Commutative Languages and their Composition by Consensual Methods

TL;DR

This paper explores the properties of commutative languages with the semilinear property, demonstrating how unions and concatenations can be recognized using consensual methods and introducing new techniques for language composition.

Contribution

It extends the family of consensually regular languages by providing new conditions for language closure under union and concatenation, especially for commutative SLIP languages.

Findings

Unions and concatenations of SLIP languages are recognizable by multi-counter machines.

New conditions for closure of CREG languages are established.

Arithmetic congruences are used as language signatures for composition.

Abstract

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.