Finite monoids and the fine structure of NC1
Journal of the ACM (JACM)
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We study the computational complexity of determining whether a systems of equations over a fixed finite monoid has a solution. In [6], it was shown that in the restricted case of groups the problem is tractable if the group is Abelian and NP-complete otherwise. We prove that in the case of an arbitrary finite monoid, the problem is in P if the monoid divides the direct product of an Abelian group and a commutative idempotent monoid, and is NP-complete otherwise. In the restricted case where only constants appear on the right-hand side, we show that the problem is in P if the monoid is in the class R1 V L1, and is NP-complete otherwise. Furthermore interesting connections to the well known CONSTRAINT SATISFIABILITY PROBLEM are uncovered and exploited.