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Polynomial time machines equipped with word problems over algebraic structures as their acceptance criteria

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Fundamentals of Computation Theory (FCT 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1279))

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Abstract

We investigate the power of polynomial time machines whose acceptance mechanism is defined by a word problem over some finite semigroup, monoid, or group. For the case of non-solvable groups or monoids (semigroups, resp.) containing non-solvable groups it follows from [21] that the according complexity class is PSPACE. For solvable monoids it was shown there that the according class is always a subclass of MOD-PH.

We obtain the following results for finite groups: Commutative groups with k elements exactly characterize co-MODkP, solvable groups with k elements, having a composition chain of length r, characterize a class that contains co-MODkP and is contained in (co-MODk)rP, the class obtained by r-fold iterated application of the co-MODk-operator to P. Our results for finite monoids are the following: The classes characterized by commutative finite monoids are the eventually periodic counting classes (see Section 2 for definitions). If we restrict our attention to aperiodic commutative finite monoids, we obtain exactly the classes of bounded counting type [15, 18], and if we consider idempotent commutative finite monoids, we obtain the classes of the Boolean Hierarchy over NP.

Finally, our results for finite semigroups are: The class characterized by a commutative finite semigroup is representable as a P-disjoint union of classes characterized by commutative finite monoids. Thus the aperiodic and idempotent commutative cases have similar solutions as for monoids.

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  1. Institut für Informatik, Universität Stuttgart, Breitwiesenstr. 20-22, D-70565, Stuttgart, Germany

    Ulrich Hertrampf

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  1. Ulrich Hertrampf
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Bogdan S. Chlebus Ludwik Czaja

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© 1997 Springer-Verlag Berlin Heidelberg

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Hertrampf, U. (1997). Polynomial time machines equipped with word problems over algebraic structures as their acceptance criteria. In: Chlebus, B.S., Czaja, L. (eds) Fundamentals of Computation Theory. FCT 1997. Lecture Notes in Computer Science, vol 1279. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0036187

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  • DOI: https://doi.org/10.1007/BFb0036187

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  • Print ISBN: 978-3-540-63386-0

  • Online ISBN: 978-3-540-69529-5

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