Parikh matrices and Parikh Rewriting Systems

TL;DR

This paper introduces Parikh rewriting systems, generalizing Salomaa's approach, to address the injectivity problem of Parikh matrix mappings by deriving counter-free Thue systems.

Contribution

It proposes a new framework called Parikh rewriting systems that systematically generalizes previous solutions to the injectivity problem.

Findings

Every Parikh rewriting system induces a counter-free Thue system.

The framework provides a systematic approach to the injectivity problem.

It extends Salomaa's 2010 solution for the ternary alphabet.

Abstract

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Parikh rewriting system as a generalization and systematization of Salomaa's result. It will be shown that every Parikh rewriting system induces a Thue system without counters that serves as a feasible solution to the injectivity problem.