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
