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User:Jon Awbrey/Cactus Language Semantics Work (view source)
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| width="20%" | <math>\mathrm{Node}^0</math>
| width="20%" | <math>\mathrm{Node}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\underline{1}</math>
| width="20%" | <math>1</math>
|-
|-
| width="20%" | <math>\mathrm{Conc}^k_j s_j</math>
| width="20%" | <math>\mathrm{Conc}^k_j s_j</math>
| width="20%" | <math>\mathrm{Lobe}^0</math>
| width="20%" | <math>\mathrm{Lobe}^0</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}</math>
| width="20%" | <math>\underline{0}</math>
| width="20%" | <math>0</math>
|-
|-
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
| width="20%" | <math>\mathrm{Surc}^k_j s_j</math>
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\underline{1}</math>
| width="20%" | <math>1</math>
|-
|-
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright</math>
| width="20%" | <math>=</math>
| width="20%" | <math>=</math>
| width="20%" | <math>\underline{0}</math>
| width="20%" | <math>0</math>
|-
|-
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright</math>
A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables. Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.</math>
A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables. Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.</math>
A boolean function <math>F^{(0)}</math> on <math>0</math> variables is just an element of the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math> Table 16 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.
A boolean function <math>F^{(0)}</math> on <math>0</math> variables is just an element of the boolean domain <math>\underline\mathbb{B} = \{ 0, 1 \}.</math> Table 16 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.
<br>
<br>
| width="24%" | <math>F</math>
| width="24%" | <math>F</math>
|-
|-
| <math>\underline{0}</math>
| <math>0</math>
| <math>F_0^{(0)}</math>
| <math>F_0^{(0)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
|-
|-
| <math>\underline{1}</math>
| <math>1</math>
| <math>F_1^{(0)}</math>
| <math>F_1^{(0)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}</math>
|}
|}
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
| width="24%" | <math>F(\underline{1})</math>
| width="24%" | <math>F(1)</math>
| width="24%" | <math>F(\underline{0})</math>
| width="24%" | <math>F(0)</math>
| width="24%" |
| width="24%" |
|-
|-
| <math>F_0^{(1)}</math>
| <math>F_0^{(1)}</math>
| <math>F_{00}^{(1)}</math>
| <math>F_{00}^{(1)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
|-
|-
| <math>F_1^{(1)}</math>
| <math>F_1^{(1)}</math>
| <math>F_{01}^{(1)}</math>
| <math>F_{01}^{(1)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)}</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
|-
| <math>F_2^{(1)}</math>
| <math>F_2^{(1)}</math>
| <math>F_{10}^{(1)}</math>
| <math>F_{10}^{(1)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>x</math>
| <math>x</math>
|-
|-
| <math>F_3^{(1)}</math>
| <math>F_3^{(1)}</math>
| <math>F_{11}^{(1)}</math>
| <math>F_{11}^{(1)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}</math>
|}
|}
F_{00}^{(1)}
F_{00}^{(1)}
& = &
& = &
0 ~:~ \underline\mathbb{B} \to \underline\mathbb{B}
\\
\\
\\
\\
F_{11}^{(1)}
F_{11}^{(1)}
& = &
& = &
1 ~:~ \underline\mathbb{B} \to \underline\mathbb{B}
\end{matrix}</math>
\end{matrix}</math>
|}
|}
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
| width="12%" | <math>F(\underline{1}, \underline{1})</math>
| width="12%" | <math>F(1, 1)</math>
| width="12%" | <math>F(\underline{1}, \underline{0})</math>
| width="12%" | <math>F(1, 0)</math>
| width="12%" | <math>F(\underline{0}, \underline{1})</math>
| width="12%" | <math>F(0, 1)</math>
| width="12%" | <math>F(\underline{0}, \underline{0})</math>
| width="12%" | <math>F(0, 0)</math>
| width="24%" |
| width="24%" |
|-
|-
| <math>F_{0}^{(2)}</math>
| <math>F_{0}^{(2)}</math>
| <math>F_{0000}^{(2)}</math>
| <math>F_{0000}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}</math>
|-
|-
| <math>F_{1}^{(2)}</math>
| <math>F_{1}^{(2)}</math>
| <math>F_{0001}^{(2)}</math>
| <math>F_{0001}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
| <math>\texttt{(} x \texttt{)(} y \texttt{)}</math>
|-
|-
| <math>F_{2}^{(2)}</math>
| <math>F_{2}^{(2)}</math>
| <math>F_{0010}^{(2)}</math>
| <math>F_{0010}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{(} x \texttt{)} y</math>
| <math>\texttt{(} x \texttt{)} y</math>
|-
|-
| <math>F_{3}^{(2)}</math>
| <math>F_{3}^{(2)}</math>
| <math>F_{0011}^{(2)}</math>
| <math>F_{0011}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{)}</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
|-
| <math>F_{4}^{(2)}</math>
| <math>F_{4}^{(2)}</math>
| <math>F_{0100}^{(2)}</math>
| <math>F_{0100}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>x \texttt{(} y \texttt{)}</math>
| <math>x \texttt{(} y \texttt{)}</math>
|-
|-
| <math>F_{5}^{(2)}</math>
| <math>F_{5}^{(2)}</math>
| <math>F_{0101}^{(2)}</math>
| <math>F_{0101}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} y \texttt{)}</math>
| <math>\texttt{(} y \texttt{)}</math>
|-
|-
| <math>F_{6}^{(2)}</math>
| <math>F_{6}^{(2)}</math>
| <math>F_{0110}^{(2)}</math>
| <math>F_{0110}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math>
|-
|-
| <math>F_{7}^{(2)}</math>
| <math>F_{7}^{(2)}</math>
| <math>F_{0111}^{(2)}</math>
| <math>F_{0111}^{(2)}</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} x y \texttt{)}</math>
| <math>\texttt{(} x y \texttt{)}</math>
|-
|-
| <math>F_{8}^{(2)}</math>
| <math>F_{8}^{(2)}</math>
| <math>F_{1000}^{(2)}</math>
| <math>F_{1000}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>x y</math>
| <math>x y</math>
|-
|-
| <math>F_{9}^{(2)}</math>
| <math>F_{9}^{(2)}</math>
| <math>F_{1001}^{(2)}</math>
| <math>F_{1001}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math>
|-
|-
| <math>F_{10}^{(2)}</math>
| <math>F_{10}^{(2)}</math>
| <math>F_{1010}^{(2)}</math>
| <math>F_{1010}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>y</math>
| <math>y</math>
|-
|-
| <math>F_{11}^{(2)}</math>
| <math>F_{11}^{(2)}</math>
| <math>F_{1011}^{(2)}</math>
| <math>F_{1011}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
| <math>\texttt{(} x \texttt{(} y \texttt{))}</math>
|-
|-
| <math>F_{12}^{(2)}</math>
| <math>F_{12}^{(2)}</math>
| <math>F_{1100}^{(2)}</math>
| <math>F_{1100}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>x</math>
| <math>x</math>
|-
|-
| <math>F_{13}^{(2)}</math>
| <math>F_{13}^{(2)}</math>
| <math>F_{1101}^{(2)}</math>
| <math>F_{1101}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
| <math>\texttt{((} x \texttt{)} y \texttt{)}</math>
|-
|-
| <math>F_{14}^{(2)}</math>
| <math>F_{14}^{(2)}</math>
| <math>F_{1110}^{(2)}</math>
| <math>F_{1110}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{0}</math>
| <math>0</math>
| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
| <math>\texttt{((} x \texttt{)(} y \texttt{))}</math>
|-
|-
| <math>F_{15}^{(2)}</math>
| <math>F_{15}^{(2)}</math>
| <math>F_{1111}^{(2)}</math>
| <math>F_{1111}^{(2)}</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\underline{1}</math>
| <math>1</math>
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}</math>
|}
|}
As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations. Just to acknowledge a few of the more notable pseudonyms:
As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations. Just to acknowledge a few of the more notable pseudonyms:
: The constant function <math>\underline{0} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math>
: The constant function <math>0 ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math>
: The constant function <math>\underline{1} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math>
: The constant function <math>1 ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math>
: The negation and identity of the first variable are <math>F_{3}^{(2)}</math> and <math>F_{12}^{(2)},</math> respectively.
: The negation and identity of the first variable are <math>F_{3}^{(2)}</math> and <math>F_{12}^{(2)},</math> respectively.