Generalized Results on Monoids as Memory

TL;DR

This paper extends the theory of monoid automata, showing new limitations and capabilities of these automata over various classes of monoids, and relates them to context-free languages and valence grammars.

Contribution

It generalizes previous results to broader classes of monoids, demonstrating new properties of rational monoid automata and their relation to context-free languages.

Findings

Certain languages cannot be recognized by rational monoid automata over specific monoids.

The class of languages recognized by these automata forms a semi-linear full trio.

Valence pushdown automata are equivalent in power to finite valence automata.

Abstract

We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and prove that they are only as powerful as (finite) valence automata. We observe that certain results proven for monoid automata can be easily lifted to the case of context-free valence grammars.