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
