On the decidability of semigroup freeness∗

Julien Cassaigne; Francois Nicolas

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 3, page 355-399
  • ISSN: 0988-3754

Abstract

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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 the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about 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.

How to cite

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Cassaigne, Julien, and Nicolas, Francois. "On the decidability of semigroup freeness∗." RAIRO - Theoretical Informatics and Applications 46.3 (2012): 355-399. <http://eudml.org/doc/277830>.

@article{Cassaigne2012,
abstract = {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 the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about 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.},
author = {Cassaigne, Julien, Nicolas, Francois},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Decidability; semigroup freeness; matrix semigroups; Post correspondence problem; decidability; free monoids; freeness problem; undecidability},
language = {eng},
month = {8},
number = {3},
pages = {355-399},
publisher = {EDP Sciences},
title = {On the decidability of semigroup freeness∗},
url = {http://eudml.org/doc/277830},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Nicolas, Francois
TI - On the decidability of semigroup freeness∗
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/8//
PB - EDP Sciences
VL - 46
IS - 3
SP - 355
EP - 399
AB - 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 the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about 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.
LA - eng
KW - Decidability; semigroup freeness; matrix semigroups; Post correspondence problem; decidability; free monoids; freeness problem; undecidability
UR - http://eudml.org/doc/277830
ER -

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