!$
$F_{9}^{(2)}\!$	$F_{1001}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$\underline{1}$	$((x, y))\!$
$F_{10}^{(2)}\!$	$F_{1010}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$\underline{0}$	$y\!$
$F_{11}^{(2)}\!$	$F_{1011}^{(2)}\!$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$\underline{1}$	$(x (y))\!$
$F_{12}^{(2)}\!$	$F_{1100}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$\underline{0}$	$x\!$
$F_{13}^{(2)}\!$	$F_{1101}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$\underline{1}$	$((x)y)\!$
$F_{14}^{(2)}\!$	$F_{1110}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$\underline{0}$	$((x)(y))\!$
$F_{15}^{(2)}\!$	$F_{1111}^{(2)}\!$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$\underline{1}$	$((~))$

---

fi‹x, y›	u = v =	1 1 0 0 1 0 1 0	= u = v	fj‹u, v›
	x = y =	1 1 1 0 1 0 0 1	= f‹u, v› = g‹u, v›

A	u = v =	1 1 0 0 1 0 1 0	= u = v	B
	x = y =	1 1 1 0 1 0 0 1	= f‹u, v› = g‹u, v›

u = v =	1 1 0 0 1 0 1 0	= u = v
x = y =	1 1 1 0 1 0 0 1	= f‹u, v› = g‹u, v›

u = v = x = y =

1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1

= u = v = f‹u, v› = g‹u, v›