Theory of Computing Systems

, Volume 40, Issue 3, pp 263–297

Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups


DOI: 10.1007/s00224-005-1279-2

Cite this article as:
Klima, O., Tesson, P. & Therien, D. Theory Comput Syst (2007) 40: 263. doi:10.1007/s00224-005-1279-2


We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup SΓ such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over SΓ.

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© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Masaryk University, BrnoCzech Republic
  2. 2.Departement d'Informatique et de Genie Logiciel, Universite Laval, Quebec, QuebecCanada
  3. 3.School of Computer Science, McGill University, Montreal, QuebecCanada