On finite semigroups of matrices

Mandel, Arnaldo; Simon, Imre

doi:10.1016/0304-3975(77)90001-9

Cited by 77 publications

(52 citation statements)

References 5 publications

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“…Note that since a counter system has a finite number of control states and of transitions, one can decide whether a counter system is flat or not. This also holds for the finite monoid property, in fact using a result of [25], one can prove that the problem of knowing whether the monoid of a counter system is finite is decidable .…”

Section: Definition 1 (Counter System)mentioning

confidence: 84%

Towards Model-Checking Programs with Lists

Finkel

Lozes

2009

Infinity in Logic and Computation

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Abstract. We aim at checking safety and temporal properties over models representing the behavior of programs manipulating dynamic singly-linked lists. The properties we consider not only allow to perform a classical shape analysis, but we also want to check quantitative aspect on the manipulated memory heap. We first explain how a translation of programs into counter systems can be used to check safety problems and temporal properties. We then study the decidability of these two problems considering some restricted classes of programs, namely flat programs without destructive update. We obtain the following results : (1) the model-checking problem is decidable if the considered program works over acyclic lists (2) the safety problem is decidable for programs without alias test. We finally explain the limit of our decidability results, showing that relaxing one of the hypothesis leads to undecidability results.

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Section: Definition 1 (Counter System)mentioning

confidence: 84%

Towards Model-Checking Programs with Lists

Finkel

Lozes

2009

Infinity in Logic and Computation

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“…The identity element is naturally the identity matrix A 0 = I. Given a matrix A ∈ Z n×n , checking whether the monoid generated by A is finite, is decidable [MS77]. By way of example, with A = 1 0 1 1 , we have…”

Section: Finite Monoid Property In Affine Presburger Counter Systemsmentioning

confidence: 99%

Witness Runs for Counter Machines

Barrett

Demri

Deters

2013

Frontiers of Combining Systems

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Abstract. In this paper, we present recent results about the verification of counter machines by using decision procedures for Presburger arithmetic. We recall several known classes of counter machines for which the reachability sets are Presburger-definable as well as temporal logics with arithmetical constraints. We discuss issues related to flat counter machines, path schema enumeration, and the use of SMT solvers.

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“…We do not discuss the well-known problem of the finiteness of a semigroup of matrices (the "Burnside problem"). The reader is referred to [9,14].…”

Section: Preliminariesmentioning

confidence: 99%

Some decision problems on integer matrices

Choffrut

Karhumäki

2005

RAIRO-Theor. Inf. Appl.

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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.

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On finite semigroups of matrices

Cited by 77 publications

References 5 publications

Towards Model-Checking Programs with Lists

Towards Model-Checking Programs with Lists

Witness Runs for Counter Machines

Some decision problems on integer matrices

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