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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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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.
M.: Positivity of second order linear recurrent sequences
 Discrete Appl. Math
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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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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
Decision Questions on Integer Matrices
"... 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.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
On Markov's . . . Integer Matrices
, 2006
"... We study a problem considered originally by A. Markov in 1947: Given two matrix semigroups, determine whether or not they contain a common element. This problem was proved undecidable by Markov for 4×4 matrices, even in the very restrict form, and for 3 × 3 matrices by Krom in 1981. Here we give a n ..."
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We study a problem considered originally by A. Markov in 1947: Given two matrix semigroups, determine whether or not they contain a common element. This problem was proved undecidable by Markov for 4×4 matrices, even in the very restrict form, and for 3 × 3 matrices by Krom in 1981. Here we give a new proof in the 3 × 3 case which gives undecidability in an almost as restricted form as the result of Markov.
Computational Problems in Matrix Semigroups
, 2007
"... This thesis deals with computational problems that are defined on matrix semigroups, which play a pivotal role in Mathematics and Computer Science in such areas as control theory, dynamical systems, hybrid systems, computational geometry and both classical and quantum computing to name but a few. P ..."
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This thesis deals with computational problems that are defined on matrix semigroups, which play a pivotal role in Mathematics and Computer Science in such areas as control theory, dynamical systems, hybrid systems, computational geometry and both classical and quantum computing to name but a few. Properties that researchers wish to study in such fields often turn out to be questions regarding the structure of the underlying matrix semigroup and thus the study of computational problems on such algebraic structures in linear algebra is of intrinsic importance. Many natural problems concerning matrix semigroups can be proven to be intractable or indeed even unsolvable in a formal mathematical sense. Thus, related problems concerning physical, chemical and biological systems modelled by such structures have properties which are not amenable to algorithmic procedures to determine their values. With such recalcitrant problems we often find that there exists a tight
Recurrent Sequences
"... We prove that the following problem, which is called the Positivity Problem in the literature, is algorithmically solvable: Given a sequence (un) ∞ n=0 of integers satisfying a linear second order recurrence relation, determine whether or not un is nonnegative for all n. This problem has connectio ..."
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We prove that the following problem, which is called the Positivity Problem in the literature, is algorithmically solvable: Given a sequence (un) ∞ n=0 of integers satisfying a linear second order recurrence relation, determine whether or not un is nonnegative for all n. This problem has connection to other fields of mathematics through the theory of matrices.