Results 1  10
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12
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
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Cited by 116 (21 self)
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The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
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.
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.
Decidable and Undecidable Problems in Matrix Theory
, 1997
"... This work is a survey on decidable and undecidable problems in matrix theory. The problems studied are simply formulated, however most of them are undecidable. The method to prove undecidabilities is the one found by Paterson [Pat] in 1970 to prove that the mortality of finitely generated matrix mon ..."
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Cited by 6 (2 self)
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This work is a survey on decidable and undecidable problems in matrix theory. The problems studied are simply formulated, however most of them are undecidable. The method to prove undecidabilities is the one found by Paterson [Pat] in 1970 to prove that the mortality of finitely generated matrix monoids is undecidable. This method is based on the undecidability of the Post Correspondence Problem. We shall present a new proof to this mortality problem, which still uses the method of Paterson, but is a bit simpler.
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
Decision Questions on Integer Matrices
 Number
, 2002
"... We give a survey of simple undecidability results and open problems concerning matrices of low order with integer entries. Connections to the theory of finite automata (with multiplicities) are also provided. ..."
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Cited by 1 (0 self)
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We give a survey of simple undecidability results and open problems concerning matrices of low order with integer entries. Connections to the theory of finite automata (with multiplicities) are also provided.
On the decidability of semigroup freeness
, 2008
"... This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been clos ..."
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Cited by 1 (1 self)
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This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over threebythree integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is threefold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic. 1