Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Klima, O., Tesson, P. & Therien, D. Dichotomies in the Complexity of Solving Systems of Equations over Finite Semigroups. Theory Comput Syst 40, 263–297 (2007). https://doi.org/10.1007/s00224-005-1279-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-005-1279-2