Minimal Negation Operators

It will take a few more rounds of stage-setting before we are able to entertain concrete examples of applications but the following may indicate the direction of generalization embodied in minimal negation operators.

To begin, let’s observe two ways of generalizing the logical operation commonly known as exclusive disjunction $(\textsc{xor})$ or symmetric difference $(\Delta).$

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

Exclusive disjunction is a boolean function $\Delta : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ isomorphic to the algebraic field addition $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ also known as addition mod 2. Adding the language of minimal negation operators to the mix we have the following equivalent expressions.

$\begin{matrix} \textsc{xor}(p, q) & = & \Delta (p, q) & = & p + q & = & \nu (p, q) & = & \texttt{(} p \texttt{,} q \texttt{)} \end{matrix}$

Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Parity Indicator

Generalizing the function $p + q$ of two variables to more variables extends the sequence of functions in the fashion $p\!+\!q\!+\!r,$ $p\!+\!q\!+\!r\!+\!s,$ $p\!+\!q\!+\!r\!+\!s\!+\!t,$ and so on. These are known as parity sums, returning a value of $0$ when there are an even number of $1$ ’s in the sum and returning a value of $1$ when there are an odd number of $1$ ’s in the sum.

Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Border Indicator

The equivalent expressions $\texttt{(} p \texttt{,} q \texttt{)} = \nu(p, q) = p + q = p \,\Delta\, q = p ~\textsc{xor}~ q$ may be read with a different connotation, indicating the venn diagram cells adjacent to the conjunction $p \land q.$ Generalizing the function $\texttt{(} p \texttt{,} q \texttt{)}$ of two variables to more variables extends the sequence of functions in the fashion $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)},$ $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{)},$ $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{,} t \texttt{)},$ and so on. That sequence of operators differs from the sequence of parity sums once it passes the 2-variable case.

The triple sum may be written in terms of 2-place minimal negations as follows.

$\begin{matrix} p + q + r & = & \texttt{((} p \texttt{,} q \texttt{)}\!\texttt{,} r \texttt{)} & = & \texttt{(} p \texttt{,} \texttt{(} q \texttt{,} r \texttt{))} \end{matrix}$

It is important to recognize the triple sum expressions and the 3-place minimal negation $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}$ have very different meanings.

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