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

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

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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›