Results 1  10
of
15
Undecidable Problems for Probabilistic Automata of Fixed Dimension
 Theory of Computing Systems
, 2001
"... We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matrices ..."
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Cited by 35 (5 self)
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We prove that several problems associated to probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 × 47 matrices with nonnegative rational entries is bounded is undecidable.
Complexity of Stability and Controllability of Elementary Hybrid Systems
, 1997
"... this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into tw ..."
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Cited by 34 (10 self)
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this paper, weconsider simple classes of nonlinear systems and provethatbasic questions related to their stabilityandcontrollabilityare either undecidable or computationally intractable (NPhard). As a special case, weconsider a class of hybrid systems in which the state space is partitioned into two halfspaces, and the dynamics in eachhalfspace correspond to a differentlinear system
When is a Pair of Matrices Mortal?
, 1996
"... A set of matrices over the integers is said to be lengthkmortal (with positive integer) if the zero matrix can be expressed as a product of length of matrices in the set. The set is said to be mortal if it is lengthkmortal for some finite k. ..."
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Cited by 25 (12 self)
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A set of matrices over the integers is said to be lengthkmortal (with positive integer) if the zero matrix can be expressed as a product of length of matrices in the set. The set is said to be mortal if it is lengthkmortal for some finite k.
Skolem's Problem  On the Border between Decidability and Undecidability
, 2005
"... We give a survey of Skolem’s problem for linear recurrence sequences. We cover the known decidable cases for recurrence depths of at most 4, and give detailed proofs for these cases. Moreover, we shall prove that the problem is decidable for linear recurrences of depth 5. ..."
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Cited by 14 (1 self)
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We give a survey of Skolem’s problem for linear recurrence sequences. We cover the known decidable cases for recurrence depths of at most 4, and give detailed proofs for these cases. Moreover, we shall prove that the problem is decidable for linear recurrences of depth 5.
Multiplicative Equations Over Commuting Matrices
, 1995
"... We consider the solvability of the equation k Y i=1 A i x i = B and generalizations, where the A i and B are given commuting matrices over an algebraic number field F . In the semigroup membership problem, the variables x i are constrained to be nonnegative integers. While this problem is NPco ..."
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Cited by 11 (4 self)
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We consider the solvability of the equation k Y i=1 A i x i = B and generalizations, where the A i and B are given commuting matrices over an algebraic number field F . In the semigroup membership problem, the variables x i are constrained to be nonnegative integers. While this problem is NPcomplete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x i may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The results generalize recent work of Cai, Lipton, and Zalcstein [CLZ] where the case k = 2 is solved using Jordan Normal Forms (JNF). We achieve greater clarity, simplicity, and generality by eliminating the use of JNF's and referring to elementary concepts of the structure theory of algebras instead (notably, the radical and the local decomposit...
Undecidability bounds for integer matrices using Claus instances
 DOI 10.1142/S0129054107005066. MR2363737 (2008m:03091) ↑5.1, 5.2
"... There are several known undecidability problems for 3 × 3 integer matrices the proof of which uses a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the numbers of matrices for the mortality, zero in left upper corner, vector reachability, matrix reachability, ..."
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Cited by 9 (1 self)
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There are several known undecidability problems for 3 × 3 integer matrices the proof of which uses a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the numbers of matrices for the mortality, zero in left upper corner, vector reachability, matrix reachability, scaler reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ Z 4×4 whether or not kI4 ∈ R for any given diagonal matrix kI4 with k > 1. These bounds are obtained by using Claus instances of the PCP. 1
On the Mortality Problem for Matrices of Low Dimensions
, 1999
"... In this paper, we discuss the existence of an algorithm to decide if a given set of 2 2 matrices is mortal: a set F = fA 1 ; : : : ; Am g of 2 2 matrices is said to be mortal if there exist an integer k 1 and some integers i 1 ; i 2 ; : : : ; i k 2 f1; : : : ; mg with A i 1 A i 2 A i k = 0. We surve ..."
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Cited by 9 (1 self)
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In this paper, we discuss the existence of an algorithm to decide if a given set of 2 2 matrices is mortal: a set F = fA 1 ; : : : ; Am g of 2 2 matrices is said to be mortal if there exist an integer k 1 and some integers i 1 ; i 2 ; : : : ; i k 2 f1; : : : ; mg with A i 1 A i 2 A i k = 0. We survey this problem and propose some new extensions: we prove the problem to be BSSundecidable for real matrices and Turingdecidable for two rational matrices. We relate the problem for rational matrices to the entry equivalence problem, to the zero in the left upper corner problem and to the reachability problem for piecewise a ne functions. Finally, we state some NPcompleteness results.
Some decision problems on integer matrices
 RAIRO Theor. Inform. Appl
"... Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimensi ..."
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Cited by 5 (1 self)
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Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free, 2) contains the identity matrix, 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup. 1
Computing Jordan Normal Forms Exactly for Commuting Matrices in Polynomial Time
"... We prove that the Jordan Normal Form of a rational matrix can be computed exactly in polynomial time. We obtain the transformation matrix and its inverse exactly, and we show how to apply the basis transformation to any commuting matrices. 1 Introduction There are two motivations for this work on c ..."
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Cited by 4 (1 self)
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We prove that the Jordan Normal Form of a rational matrix can be computed exactly in polynomial time. We obtain the transformation matrix and its inverse exactly, and we show how to apply the basis transformation to any commuting matrices. 1 Introduction There are two motivations for this work on computing the Jordan Normal Form of a rational matrix exactly. The first is related to the resolution of the complexity of the A B C problem [4], and its application to the complexity problem in finitely generated commutative linear groups and semigroups in general. The second motivation is concerned with the design and analysis of uncheatable benchmarks for numerical algorithms, especially matrix multiplication [3, 1]. Our problem is the following. Given a finite set of commuting matrices over the rational numbers, A; B; : : : ; can we compute, in polynomial time, a basis transformation T , and the matrices under the similarity transformation T \Gamma1 AT ; T \Gamma1 BT ; : : : ; so tha...
Matrix Mortality and the Čern´yPin Conjecture
"... Abstract. In this paper, we establish the Čern´yPin conjecture for automata with the property that their transition monoid cannot recognize the language {a, b} ∗ ab{a, b} ∗. For the subclass of automata whose transition monoids have the property that each regular Jclass is a subsemigroup, we giv ..."
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Cited by 1 (1 self)
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Abstract. In this paper, we establish the Čern´yPin conjecture for automata with the property that their transition monoid cannot recognize the language {a, b} ∗ ab{a, b} ∗. For the subclass of automata whose transition monoids have the property that each regular Jclass is a subsemigroup, we give a tight bound on lengths of reset words for synchronizing automata thereby answering a question of Volkov. 1