Two-variable descriptions of regularity

Abstract

We prove that the class of all languages that are definable in /spl Sigma//sub 1//sup 1/(FO/sup 2/), that is, in (non-monadic) existential second-order logic with only two first-order variables, coincides with the regular languages. This provides an alternative logical description of regularity to both the traditional one in terms of monadic second-order logic, due to Buchi and Trakhtenbrot, and the more recent ones in terms of prefix fragments of /spl Sigma//sub 1//sup 1/, due to Eiter, Gottlob and Gurevich. Our result extends to more general settings than words. Indeed, definability in /spl Sigma//sub 1//sup 1/(FO/sup 2/) coincides with recognizability by appropriate notions of automata on a large class of objects, including /spl omega/-words, trees, pictures and, more generally, all weakly deterministic, triangle-free transition systems.