Theory of Computing Systems

, Volume 40, Issue 3, pp 263-297

First online:

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

  • O. KlimaAffiliated withDepartment of Mathematics, Masaryk University, Brno Email author 
  • , P. TessonAffiliated withDepartement d'Informatique et de Genie Logiciel, Universite Laval, Quebec, Quebec Email author 
  • , D. TherienAffiliated withSchool of Computer Science, McGill University, Montreal, Quebec Email author 

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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.