# 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

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topCassaigne, 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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