Formal Languages and Groups as Memory

Abstract

We present an exposition of the theory of M -automata and G-automata, or finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schützenberger from formal language theory.

Towards a Coalgebraic Chomsky Hierarchy

“…In this form it is rather similar to another example of a machine previously studied in the literature (e.g. [28,16]) and which we also can formalize as a T-automaton. We present the corresponding algebraic theories side by side.…”

“…Contrasting [16] we cannot replace the polycyclic monoid in Example 7.2 by a free group and conclude by Theorem 7.3 that automata with free group memory recognize context-free languages. But as shown in [16] the latter is true once internal transitions are allowed.…”

On the rational subset problem for groups

“…We show that if G is a finitely generated group and M a finitely generated monoid, then the rational subset problem for G × M is decidable exactly if the subsets of M defined by G-automata have uniformly decidable membership problem. This combines with a group-theoretic interpretation [25] of a well-known theorem of Chomsky and Schützenberger [9] to give a characterisation of rational subset membership in direct products of the form F × M with F a free group, in terms of the uniform decidability of membership for context-free subsets of M.…”

“…By [25,Theorem 7], which is essentially a group-theoretic restatement of the Chomsky-Schützenberger theorem [9], the languages accepted by F -automata are exactly the context-free languages. Combining with Theorem 6.1, we immediately obtain the following corollary, where F n denotes a free group of rank n. Corollary 6.3.…”

“…It follows, in particular, that it passes through amalgamated free products over finite subgroups, and HNN extensions with finite associated subgroups. Section 6 introduces a connection between decidability of rational subsets and the theory of G-automata (see, for example, [17,25]), that is, rational transductions of group word problems. Specifically, we show that a direct product of the form G × M has decidable rational subset membership exactly if membership is (uniformly) decidable for G-automaton subsets of M. This combines with a group-theoretic interpretation [25] of a theorem of Chomsky and Schützen-berger [9] and some classical results on commutative monoids [14,36] to show that any direct product of a free group with an abelian group (or commutative monoid) has decidable rational subset membership.…”