Finite semigroup varieties defined by programs

Abstract

We study the regular languages recognized by polynomial-length programs over finite semigroups belonging to product varieties V ∗ LI, where V is a variety of finite monoids, and LI is the variety of finite locally trivial semigroups. In the case where the semigroup variety has a particular closure property with respect to programs, we are able to give precise characterizations of these regular languages. As a corollary we obtain new proofs of the results of Barrington, Compton, Straubing and Therien characterizing the regular languages in certain circuit complexity classes.