Periodic and Infinite Traces in Matrix Semigroups

In this paper we provide several new results concerning word and matrix semigroup problems using counter automaton models. As a main result, we prove a new version of Post’s correspondence problem to be undecidable and show its application to matrix semigroup problems, such as Any Diagonal Matrix Problem and Recurrent Matrix Problem. We also use infinite periodic traces in counter automaton models to show the undecidability of a new variation of the Infinite Post Correspondence Problem and Vector Ambiguity Problem for matrix semigroups.