The Identity Problem for Matrix Semigroups in SL₂(ℤ) is NP-complete

Abstract

In this paper, 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 PSL 2 (Z) and thus the Special Linear group SL 2 (Z) is solvable in NP. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from SL 2 (Z) or PSL 2 (Z) generates a group or free semigroup is also decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet {s, r} and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in SL 2 (Z) are NP-complete.

Matrix Semigroup Freeness Problems in SL $$(2,\mathbb {Z})$$

“…The freeness problem was already known to be decidable in SL(2, Z). Here we have shown the that the freeness problem in SL(2, Z) is NP-hard which, along with the fact that the problem is in NP [2], proves that the freeness problem in SL(2, Z) is NP-complete. We also have studied a relaxed variant called the finite freeness problem in which we decide whether or not every matrix in the semigroup has a finite number of factorizations.…”

Decidability of the Membership Problem for 2 × 2 integer matrices

“…As we have shown, ϕ(w) D ∈ GL(2, Z) and hence w is in L H (2) . Let u 0 , u 1 , u 1 , u 2 , u 0 be an accepting run 2 of A H(2) on w. Now, we rewrite ϕ(RSRS ) as follows…”

“…Recently a simpler case of the membership of the identity matrix in SL(2, Z) was shown to be NPcomplete [2]. However we do not know what is the exact complexity of the membership problem for nonsingular matrices from Z 2×2 .…”

On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond

“…A large number of naturally defined matrix problems are still unanswered, despite the long history of matrix theory. Some of these questions have recently drawn renewed interest in the context of the analysis of digital processes, verification problems, and links with several fundamental questions in mathematics [11,7,37,39,38,35,17,14,15,36,5,41,27]. One of these challenging problems is the Mortality Problem of whether the zero matrix belongs to a finitely generated matrix semigroup.…”