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