Learning arithmetic read-once formulas

Nader H. Bshouty, Thomas R. Hancock, Lisa Hellerstein

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

    Abstract

    A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exactly learning (or interpolating) arithmetic read-once formulas computing functions over a field. We present an algorithm that uses randomized membership queries (or substitutions) to identify such formulas over large finite fields and infinite fields. We also present a deterministic algorithm that uses equivalence queries as well as membership queries to identify arithmetic read-once formulas over small finite fields. We then non-constructively show the existence of deterministic membership query (interpolation) algorithms for arbitrary formulas over fields of characteristic 0 and for division-free formulas over large or infinite fields. Our algorithms assume we are able to efficiently perform arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary, in the sense that the problem of computing n - 1 square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly non-standard form, in which counterexamples are required to not be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n/ log n), for which it is shown that there is no polynomial time identification algorithm that uses just membership and standard equivalence queries.

    Original languageEnglish (US)
    Title of host publicationProceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992
    PublisherAssociation for Computing Machinery
    Pages370-381
    Number of pages12
    ISBN (Electronic)0897915119
    DOIs
    StatePublished - Jul 1 1992
    Event24th Annual ACM Symposium on Theory of Computing, STOC 1992 - Victoria, Canada
    Duration: May 4 1992May 6 1992

    Publication series

    NameProceedings of the Annual ACM Symposium on Theory of Computing
    VolumePart F129722
    ISSN (Print)0737-8017

    Conference

    Conference24th Annual ACM Symposium on Theory of Computing, STOC 1992
    CountryCanada
    CityVictoria
    Period5/4/925/6/92

    ASJC Scopus subject areas

    • Software

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