Decidability of the Membership Problem for 2 <b>×</b> 2 integer matrices

Potapov, Igor; Semukhin, Pavel

doi:10.1137/1.9781611974782.12

Cited by 23 publications

(27 citation statements)

References 24 publications

(39 reference statements)

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“…For example, even the decidability of the freeness problem for 2 × 2 matrices over natural numbers still remains a long-standing open problem [5]. Recently progress was made to show the decidability of the vector reachability problem for SL 2 (Z), see [21] and the decidability of the membership problem for non-singular integer 2 × 2 matrices see [22]. However, the exact complexity of these problems is not yet known.…”

Section: Resultsmentioning

confidence: 99%

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

Bell¹,

Hirvensalo²,

Potapov³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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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.

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Section: Resultsmentioning

confidence: 99%

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

Bell¹,

Hirvensalo²,

Potapov³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…It would be interesting to look at possibly decidable restrictions of Matrix-Exponential Semigroup Membership Problem: For example, the case with two non-commuting generators, which was shown to be decidable in the case of the analogous discrete problem in Reference [5]. Bounding the dimension of the ambient vector space might also yield decidability, along the lines of results in the discrete case in Reference [24]. Finally, deriving upper and lower bounds for the computational complexity of the commutative case remains to be addressed.…”

Section: Undecidability Of the Semigroup Membership Problemmentioning

confidence: 99%

On the Decidability of Membership in Matrix-exponential Semigroups

Ouaknine

Pouly

Sousa-Pinto

et al. 2019

J. ACM

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We consider the decidability of the membership problem for matrix-exponential semigroups: Given k ∈ N and square matrices A 1 ,. .. , A k , C, all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp(A i t), where i ∈ {1,. .. , k } and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A 1 ,. .. , A k commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker's theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert's Tenth Problem.

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“…In this framework, primitivity can be also rephrased as a membership problem (see e.g. [24,26]), where we ask whether the all-ones matrix belongs to the semigroup generated by the matrix set. The following example reports a primitive set M of NZ matrices and the synchronizing automata Aut(M) and Aut(M T ).…”

Section: Introductionmentioning

confidence: 99%

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

Azfar

Catalano

Charlier

et al. 2019

Developments in Language Theory

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A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods.Keywords: Primitive set of matrices, synchronizing automaton, Černý conjecture.Deciding whether a set is primitive is an NP-hard problem if the set contains at least three matrices (while it is still of unknown complexity for sets of two matrices) [3], and so is computing its exponent. For sets of matrices having at least a positive entry in every row and every column (called NZ [13] or allowable matrices [16,18]), the primitivity problem becomes decidable in polynomial-time [28], although computing the exponent remains NP-hard [13]. Methods for approximating the exponent have been proposed [6] as well as upper bounds that depend just on the matrix size; in particular, if we denote with exp N Z (n) the maximal exponent among all the primitive sets of n × n NZ matrices, it is known that exp N Z (n) ≤ (15617n 3 + 7500n 2 + 56250n − 78125)/46875 [3,33]. Better upper bounds have been found for some classes of primitive sets [13,17]. The NZ condition is often met in applications and in particular in the connection with synchronizing automata.

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Decidability of the Membership Problem for 2 × 2 integer matrices

Cited by 23 publications

References 24 publications

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

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

On the Decidability of Membership in Matrix-exponential Semigroups

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

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Decidability of the Membership Problem for 2 × 2 integer matrices

Cited by 23 publications

References 24 publications

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

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

On the Decidability of Membership in Matrix-exponential Semigroups

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

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Product

Resources

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

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