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User:Jon Awbrey/SANDBOX

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Revision as of 16:32, 22 January 2009 by Jon Awbrey (talk | contribs) (→‎Table Stuff)

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

\(\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}\)

\(\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}\)

\(~\!\) \(x\colon\!\) \(1~1~0~0\!\) \(~\!\) \(~\!\) \(~\!\)
\(~\!\) \(y\colon\!\) \(1~0~1~0\!\) \(~\!\) \(~\!\) \(~\!\)
\(f_{0}\!\) \(f_{0000}\!\) \(0~0~0~0\!\) \((~)\!\) \(\mbox{false}\!\) \(0\!\)
\(f_{1}\!\) \(f_{0001}\!\) \(0~0~0~1\!\) \((x)(y)\!\) \(\mbox{neither}\ x\ \mbox{nor}\ y\!\) \(\lnot x \land \lnot y\!\)
\(f_{2}\!\) \(f_{0010}\!\) \(0~0~1~0\!\) \((x)\ y\!\) \(y\ \mbox{without}\ x\!\) \(\lnot x \land y\!\)
\(f_{3}\!\) \(f_{0011}\!\) \(0~0~1~1\!\) \((x)\!\) \(\mbox{not}\ x\!\) \(\lnot x\!\)
\(f_{4}\!\) \(f_{0100}\!\) \(0~1~0~0\!\) \(x\ (y)\!\) \(x\ \mbox{without}\ y\!\) \(x \land \lnot y\!\)
\(f_{5}\!\) \(f_{0101}\!\) \(0~1~0~1\!\) \((y)\!\) \(\mbox{not}\ y\!\) \(\lnot y\!\)
\(f_{6}\!\) \(f_{0110}\!\) \(0~1~1~0\!\) \((x, y)\!\) \(x\ \mbox{not equal to}\ y\!\) \(x \ne y\!\)
\(f_{7}\!\) \(f_{0111}\!\) \(0~1~1~1\!\) \((x\ y)\!\) \(\mbox{not both}\ x\ \mbox{and}\ y\!\) \(\lnot x \lor \lnot y\!\)
\(f_{8}\!\) \(f_{1000}\!\) \(1~0~0~0\!\) \(x\ y\!\) \(x\ \mbox{and}\ y\!\) \(x \land y\!\)
\(f_{9}\!\) \(f_{1001}\!\) \(1~0~0~1\!\) \(((x, y))\!\) \(x\ \mbox{equal to}\ y\!\) \(x = y\!\)
\(f_{10}\!\) \(f_{1010}\!\) \(1~0~1~0\!\) \(y\!\) \(y\!\) \(y\!\)
\(f_{11}\!\) \(f_{1011}\!\) \(1~0~1~1\!\) \((x\ (y))\!\) \(\mbox{not}\ x\ \mbox{without}\ y\!\) \(x \Rightarrow y\!\)
\(f_{12}\!\) \(f_{1100}\!\) \(1~1~0~0\!\) \(x\!\) \(x\!\) \(x\!\)
\(f_{13}\!\) \(f_{1101}\!\) \(1~1~0~1\!\) \(((x)\ y)\!\) \(\mbox{not}\ y\ \mbox{without}\ x\!\) \(x \Leftarrow y\!\)
\(f_{14}\!\) \(f_{1110}\!\) \(1~1~1~0\!\) \(((x)(y))\!\) \(x\ \mbox{or}\ y\!\) \(x \lor y\!\)
\(f_{15}\!\) \(f_{1111}\!\) \(1~1~1~1\!\) \(((~))\!\) \(\mbox{true}\!\) \(1\!\)



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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›


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