2008/08/22 by Julien Cassaigne, Cassaigne, Julien, François Nicolas +2
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Geometric and Algebraic Topology #cs.DM #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0808.3112
46 pages. 1 table. To appear in RAIRO
openalex publication_date 2008/08/22 · arxiv created 2012/05/04 · arxiv updated 2012/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 of S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. 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 three-fold: (i) to present general results concerning 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.