## Alpha Now, Omega Later : 6

In the late 1970s a number of problems in combinatorics and graph theory that I really wanted to know the answers to had driven me to the desperate measures of trying to write a theorem-proving program to help with the work.  Being familiar with the conceptual efficiencies of Peirce’s logical graphs and inspired by George Spencer Brown’s more recent resurrection of Peirce’s ideas, I naturally turned to those resources for the initial implements of my computational prospecting. The succession of computers and programming languages that I quested with over the years taught me a lot about the things that work and the things that do not. Dissolving for now to the present scene, I will use the next few posts to outline, as succinctly as I can, the basic constructs that developed through that line of inquiry.

### Minimal Negation Operators and Painted Cacti

Let $\mathbb{B} = \{ 0, 1 \}.$

The objects of penultimate interest are the boolean functions $f : \mathbb{B}^n \to \mathbb{B}$ for $n \in \mathbb{N}.$

A minimal negation operator $\nu_k$ for $k \in \mathbb{N}$ is a boolean function $\nu_k : \mathbb{B}^k \to \mathbb{B}$ defined as follows:

• $\nu_0 = 0.$
• $\nu_k (x_1, \ldots, x_k) = 1$ if and only if exactly one of the arguments $x_j$ equals $0.$

The first few of these operators are already enough to generate all boolean functions $f : \mathbb{B}^n \to \mathbb{B}$ via functional composition but the rest of the family is worth keeping around for many practical purposes.

In most contexts $\nu (x_1, \ldots, x_k)$ may be written for $\nu_k (x_1, \ldots, x_k)$ since the number of arguments determines the rank of the operator.  In some contexts even the letter $\nu$ may be omitted, writing just the argument list $(x_1, \ldots, x_k),$ in which case it helps to use a distinctive typeface for the list delimiters, as $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}.$

A logical conjunction of $k$ arguments can be expressed in terms of minimal negation operators as $\nu_{k+1} (x_1, x_2, \ldots, x_{k-1}, x_k, 0)$ and this is conveniently abbreviated as a concatenation of arguments $x_1 x_2 \ldots x_{k-1} x_k.$