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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.
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
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.
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. 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. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. 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. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.