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User:Jon Awbrey/Cactus Language Semantics Work (view source)

Revision as of 12:54, 2 November 2025

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→‎Cactus Language • Semantics: replace wiki table with png File:Boolean Functions on Two Variables • Truth Table.png
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
 
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
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| style="height:25px; font-size:large" | <math>\text{Table 14. Semantic Translation}</math> &bull; <math>\text{Functional Form}</math>
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|+ style="height:25px; font-size:large" | <math>\text{Table 14. Semantic Translation}</math> &bull; <math>\text{Functional Form}</math>
 
|-
 
|-
 
| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]
 
| [[File:Cactus Language Semantic Translation Functional Form.png|600px]]
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
 
{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
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| style="height:25px; font-size:large" | <math>\text{Table 15. Semantic Translation}</math> &bull; <math>\text{Equational Form}</math>
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|+ style="height:25px; font-size:large" | <math>\text{Table 15. Semantic Translation}</math> &bull; <math>\text{Equational Form}</math>
 
|-
 
|-
 
| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]
 
| [[File:Cactus Language Semantic Translation Equational Form.png|600px]]
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A boolean function <math>F^{(0)}</math> on zero variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math>&nbsp; The&nbsp;following Table shows several ways of referring to those elements, for the sake of consistency using the same format we'll use in subsequent Tables, however degenerate it appears in this case.
 
A boolean function <math>F^{(0)}</math> on zero variables is just an element of the boolean domain <math>\mathbb{B} = \{ 0, 1 \}.</math>&nbsp; The&nbsp;following Table shows several ways of referring to those elements, for the sake of consistency using the same format we'll use in subsequent Tables, however degenerate it appears in this case.
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<br>
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
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|+ style="height:25px; font-size:large" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
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|+ style="height:30px" | <math>\text{Table 16. Boolean Functions on Zero Variables}</math>
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|- style="height:40px; background:ghostwhite"
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| width="14%" | <math>F</math>
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| width="14%" | <math>F</math>
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| width="48%" | <math>F()</math>
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| width="24%" | <math>F</math>
   
|-
 
|-
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| <math>0</math>
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| [[File:Boolean Functions on Zero Variables.png|600px]]
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| <math>F_0^{(0)}</math>
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| <math>0</math>
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| <math>\texttt{( )}</math>
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|-
  −
| <math>1</math>
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| <math>F_1^{(0)}</math>
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| <math>1</math>
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| <math>\texttt{(( ))}</math>
   
|}
 
|}
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  −
<br>
      
<ul>
 
<ul>
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The next Table shows the four boolean functions on one variable, <math>F^{(1)} : \mathbb{B} \to \mathbb{B}.</math>
 
The next Table shows the four boolean functions on one variable, <math>F^{(1)} : \mathbb{B} \to \mathbb{B}.</math>
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<br>
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
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+
|+ style="height:25px; font-size:large" | <math>\text{Table 17. Boolean Functions on One Variable}</math>
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:60%"
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|+ style="height:30px" | <math>\text{Table 17. Boolean Functions on One Variable}</math>
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|- style="height:40px; background:ghostwhite"
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| width="14%" | <math>F</math>
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| width="14%" | <math>F</math>
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| colspan="2" | <math>F(x)</math>
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| width="24%" | <math>F</math>
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|- style="height:40px; background:ghostwhite"
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| width="14%" | &nbsp;
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| width="14%" | &nbsp;
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| width="24%" | <math>F(1)</math>
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| width="24%" | <math>F(0)</math>
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| width="24%" | &nbsp;
   
|-
 
|-
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| <math>F_0^{(1)}</math>
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| [[File:Boolean Functions on One Variable.png|600px]]
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| <math>F_{00}^{(1)}</math>
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| <math>0</math>
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| <math>0</math>
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| <math>\texttt{( )}</math>
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|-
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| <math>F_1^{(1)}</math>
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| <math>F_{01}^{(1)}</math>
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| <math>0</math>
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| <math>1</math>
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| <math>\texttt{(} x \texttt{)}</math>
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|-
  −
| <math>F_2^{(1)}</math>
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| <math>F_{10}^{(1)}</math>
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| <math>1</math>
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| <math>0</math>
  −
| <math>x</math>
  −
|-
  −
| <math>F_3^{(1)}</math>
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| <math>F_{11}^{(1)}</math>
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| <math>1</math>
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| <math>1</math>
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| <math>\texttt{(( ))}</math>
   
|}
 
|}
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<br>
      
<ul><li>Column&nbsp;1 lists the contents of Column&nbsp;2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.&nbsp; Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.&nbsp; The constant functions are thus expressible in the following equivalent ways.</li></ul>
 
<ul><li>Column&nbsp;1 lists the contents of Column&nbsp;2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.&nbsp; Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.&nbsp; The constant functions are thus expressible in the following equivalent ways.</li></ul>
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The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table.
 
The 16 boolean functions on two variables, <math>F^{(2)} : \mathbb{B}^2 \to \mathbb{B},</math> are shown in the following Table.
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<br>
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"
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|+ style="height:25px; font-size:large" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:60%"
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|+ style="height:30px" | <math>\text{Table 18. Boolean Functions on Two Variables}</math>
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|- style="height:40px; background:ghostwhite"
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| width="14%" | <math>F</math>
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| width="14%" | <math>F</math>
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| colspan="4" | <math>F(x, y)</math>
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| width="24%" | <math>F</math>
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|- style="height:40px; background:ghostwhite"
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| width="14%" | &nbsp;
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| width="14%" | &nbsp;
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| width="12%" | <math>F(1, 1)</math>
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| width="12%" | <math>F(1, 0)</math>
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| width="12%" | <math>F(0, 1)</math>
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| width="12%" | <math>F(0, 0)</math>
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| width="24%" | &nbsp;
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|-
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| <math>F_{0}^{(2)}</math>
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| <math>F_{0000}^{(2)}</math>
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| <math>0</math>
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| <math>0</math>
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| <math>0</math>
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| <math>0</math>
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| <math>\texttt{( )}</math>
   
|-
 
|-
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| <math>F_{1}^{(2)}</math>
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| [[File:Boolean Functions on Two Variables &bull; Truth Table.png|600px]]
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| <math>F_{0001}^{(2)}</math>
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| <math>0</math>
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| <math>0</math>
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| <math>0</math>
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| <math>1</math>
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| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
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|-
  −
| <math>F_{2}^{(2)}</math>
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| <math>F_{0010}^{(2)}</math>
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| <math>0</math>
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| <math>0</math>
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| <math>1</math>
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| <math>0</math>
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| <math>\texttt{(} x \texttt{)} y</math>
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|-
  −
| <math>F_{3}^{(2)}</math>
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| <math>F_{0011}^{(2)}</math>
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| <math>0</math>
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| <math>0</math>
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| <math>1</math>
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| <math>1</math>
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| <math>\texttt{(} x \texttt{)}</math>
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|-
  −
| <math>F_{4}^{(2)}</math>
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| <math>F_{0100}^{(2)}</math>
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| <math>0</math>
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| <math>1</math>
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| <math>0</math>
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| <math>0</math>
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| <math>x \texttt{(} y \texttt{)}</math>
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|-
  −
| <math>F_{5}^{(2)}</math>
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| <math>F_{0101}^{(2)}</math>
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| <math>0</math>
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| <math>1</math>
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| <math>0</math>
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| <math>1</math>
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| <math>\texttt{(} y \texttt{)}</math>
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|-
  −
| <math>F_{6}^{(2)}</math>
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| <math>F_{0110}^{(2)}</math>
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| <math>0</math>
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| <math>1</math>
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| <math>1</math>
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| <math>0</math>
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| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
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|-
  −
| <math>F_{7}^{(2)}</math>
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| <math>F_{0111}^{(2)}</math>
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| <math>0</math>
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| <math>1</math>
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| <math>1</math>
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| <math>1</math>
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| <math>\texttt{(} x y \texttt{)}</math>
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|-
  −
| <math>F_{8}^{(2)}</math>
  −
| <math>F_{1000}^{(2)}</math>
  −
| <math>1</math>
  −
| <math>0</math>
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| <math>0</math>
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| <math>0</math>
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| <math>x y</math>
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|-
  −
| <math>F_{9}^{(2)}</math>
  −
| <math>F_{1001}^{(2)}</math>
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| <math>1</math>
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| <math>0</math>
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| <math>0</math>
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| <math>1</math>
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| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
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|-
  −
| <math>F_{10}^{(2)}</math>
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| <math>F_{1010}^{(2)}</math>
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| <math>1</math>
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| <math>0</math>
  −
| <math>1</math>
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| <math>0</math>
  −
| <math>y</math>
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|-
  −
| <math>F_{11}^{(2)}</math>
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| <math>F_{1011}^{(2)}</math>
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| <math>1</math>
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| <math>0</math>
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| <math>1</math>
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| <math>1</math>
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| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
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|-
  −
| <math>F_{12}^{(2)}</math>
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| <math>F_{1100}^{(2)}</math>
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| <math>1</math>
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| <math>1</math>
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| <math>0</math>
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| <math>0</math>
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| <math>x</math>
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|-
  −
| <math>F_{13}^{(2)}</math>
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| <math>F_{1101}^{(2)}</math>
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| <math>1</math>
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| <math>1</math>
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| <math>0</math>
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| <math>1</math>
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| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
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|-
  −
| <math>F_{14}^{(2)}</math>
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| <math>F_{1110}^{(2)}</math>
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| <math>1</math>
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| <math>1</math>
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| <math>1</math>
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| <math>0</math>
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| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
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|-
  −
| <math>F_{15}^{(2)}</math>
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| <math>F_{1111}^{(2)}</math>
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| <math>1</math>
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| <math>1</math>
  −
| <math>1</math>
  −
| <math>1</math>
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| <math>\texttt{(( ))}</math>
   
|}
 
|}
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<br>
      
As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand.&nbsp; In particular, we have the following inclusions.
 
As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand.&nbsp; In particular, we have the following inclusions.
Jon Awbrey
12,154

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