Thomas R. Hancock

Learning monotone ku DNF formulas on product distributions

We show how to learn in polynomial time monotone kμ DNF formulas with respect to the uniform distribution. Our result generalizes to product distributions where input bits take on either value with a constant probability.

Learning Boolean Read-Once Formulas over Generalized Bases

A formula is read-once if each variable appears on at most a single input. Previously, Angluin, Hellerstein, and Karpinski gave a polynomial time algorithm hat uses membership and equivalence queries to identify exactly read once boolean formulas over the basis {AND, OR, NOT}. In this paper we consider natural generalizations of this basis, and develop exact identification algorithms for more powerful classes of read-once formulas. We show that read-once formulas over the basis of arbitrary boolean functions of constant fan-in L (i.e., any ?: {0,1}1 ? c ? k ? {0,1}, where k is a constant) are exactly identifiable i polynomial time using membership and equivalence queries. We also show that read-once formulas over the basis of arbitrary symmetric boolean functions are exactly identifiable in polynomial time in this model. Given standard cryptographic assumptions, there is no polynomial time identification algorithm for read-twice formulas over either of these bases in the model. We further show that for any basis class B meeting certain technical conditions, any polynomial time identification algorithm for read-once formulas over B can be extended to a polynomial time identification algorithm for read-once formulas over the union of B and the arbitrary functions of constant fan-in. As a result, read-once formulas over the union of arbitrary symmetric and arbitrary constant fan-in gates are also exactly identifiable in polynomial time using membership and equivalence queries.

Learning read-once formulas over fields and extended bases

A formula is read-once if each variable in it occurs at most once. Angluin, Hellerstein, and Karpinski [AHK89] have shown that read-once formulas over the basis (AND, OR, NOT) are identifiable in polynomial time with membership and equivalence queries. We extend this result for boolean formulas to a larger basis including arbitrary threshold functions (generalizing AND and OR), NOT, parity, and functions computing congruence to a residue in some modulus up to a constant k. Note these functions are all symmetric, but are not all unate. We further examine arithmetic read-once formulas over multiplication and addition on an arbitrary field. We show these are identifiable in time polynomial in the number of variables using equivalence queries and the natural extension of membership queries to a non-boolean domain.

Learning Arithmetic Read-Once Formulas

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 exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i. e. a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least

Learning Boolean read-once formulas with arbitrary symmetric and constant fan-in gates

2n+5

Learning arithmetic read-once formulas

elements (where

Learning k μ decision trees on the uniform distribution

n

Learning 2 μ DNF Formulas and kμ Decision Trees

is the number of variables), and over infinite fields. We also show the existence of non-uniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and for division-free read-once formulas over fields that have at least

Identifying μ-formula decision trees with queries

2n^3+1

Learning 2u DNF formulas and ku decision trees

elements. For our algorithms, we assume we are able to efficiently perform arithmetic operations on field elements and to 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