Since the undecidability of the mortality problem for 3 × 3 matrices over integers was proved using the Post Correspondence Problem, various studies on decision problems of matrix semigroups have emerged. The freeness problem in particular has received much attention but decidability remains open even for 2 × 2 upper triangular matrices over nonnegative integers. Parikh matrices are upper triangular matrices introduced as a generalization of Parikh vectors and have become useful tools in studying of subword occurrences. In this work, we focus on semigroups of Parikh matrices and study the freeness problem in this context.

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.