On the power of finite automata with both nondeterministic and probabilistic states (Preliminary version)

Anne Condon, Lisa Hellerstein, Samuel Pottle, Avi Wigderson

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where the players are limited to polynomial time and constant space. Dwork and Stockmeyer asked whether the above class of npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a l-way input head. We also show that if L is a nonregular language, then either L or is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its l-Tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the "characteristic matrix" of L, namely the binary matrix with a row and column for each string of length n, where entry [z, y] = 1 if and only if the string zy ϵ L. We show that a language has constant l-Tiling complexity if and only if it is regular, from which the result on l-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: An upper bound of polylog(n) on the l-Tiling complexity of every language computed by our model, and a lower bound stating that the l-Tiling complexity of a nonregular language or its complement exceeds a function in 2Ω√(logn) infinitely often. The last lower bound follows by proving that the characteristic matrix of ever-y nonregular language has rank n for infinitely many n. This is our main technical result, and its proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices.

    Original languageEnglish (US)
    Title of host publicationProceedings of the 26th Annual ACM Symposium on Theory of Computing, STOC 1994
    PublisherAssociation for Computing Machinery
    Pages676-685
    Number of pages10
    ISBN (Electronic)0897916638
    DOIs
    StatePublished - May 23 1994
    Event26th Annual ACM Symposium on Theory of Computing, STOC 1994 - Montreal, Canada
    Duration: May 23 1994May 25 1994

    Publication series

    NameProceedings of the Annual ACM Symposium on Theory of Computing
    VolumePart F129502
    ISSN (Print)0737-8017

    Conference

    Conference26th Annual ACM Symposium on Theory of Computing, STOC 1994
    CountryCanada
    CityMontreal
    Period5/23/945/25/94

    ASJC Scopus subject areas

    • Software

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