Conjunctions of Unate DNF Formulas

Learning and Structure

Aaron Feigelson, Lisa Hellerstein

    Research output: Contribution to journalArticle

    Abstract

    A central topic in query learning is to determine which classes of Boolean formulas are efficiently learnable with membership and equivalence queries. We consider the class ℛk consisting of conjunctions of k unate DNF formulas. This class generalizes the class of k-clause CNF formulas and the class of unate DNF formulas, both of which are known to be learnable in polynomial time with membership and equivalence queries. We prove that ℛ2 can be properly learned with a polynomial number of polynomial-size membership and equivalence queries, but can be properly learned in polynomial time with such queries if and only if P = NP. Thus the barrier to properly learning ℛ2 with membership and equivalence queries is computational rather than informational. Few results of this type are known. In our proofs, we use recent results of Hellerstein et al. (1997, J. Assoc. Comput. Mach. 43 (5), 840-862), characterizing the classes that are polynomial-query learnable, together with work of Bshouty on the monotone dimension of Boolean functions. We extend some of our results to ℛ2 and pose open questions on learning DNF formulas of small monotone dimension. We also prove structural results for ℛk. We construct, for any fixed k ≥ 2, a class of functions f that cannot be represented by any formula in ℛk, but which cannot be "easily" shown to have this property. More precisely, for any function f on n variables in the class, the value of f on any polynomial-size set of points in its domain is not a witness that f cannot be represented by a formula in ℛk. Our construction is based on BCH codes.

    Original languageEnglish (US)
    Pages (from-to)203-228
    Number of pages26
    JournalInformation and Computation
    Volume140
    Issue number2
    StatePublished - Feb 1 1998

    Fingerprint

    Polynomials
    Query
    Equivalence
    Polynomial
    Monotone
    Polynomial time
    Boolean functions
    BCH Codes
    Mach number
    Learning
    Class
    Boolean Functions
    Set of points
    If and only if
    Generalise

    ASJC Scopus subject areas

    • Computational Theory and Mathematics

    Cite this

    Feigelson, A., & Hellerstein, L. (1998). Conjunctions of Unate DNF Formulas: Learning and Structure. Information and Computation, 140(2), 203-228.

    Conjunctions of Unate DNF Formulas : Learning and Structure. / Feigelson, Aaron; Hellerstein, Lisa.

    In: Information and Computation, Vol. 140, No. 2, 01.02.1998, p. 203-228.

    Research output: Contribution to journalArticle

    Feigelson, A & Hellerstein, L 1998, 'Conjunctions of Unate DNF Formulas: Learning and Structure', Information and Computation, vol. 140, no. 2, pp. 203-228.
    Feigelson A, Hellerstein L. Conjunctions of Unate DNF Formulas: Learning and Structure. Information and Computation. 1998 Feb 1;140(2):203-228.
    Feigelson, Aaron ; Hellerstein, Lisa. / Conjunctions of Unate DNF Formulas : Learning and Structure. In: Information and Computation. 1998 ; Vol. 140, No. 2. pp. 203-228.
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