Parallel Hyperedge Replacement String Languages

TL;DR

This paper introduces a new class of formal languages called parallel hyperedge replacement string languages, which encompasses multiple context-free and ET0L languages, aiming to classify the word problems of hyperbolic groups within this hierarchy.

Contribution

It defines a novel language class and establishes its relation to existing classes, providing a foundation for future classification of hyperbolic group word problems.

Findings

Contains all multiple context-free languages

Includes all ET0L languages

Provides groundwork for classifying hyperbolic group word problems

Abstract

There are many open questions surrounding the characterisation of groups with context-sensitive word problem. Only in 2018 was it shown that all finitely generated virtually Abelian groups have multiple context-free word problems, and it is a long-standing open question as to where to place the word problems of hyperbolic groups in the formal language hierarchy. In this paper, we introduce a new language class called the parallel hyperedge replacement string languages, show that it contains all multiple context-free and ET0L languages, and lay down the foundations for future work that may be able to place the word problems of many hyperbolic groups in this class.