A generic algorithm for the word problem in semigroups and groups

Abstract

Abstract Kapovich, Myasnikov, Schupp and Shpilrain in 2003 developed generic approach to algorithmic problems, which considers an algorithmic problem on “most” of the inputs instead of the entire domain and ignores it on the rest of inputs. Myasnikov, Ushakov and Won in 2008 constructed a simple generic algorithm for the word problem in some finitely defined semigroups, including classical semigroups with undecidable word problem: Tseitin semigroup, Makanin semigroup and Matiyasevich semigroup. But this algorithm does not work for wider classes of semigroups: for example, semigroups with one relation. In this paper we present new generic algorithm for a wide class of finitely defined semigroups, including Tseitin and Makanin semigroups, semigroups with one relation. This algorithm can be viewed as advanced development of the Myasnikov-Ushakov-Won algorithm. Also we adopt our generic algorithm for the word problem in some finitely defined groups, including classical examples of groups with undecidable word problem. The work is supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613.