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Tropical Semirings
"... this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min; +)semirings ..."
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this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min; +)semirings
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 8 (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.
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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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
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
A Study on Unique Rational Operations
"... Abstract. For each basic language operation we define its “unique ” counterpart as being the operation which results in a language whose words can be obtain uniquely through the given operation. As shown in the preliminaries of this paper, these unique operations can arguably be viewed as combined b ..."
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Abstract. For each basic language operation we define its “unique ” counterpart as being the operation which results in a language whose words can be obtain uniquely through the given operation. As shown in the preliminaries of this paper, these unique operations can arguably be viewed as combined basic operations, placing this work in the popular area of state complexity of combined language operations. Considering unique rational operations, we are questioning about their state complexity. For an answer, we provide upper bounds and empirical results meant to cast light into this matter. Equally important, we hope to have provided a generic methodology for estimating their state complexity. Yet, the core value of this work may lay more in its initiative and approach rather than any particular result.
Rational languages and the Burnside problem
 Theoret. Comput. Sci
, 1985
"... Abstract. The problem of finding regularity conditions for languages is, via the syntactic monoid, closely related to the classical Burnside problem. This survey paper presents several results and conjectures in this direction as well as on related subjects, including bounded languages, pumping, squ ..."
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Abstract. The problem of finding regularity conditions for languages is, via the syntactic monoid, closely related to the classical Burnside problem. This survey paper presents several results and conjectures in this direction as well as on related subjects, including bounded languages, pumping, squarefree words, commutativity, and rational power series. 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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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.
Theory of Computing Systems
"... Abstract. We prove that several problems associated with 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 ma ..."
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Abstract. We prove that several problems associated with 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. 1.