Some decision problems on integer matrices

Choffrut, Christian; Karhumäki, Juhani

doi:10.1051/ita:2005007

Cited by 35 publications

(58 citation statements)

References 13 publications

(12 reference statements)

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“…Bounding the dimension of the ambient vector space could also yield decidability, which has been partly accomplished in the discrete case in (Choffrut and Karhumäki 2005). Finally, upper bounding the complexity of our decision procedure for the commutative case would also be a worthwhile task.…”

Section: Theorem 16 the Matrix-exponential Semigroup Problem Is Undementioning

confidence: 99%

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Solvability of Matrix-Exponential Equations

Ouaknine

Pouly

Sousa-Pinto

et al. 2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider a continuous analogue of (Babai et al. 1996) 's and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, . . . , A k , C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, . . . , t k such thatWe show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, . . . , A k commute. Our results have applications to reachability problems for linear hybrid automata.Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the MinkowskiWeyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.

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Section: Theorem 16 the Matrix-exponential Semigroup Problem Is Undementioning

confidence: 99%

“…See (Halava 1997) for a relevant survey, and (Choffrut and Karhumäki 2005) for some interesting related problems.…”

Section: Definition 2 (Solvability Of Multiplicative Matrix Equations)mentioning

confidence: 99%

Solvability of Matrix-Exponential Equations

Ouaknine

Pouly

Sousa-Pinto

et al. 2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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“…In contrast to the Mortality(3 × 3), which was one of the first matrix problems, shown to be undecidable several decades ago, the Identity Problem 4 was shown to be undecidable for dimension 4 only a few years ago [2]. Moreover it has been recently proven in [9] that the Identity Problem for integral matrices of dimension 2 is decidable and later in [3] that the problem for SL 2 (Z) is NP-hard. The NPhardness result of this paper about the Mortality(2 × 2) corresponds very well with the Identity situation.…”

Section: Introductionmentioning

confidence: 99%

“…The mortality problem was shown to be decidable for a pair of rational 2 × 2 matrices in [7]. Also, it was recently shown in [12] that the Mortality Problem is decidable for any set of 2 × 2 integer matrices whose determinants assume the values 0, ±1, by adapting a technique from [9]. The main goal of this paper is to show that the Mortality Problem for the same set of 2 × 2 integer matrices (whose determinants assume the values 0, ±1) is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

Mortality for 2 ×2 Matrices Is NP-Hard

Bell

Hirvensalo

Potapov

2012

Mathematical Foundations of Computer Science 2012

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Abstract. We study the computational complexity of determining whether the zero matrix belongs to a finitely generated semigroup of two dimensional integer matrices (the mortality problem). We show that this problem is NP-hard to decide in the two-dimensional case by using a new encoding and properties of the projective special linear group. The decidability of the mortality problem in two dimensions remains a long standing open problem although in dimension three is known to be undecidable as was shown by Paterson in 1970. We also show a lower bound on the minimum length solution to the Mortality Problem, which is exponential in the number of matrices of the generator set and the maximal element of the matrices.

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“…There are many open problems related to freeness in 2×2 matrices, see [8][9][10] for good surveys. The freeness problem over H 2×2 is undecidable [4], where H is the skew-field of quaternions (in fact the result even holds when all entries of the quaternions are rationals).…”

Section: Introductionmentioning

confidence: 99%