User:Jon Awbrey/SANDBOX

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

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

Table 16. Boolean Functions on One Variable

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

Table 17. Boolean Functions on Two Variables

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

Table 17.  Boolean Functions on Two Variables
o----------o----------o-------------------------------------------o----------o
| Function | Function |                  F(x, y)                  | Function |
o----------o----------o----------o----------o----------o----------o----------o
|          |          | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% |          |
o----------o----------o----------o----------o----------o----------o----------o
|          |          |          |          |          |          |          |
| F^2_00   | F^2_0000 |   %0%    |   %0%    |   %0%    |   %0%    |    ()    |
|          |          |          |          |          |          |          |
| F^2_01   | F^2_0001 |   %0%    |   %0%    |   %0%    |   %1%    |  (x)(y)  |
|          |          |          |          |          |          |          |
| F^2_02   | F^2_0010 |   %0%    |   %0%    |   %1%    |   %0%    |  (x) y   |
|          |          |          |          |          |          |          |
| F^2_03   | F^2_0011 |   %0%    |   %0%    |   %1%    |   %1%    |  (x)     |
|          |          |          |          |          |          |          |
| F^2_04   | F^2_0100 |   %0%    |   %1%    |   %0%    |   %0%    |   x (y)  |
|          |          |          |          |          |          |          |
| F^2_05   | F^2_0101 |   %0%    |   %1%    |   %0%    |   %1%    |     (y)  |
|          |          |          |          |          |          |          |
| F^2_06   | F^2_0110 |   %0%    |   %1%    |   %1%    |   %0%    |  (x, y)  |
|          |          |          |          |          |          |          |
| F^2_07   | F^2_0111 |   %0%    |   %1%    |   %1%    |   %1%    |  (x  y)  |
|          |          |          |          |          |          |          |
| F^2_08   | F^2_1000 |   %1%    |   %0%    |   %0%    |   %0%    |   x  y   |
|          |          |          |          |          |          |          |
| F^2_09   | F^2_1001 |   %1%    |   %0%    |   %0%    |   %1%    | ((x, y)) |
|          |          |          |          |          |          |          |
| F^2_10   | F^2_1010 |   %1%    |   %0%    |   %1%    |   %0%    |      y   |
|          |          |          |          |          |          |          |
| F^2_11   | F^2_1011 |   %1%    |   %0%    |   %1%    |   %1%    |  (x (y)) |
|          |          |          |          |          |          |          |
| F^2_12   | F^2_1100 |   %1%    |   %1%    |   %0%    |   %0%    |   x      |
|          |          |          |          |          |          |          |
| F^2_13   | F^2_1101 |   %1%    |   %1%    |   %0%    |   %1%    | ((x) y)  |
|          |          |          |          |          |          |          |
| F^2_14   | F^2_1110 |   %1%    |   %1%    |   %1%    |   %0%    | ((x)(y)) |
|          |          |          |          |          |          |          |
| F^2_15   | F^2_1111 |   %1%    |   %1%    |   %1%    |   %1%    |   (())   |
|          |          |          |          |          |          |          |
o----------o----------o----------o----------o----------o----------o----------o

Table 7. Propositional Forms on Two Variables

\(\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}\)
\(~\!\)	\(x\colon\!\)	\(1~1~0~0\!\)	\(~\!\)	\(~\!\)	\(~\!\)
\(~\!\)	\(y\colon\!\)	\(1~0~1~0\!\)	\(~\!\)	\(~\!\)	\(~\!\)
\(f_{0}\!\)	\(f_{0000}\!\)	\(0~0~0~0\!\)	\((~)\!\)	\(\mbox{false}\!\)	\(0\!\)
\(f_{1}\!\)	\(f_{0001}\!\)	\(0~0~0~1\!\)	\((x)(y)\!\)	\(\mbox{neither}\ x\ \mbox{nor}\ y\!\)	\(\lnot x \land \lnot y\!\)
\(f_{2}\!\)	\(f_{0010}\!\)	\(0~0~1~0\!\)	\((x)\ y\!\)	\(y\ \mbox{without}\ x\!\)	\(\lnot x \land y\!\)
\(f_{3}\!\)	\(f_{0011}\!\)	\(0~0~1~1\!\)	\((x)\!\)	\(\mbox{not}\ x\!\)	\(\lnot x\!\)
\(f_{4}\!\)	\(f_{0100}\!\)	\(0~1~0~0\!\)	\(x\ (y)\!\)	\(x\ \mbox{without}\ y\!\)	\(x \land \lnot y\!\)
\(f_{5}\!\)	\(f_{0101}\!\)	\(0~1~0~1\!\)	\((y)\!\)	\(\mbox{not}\ y\!\)	\(\lnot y\!\)
\(f_{6}\!\)	\(f_{0110}\!\)	\(0~1~1~0\!\)	\((x, y)\!\)	\(x\ \mbox{not equal to}\ y\!\)	\(x \ne y\!\)
\(f_{7}\!\)	\(f_{0111}\!\)	\(0~1~1~1\!\)	\((x\ y)\!\)	\(\mbox{not both}\ x\ \mbox{and}\ y\!\)	\(\lnot x \lor \lnot y\!\)
\(f_{8}\!\)	\(f_{1000}\!\)	\(1~0~0~0\!\)	\(x\ y\!\)	\(x\ \mbox{and}\ y\!\)	\(x \land y\!\)
\(f_{9}\!\)	\(f_{1001}\!\)	\(1~0~0~1\!\)	\(((x, y))\!\)	\(x\ \mbox{equal to}\ y\!\)	\(x = y\!\)
\(f_{10}\!\)	\(f_{1010}\!\)	\(1~0~1~0\!\)	\(y\!\)	\(y\!\)	\(y\!\)
\(f_{11}\!\)	\(f_{1011}\!\)	\(1~0~1~1\!\)	\((x\ (y))\!\)	\(\mbox{not}\ x\ \mbox{without}\ y\!\)	\(x \Rightarrow y\!\)
\(f_{12}\!\)	\(f_{1100}\!\)	\(1~1~0~0\!\)	\(x\!\)	\(x\!\)	\(x\!\)
\(f_{13}\!\)	\(f_{1101}\!\)	\(1~1~0~1\!\)	\(((x)\ y)\!\)	\(\mbox{not}\ y\ \mbox{without}\ x\!\)	\(x \Leftarrow y\!\)
\(f_{14}\!\)	\(f_{1110}\!\)	\(1~1~1~0\!\)	\(((x)(y))\!\)	\(x\ \mbox{or}\ y\!\)	\(x \lor y\!\)
\(f_{15}\!\)	\(f_{1111}\!\)	\(1~1~1~1\!\)	\(((~))\!\)	\(\mbox{true}\!\)	\(1\!\)

---

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›