From Post Systems to the Reachability Problems for Matrix Semigroups and Multicounter Automata

Abstract. The main result of this paper is the reduction of PCP(n) to the vector reachability problem for a matrix semigroup generated by n 4 \Theta 4 integral matrices. It follows that the vector reachability problem is undecidable for a semigroup generated by 7 integral matrices of dimension 4. The question whether the vector reachability problem is decidable for n=2 and n=3 remains open. Also we show that proposed technique can be applied to Post's tag-systems. As a result we define new classes of counter automata that lie on the border between decidability and undecidability. 1 Introduction In this paper we show the connection between decision problems for Post systems and the reachability problems for matrix semigroups and counter automata. We start from the vector reachability problem for a matrix semigroup, which is a generalisation of the orbit problem [12]. The vector reachability problem is formulated as follows: "Let S be a given finitely generated semigroup of n \Theta n matrices from