The membership problem for subsemigroups of [formula omitted] is NP-complete

Abstract

We show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2×2 matrices from the modular group PSL2(Z), the Special Linear group SL2(Z) and the General Linear Group GL2(Z) is solvable in NP. We extend this to prove that the membership problem is decidable in NP for GL2(Z) and for any arbitrary regular expression over matrices from SL2(Z). We then derive that the problems of whether a given finite set of matrices from SL2(Z) or PSL2(Z) generates a group or a free semigroup are both decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE. Our algorithm is based on new techniques allowing us to operate on compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the identity (and thus membership) problem over GL2(Z) is NP-complete, and the group problem and the non-freeness problem in SL2(Z) are NP-complete. Thus the paper answer the long standing open question on the complexity of the membership problem in semigroups generated by matrices from GL2(Z). We develop novel techniques that can be used for solving numerical matrix problems in symbolic form, which are applicable for solving compressed word problems for groups and semigroups, bridging the gap between combinatorial group theory, computational problems on matrices and complexity theory.1