User:Jon Awbrey/Figures and Tables 1

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Format Samples

  • \(\rightsquigarrow\)
  • \(\leftrightsquigarrow\)
  • \(\xrightarrow{\mathrm{Parse}}\)
  • \(\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}\)

Table 14. Semantic Translation • Functional Form


\(\text{Table 14. Semantic Translation : Functional Form}\)
\(\mathrm{Sentence}\) \(\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}\) \(\mathrm{Graph}\) \(\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Denotation}}\) \(\mathrm{Proposition}\)
\(s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(q_j\)
\(\mathrm{Conc}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Node}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(1\)
\(\mathrm{Conc}^k_j s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Node}^k_j C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Conj}^k_j q_j\)
\(\mathrm{Surc}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Lobe}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(0\)
\(\mathrm{Surc}^k_j s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Lobe}^k_j C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Surj}^k_j q_j\)


Table 15. Semantic Translation • Equational Form


\(\text{Table 15. Semantic Translation : Equational Form}\)
\(\downharpoonleft \mathrm{Sentence} \downharpoonright\) \(\stackrel{\mathrm{Parse}}{=}\) \(\downharpoonleft \mathrm{Graph} \downharpoonright\) \(\stackrel{\mathrm{Denotation}}{=}\) \(\mathrm{Proposition}\)
\(\downharpoonleft s_j \downharpoonright\) \(=\) \(\downharpoonleft C_j \downharpoonright\) \(=\) \(q_j\)
\(\downharpoonleft \mathrm{Conc}^0 \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Node}^0 \downharpoonright\) \(=\) \(1\)
\(\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright\) \(=\) \(\mathrm{Conj}^k_j q_j\)
\(\downharpoonleft \mathrm{Surc}^0 \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Lobe}^0 \downharpoonright\) \(=\) \(0\)
\(\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright\) \(=\) \(\mathrm{Surj}^k_j q_j\)


Table 16. Boolean Functions on Zero Variables


\(\text{Table 16. Boolean Functions on Zero Variables}\)
\(F\) \(F\) \(F()\) \(F\)
\(0\) \(F_0^{(0)}\) \(0\) \(\texttt{( )}\)
\(1\) \(F_1^{(0)}\) \(1\) \(\texttt{(( ))}\)


Table 17. Boolean Functions on One Variable


\(\text{Table 17. Boolean Functions on One Variable}\)
\(F\) \(F\) \(F(x)\) \(F\)
    \(F(1)\) \(F(0)\)  
\(F_0^{(1)}\) \(F_{00}^{(1)}\) \(0\) \(0\) \(\texttt{( )}\)
\(F_1^{(1)}\) \(F_{01}^{(1)}\) \(0\) \(1\) \(\texttt{(} x \texttt{)}\)
\(F_2^{(1)}\) \(F_{10}^{(1)}\) \(1\) \(0\) \(x\)
\(F_3^{(1)}\) \(F_{11}^{(1)}\) \(1\) \(1\) \(\texttt{(( ))}\)


Table 18. Boolean Functions on Two Variables


\(\text{Table 18. Boolean Functions on Two Variables}\)
\(F\) \(F\) \(F(x, y)\) \(F\)
    \(F(1, 1)\) \(F(1, 0)\) \(F(0, 1)\) \(F(0, 0)\)  
\(F_{0}^{(2)}\) \(F_{0000}^{(2)}\) \(0\) \(0\) \(0\) \(0\) \(\texttt{( )}\)
\(F_{1}^{(2)}\) \(F_{0001}^{(2)}\) \(0\) \(0\) \(0\) \(1\) \(\texttt{(} x \texttt{)(} y \texttt{)}\)
\(F_{2}^{(2)}\) \(F_{0010}^{(2)}\) \(0\) \(0\) \(1\) \(0\) \(\texttt{(} x \texttt{)} y\)
\(F_{3}^{(2)}\) \(F_{0011}^{(2)}\) \(0\) \(0\) \(1\) \(1\) \(\texttt{(} x \texttt{)}\)
\(F_{4}^{(2)}\) \(F_{0100}^{(2)}\) \(0\) \(1\) \(0\) \(0\) \(x \texttt{(} y \texttt{)}\)
\(F_{5}^{(2)}\) \(F_{0101}^{(2)}\) \(0\) \(1\) \(0\) \(1\) \(\texttt{(} y \texttt{)}\)
\(F_{6}^{(2)}\) \(F_{0110}^{(2)}\) \(0\) \(1\) \(1\) \(0\) \(\texttt{(} x \texttt{,} y \texttt{)}\)
\(F_{7}^{(2)}\) \(F_{0111}^{(2)}\) \(0\) \(1\) \(1\) \(1\) \(\texttt{(} x y \texttt{)}\)
\(F_{8}^{(2)}\) \(F_{1000}^{(2)}\) \(1\) \(0\) \(0\) \(0\) \(x y\)
\(F_{9}^{(2)}\) \(F_{1001}^{(2)}\) \(1\) \(0\) \(0\) \(1\) \(\texttt{((} x \texttt{,} y \texttt{))}\)
\(F_{10}^{(2)}\) \(F_{1010}^{(2)}\) \(1\) \(0\) \(1\) \(0\) \(y\)
\(F_{11}^{(2)}\) \(F_{1011}^{(2)}\) \(1\) \(0\) \(1\) \(1\) \(\texttt{(} x \texttt{(} y \texttt{))}\)
\(F_{12}^{(2)}\) \(F_{1100}^{(2)}\) \(1\) \(1\) \(0\) \(0\) \(x\)
\(F_{13}^{(2)}\) \(F_{1101}^{(2)}\) \(1\) \(1\) \(0\) \(1\) \(\texttt{((} x \texttt{)} y \texttt{)}\)
\(F_{14}^{(2)}\) \(F_{1110}^{(2)}\) \(1\) \(1\) \(1\) \(0\) \(\texttt{((} x \texttt{)(} y \texttt{))}\)
\(F_{15}^{(2)}\) \(F_{1111}^{(2)}\) \(1\) \(1\) \(1\) \(1\) \(\texttt{(( ))}\)


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