Cactus Language • Semantics 7

A good way to illustrate the action of the conjunction and surjunction operators is to show how they can be used to construct the boolean functions on any finite number of variables. Though it’s not much to look at let’s start with the case of zero variables, boolean constants by any other word, partly for completeness and partly to supply an anchor for the cases in its train.

A boolean function $F^{(0)}$ on zero variables is just an element of the boolean domain $\mathbb{B} = \{ 0, 1 \}.$ The 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.

$\text{Boolean Functions on Zero Variables}$

Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.
Column 2 lists each boolean function by means of a function name $F_j^{(k)}$ of the following form. The superscript $(k)$ gives the dimension of the functional domain, in effect, the number of variables, and the subscript $j$ is a binary string formed from the functional values, using the obvious coding of boolean values into binary values.
Column 3 lists the values each function takes for each combination of its domain values.
Column 4 lists the ordinary cactus expressions for each boolean function. Here, as usual, the expression $``\texttt{(( ))}"$ renders the blank expression for logical truth more visible in context.

The next Table shows the four boolean functions on one variable, $F^{(1)} : \mathbb{B} \to \mathbb{B}.$

$\text{Boolean Functions on One Variable}$

Column 1 lists the contents of Column 2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents. Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable. The constant functions are thus expressible in the following equivalent ways.

$\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & 0 ~:~ \mathbb{B} \to \mathbb{B}. \\[4pt] F_3^{(1)} & = & F_{11}^{(1)} & = & 1 ~:~ \mathbb{B} \to \mathbb{B}. \end{matrix}$

The other two functions in the Table are easily recognized as the one‑place logical connectives or the monadic operators on $\mathbb{B}.$ Thus the function $F_1^{(1)} = F_{01}^{(1)}$ is recognizable as the negation operation and the function $F_2^{(1)} = F_{10}^{(1)}$ is obviously the identity operation.

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