Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Non-uniform automata over groups
Information and Computation
Automata, Languages, and Machines
Automata, Languages, and Machines
The Complexity of Solving Equations Over Finite Groups
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Satisfiability of Systems of Equations over Finite Monoids
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Inapproximability Results for Equations over Finite Groups
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
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We study the computational complexity of solving equations and of determining the satisfiability of programs over a fixed finite monoid. We partially answer an open problem of [4] by exhibiting quasi-polynomial time algorithms for a subclass of solvable nonnilpotent groups and relate this question to a natural circuit complexity conjecture. In the special case when M is aperiodic, we show that PROGRAM SATISFIABILITY is in P when the monoid belongs to the variety DA and is NP-complete otherwise. In contrast, we give an example of an aperiodic outside DA for which EQUATION SATISFIABILITY is computable in polynomial time and discuss the relative complexity of the two problems. We also study the closure properties of classes for which these problems belong to P and the extent to which these fail to form algebraic varieties.