Commutative Regular Languages with Product-Form Minimal Automata

TL;DR

This paper defines a new subclass of commutative regular languages with product-form minimal automata, providing bounds on state complexity for various operations and exploring related language classes in partial commutativity.

Contribution

It introduces a novel subclass of commutative regular languages characterized by Cartesian product automata and analyzes their complexity bounds and relationships.

Findings

Bound of 2nm for shuffle operation complexity.

Bound of n for upward and downward closure operations.

Characterizations and construction methods for the new class.

Abstract

We introduce a subclass of the commutative regular languages that is characterized by the property that the state set of the minimal deterministic automaton can be written as a certain Cartesian product. This class behaves much better with respect to the state complexity of the shuffle, for which we find the bound~$2nm$ if the input languages have state complexities $n$ and $m$, and the upward and downward closure and interior operations, for which we find the bound~$n$. In general, only the bounds $(2nm)^{|\Sigma|}$ and $n^{|\Sigma|}$ are known for these operations in the commutative case. We prove different characterizations of this class and present results to construct languages from this class. Lastly, in a slightly more general setting of partial commutativity, we introduce other, related, language classes and investigate the relations between them.