RAIRO - Theoretical Informatics and Applications

Research Article

On the decidability of semigroup freeness 

Julien Cassaignea1 and Francois Nicolasa2

a1 Institut de mathématiques de Luminy, case 907, 163 avenue de Luminy, 13288 Marseille Cedex 9, France. cassaigne@iml.univ-mrs.fr

a2 Lehrstuhl für Bioinformatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany; francois.nicolas@uni-jena.de


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.

(Received September 30 2008)

(Accepted February 7 2012)

(Online publication May 29 2012)

Key Words:

  • Decidability;
  • semigroup freeness;
  • matrix semigroups;
  • Post correspondence problem

Mathematics Subject Classification:

  • 20M05;
  • 03B25;
  • 15A30


  Most of the research was achieved while the authors were visiting the University of Turku. The Academy of Finland supported the work under grants 203354 and 7523004.