摘要：We consider the membership problem for matrix semigroups, which is the problem to decide whether a matrix belongs to a given finitely generated matrix semigroup. In general, the decidability and complexity of this problem for two-dimensional matrix semigroups remains open. Recently there was a significant progress with this open problem by showing that the membership is decidable for 2x2 nonsingular integer matrices. In this paper we focus on the membership for singular integer matrices and prove that this problem is decidable for 2x2 integer matrices whose determinants are equal to 0, 1, -1 (i.e. for matrices from GL(2,Z) and any singular matrices). Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words and conversion of the membership problem into decision problem on regular languages.
关键词：Matrix Semigroups; Membership Problem; General Linear Group; Singular Matrices; Automata and Formal Languages
