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
o----------o----------o-------------------------------------------o----------o
| Constant | Function |                    F()                    | Function |
o----------o----------o-------------------------------------------o----------o
|          |          |                                           |          |
| %0%      | F^0_0    |                    %0%                    |    ()    |
|          |          |                                           |          |
| %1%      | F^0_1    |                    %1%                    |   (())   |
|          |          |                                           |          |
o----------o----------o-------------------------------------------o----------o

Table 16.  Boolean Functions on One Variable
o----------o----------o-------------------------------------------o----------o
| Function | Function |                   F(x)                    | Function |
o----------o----------o---------------------o---------------------o----------o
|          |          |       F(%0%)        |       F(%1%)        |          |
o----------o----------o---------------------o---------------------o----------o
|          |          |                     |                     |          |
| F^1_0    | F^1_00   |         %0%         |         %0%         |   ( )    |
|          |          |                     |                     |          |
| F^1_1    | F^1_01   |         %0%         |         %1%         |   (x)    |
|          |          |                     |                     |          |
| F^1_2    | F^1_10   |         %1%         |         %0%         |    x     |
|          |          |                     |                     |          |
| F^1_3    | F^1_11   |         %1%         |         %1%         |  (( ))   |
|          |          |                     |                     |          |
o----------o----------o---------------------o---------------------o----------o

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

---

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›