Regular formula

A regular formula is a formula R such that $R\linequiv\wn\oc R$.

A formula L is co-regular if its dual $L\orth$ is regular, that is if $L\linequiv\oc\wn L$.

Alternative characterization

R is regular if and only if it is equivalent to a formula of the shape $\wn P$ for some positive formula P.

Proof.  If R is regular then $R\linequiv\wn\oc R$ with $\oc R$ positive. If $R\linequiv\wn P$ with P positive then R is regular since $P\linequiv\oc P$.

Regular connectives

A connective c of arity n is regular if for any regular formulas R₁,...,R_n, $c(R_1,\dots,R_n)$ is regular.

Proposition (Regular connectives)

$\parr$, $\bot$ and $\wn\oc$ define regular connectives.

Proof.  If R and S are regular, $R\parr S \linequiv \wn\oc R \parr \wn\oc S \linequiv \wn{(\oc R\plus\oc S)}$ thus it is regular since $\oc R\plus\oc S$ is positive.

$\bot\linequiv\wn\zero$ thus it is regular since $\zero$ is positive.

If R is regular then $\wn\oc R$ is regular, since $\wn\oc\wn\oc R\linequiv \wn\oc R$.

More generally, $\wn\oc A$ is regular for any formula A.