Difference between revisions of "User:Jon Awbrey/Figures and Tables 1"

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==Format Samples==
 +
 
 +
<ul>
 +
<li><math>\rightsquigarrow</math></li>
 +
 
 +
<li><math>\leftrightsquigarrow</math></li>
 +
 
 +
<li><math>\xrightarrow{\mathrm{Parse}}</math></li>
 +
 
 +
<li><math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math></li>
 +
</ul>
 +
 
 +
==Table 13. Algorithmic Translation Rules==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 13. Algorithmic Translation Rules}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; text-align:center; width:100%"
 +
| width="33%" | <math>\text{Sentence in PARCE}</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
 +
| width="33%" | <math>\text{Graph in PARC}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
| width="33%" | <math>\mathrm{Conc}^0</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
 +
| width="33%" | <math>\mathrm{Node}^0</math>
 +
|-
 +
| width="33%" | <math>\mathrm{Conc}_{j=1}^k s_j</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
 +
| width="33%" | <math>\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
| width="33%" | <math>\mathrm{Surc}^0</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
 +
| width="33%" | <math>\mathrm{Lobe}^0</math>
 +
|-
 +
| width="33%" | <math>\mathrm{Surc}_{j=1}^k s_j</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}</math>
 +
| width="33%" | <math>\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)</math>
 +
|}
 +
|}
 +
 
 +
<br>
 +
 
 +
==Table 14. Semantic Translation &bull; Functional Form==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 14. Semantic Translation : Functional Form}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
 +
| width="20%" | <math>\mathrm{Sentence}</math>
 +
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}</math>
 +
| width="20%" | <math>\mathrm{Graph}</math>
 +
| width="20%" | <math>\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Denotation}}</math>
 +
| width="20%" | <math>\mathrm{Proposition}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>s_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>C_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>q_j</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\mathrm{Conc}^0</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Node}^0</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>1</math>
 +
|-
 +
| width="20%" | <math>\mathrm{Conc}^k_j s_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Node}^k_j C_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\mathrm{Surc}^0</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Lobe}^0</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>0</math>
 +
|-
 +
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Lobe}^k_j C_j</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
 +
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
 +
|}
 +
|}
 +
 
 +
<br>
 +
 
 +
==Table 15. Semantic Translation &bull; Equational Form==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 15. Semantic Translation : Equational Form}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
 +
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright</math>
 +
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright</math>
 +
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}</math>
 +
| width="20%" | <math>\mathrm{Proposition}</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft s_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>q_j</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^0 \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>1</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\mathrm{Conj}^k_j q_j</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^0 \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>0</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright</math>
 +
| width="20%" | <math>=</math>
 +
| width="20%" | <math>\mathrm{Surj}^k_j q_j</math>
 +
|}
 +
|}
 +
 
 +
<br>
 +
 
 +
==Table 16. Boolean Functions on Zero Variables==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | <math>F</math>
 +
| width="14%" | <math>F</math>
 +
| width="48%" | <math>F()</math>
 +
| width="24%" | <math>F</math>
 +
|-
 +
| <math>0</math>
 +
| <math>F_0^{(0)}</math>
 +
| <math>0</math>
 +
| <math>\texttt{( )}</math>
 +
|-
 +
| <math>1</math>
 +
| <math>F_1^{(0)}</math>
 +
| <math>1</math>
 +
| <math>\texttt{(( ))}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
==Table 17. Boolean Functions on One Variable==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 17. Boolean Functions on One Variable}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | <math>F</math>
 +
| width="14%" | <math>F</math>
 +
| colspan="2" | <math>F(x)</math>
 +
| width="24%" | <math>F</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | &nbsp;
 +
| width="14%" | &nbsp;
 +
| width="24%" | <math>F(1)</math>
 +
| width="24%" | <math>F(0)</math>
 +
| width="24%" | &nbsp;
 +
|-
 +
| <math>F_0^{(1)}</math>
 +
| <math>F_{00}^{(1)}</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>\texttt{( )}</math>
 +
|-
 +
| <math>F_1^{(1)}</math>
 +
| <math>F_{01}^{(1)}</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} x \texttt{)}</math>
 +
|-
 +
| <math>F_2^{(1)}</math>
 +
| <math>F_{10}^{(1)}</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>x</math>
 +
|-
 +
| <math>F_3^{(1)}</math>
 +
| <math>F_{11}^{(1)}</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>\texttt{(( ))}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
==Table 18. Boolean Functions on Two Variables==
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | <math>F</math>
 +
| width="14%" | <math>F</math>
 +
| colspan="4" | <math>F(x, y)</math>
 +
| width="24%" | <math>F</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | &nbsp;
 +
| width="14%" | &nbsp;
 +
| width="12%" | <math>F(1, 1)</math>
 +
| width="12%" | <math>F(1, 0)</math>
 +
| width="12%" | <math>F(0, 1)</math>
 +
| width="12%" | <math>F(0, 0)</math>
 +
| width="24%" | &nbsp;
 +
|-
 +
| <math>F_{0}^{(2)}</math>
 +
| <math>F_{0000}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>\texttt{( )}</math>
 +
|-
 +
| <math>F_{1}^{(2)}</math>
 +
| <math>F_{0001}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
 +
|-
 +
| <math>F_{2}^{(2)}</math>
 +
| <math>F_{0010}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>\texttt{(} x \texttt{)} y</math>
 +
|-
 +
| <math>F_{3}^{(2)}</math>
 +
| <math>F_{0011}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} x \texttt{)}</math>
 +
|-
 +
| <math>F_{4}^{(2)}</math>
 +
| <math>F_{0100}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>x \texttt{(} y \texttt{)}</math>
 +
|-
 +
| <math>F_{5}^{(2)}</math>
 +
| <math>F_{0101}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} y \texttt{)}</math>
 +
|-
 +
| <math>F_{6}^{(2)}</math>
 +
| <math>F_{0110}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
 +
|-
 +
| <math>F_{7}^{(2)}</math>
 +
| <math>F_{0111}^{(2)}</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} x y \texttt{)}</math>
 +
|-
 +
| <math>F_{8}^{(2)}</math>
 +
| <math>F_{1000}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>x y</math>
 +
|-
 +
| <math>F_{9}^{(2)}</math>
 +
| <math>F_{1001}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
 +
|-
 +
| <math>F_{10}^{(2)}</math>
 +
| <math>F_{1010}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>y</math>
 +
|-
 +
| <math>F_{11}^{(2)}</math>
 +
| <math>F_{1011}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
 +
|-
 +
| <math>F_{12}^{(2)}</math>
 +
| <math>F_{1100}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>0</math>
 +
| <math>x</math>
 +
|-
 +
| <math>F_{13}^{(2)}</math>
 +
| <math>F_{1101}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>1</math>
 +
| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
 +
|-
 +
| <math>F_{14}^{(2)}</math>
 +
| <math>F_{1110}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>0</math>
 +
| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
 +
|-
 +
| <math>F_{15}^{(2)}</math>
 +
| <math>F_{1111}^{(2)}</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>1</math>
 +
| <math>\texttt{(( ))}</math>
 +
|}
 +
 
 +
<br>

Latest revision as of 12:14, 18 October 2025

Format Samples

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

Table 13. Algorithmic Translation Rules


\(\text{Table 13. Algorithmic Translation Rules}\)
\(\text{Sentence in PARCE}\) \(\xrightarrow{\mathrm{Parse}}\) \(\text{Graph in PARC}\)
\(\mathrm{Conc}^0\) \(\xrightarrow{\mathrm{Parse}}\) \(\mathrm{Node}^0\)
\(\mathrm{Conc}_{j=1}^k s_j\) \(\xrightarrow{\mathrm{Parse}}\) \(\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)\)
\(\mathrm{Surc}^0\) \(\xrightarrow{\mathrm{Parse}}\) \(\mathrm{Lobe}^0\)
\(\mathrm{Surc}_{j=1}^k s_j\) \(\xrightarrow{\mathrm{Parse}}\) \(\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)\)


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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