We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of an NP-oracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a P 3 -oracle. The hypothesis class for the above learning algorithms is the class of circuits of larger but polynomially related size. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth-3 7-6-7 formulas (by the work of Angluin this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e., depth-2 6-7 formulas). We also investigate the power of an NP-oracle in the context of learning with membership queries. We show that there are deterministic learning algorithms that use membership queries and an NP-oracle to learn: monotone boolean functions in time polynomial in the DNF size and CNF size of the target formula; and the class of O(log n)-DNF & O(log n)-CNF formulas in time polynomial in n. We also show that, with an NP-oracle and membership queries, there is a randomized expected polynomial time algorithm that learns any class that is learnable from membership queries with unlimited computational power. Using similar techniques, we show the following both for membership and for equivalence queries (when the hypotheses allowed are precisely the concepts in the class); any class learnable with unbounded computational-power is learnable in deterministic polynomial time with a p 5 -oracle. Furthermore, we identify the combinatorial properties that completely determine learnability in this information-theoretic sense. Finally we point out a consequence of our result in structural complexity theory showing that if every NP set has polynomial-size circuits then the polynomial hierarchy collapses to ZPPNP . ] 1996Academic Press, Inc.

An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4 log k/ log r. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex.We then give a randomized algorithm that runs in time poly(n) · 2 O(k·log r/r) that solves the r-SIMPLE k-PATH on a graph with n vertices with one-sided error. We also show that a randomized algorithm with running time poly(n) · 2 (c/2)k/r with c < 1 gives a randomized algorithm with running time poly(n) · 2 cn for the Hamiltonian path problem in a directed graph -an outstanding open problem. So in a sense our algorithm is optimal up to an O(log r) factor.

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We study the learnability of multiplicity automata in Angluin's exact learning model, and we investigate its applications. Our starting point is a known theorem from automata theory relating the This paper contains results from three conference papers. It is mostly based on the FOCS '96 paper by the authors. The rest of the results are from the STOC '96 paper of Bergadano, Catalano, and Varricchio and the EuroCOLT '97 paper of Beimel and Kushilevitz.Part of this research was done while A. Beimel was a Ph.D student at the Technion.number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. With this theorem in hand, we present a new simple algorithm for learning multiplicity automata with improved time and query complexity, and we prove the learnability of various concept classes. These include (among others):-The class of disjoint DNF, and more generally satisfy-O(1) DNF.-The class of polynomials over finite fields.-The class of bounded-degree polynomials over infinite fields.-The class of XOR of terms.-Certain classes of boxes in high dimensions.In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over GF(2).While multiplicity automata are shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, we study the limitations of this method. We prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulas or even k-term DNF for k ϭ (log n) or satisfy-s DNF formulas for s ϭ (1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with super-polynomial number of states.

We generalize the notion of PAC learning from an example oracle to a notion of efficient learning on a quantum computer using a quantum example oracle. This quantum example oracle is a natural extension of the traditional PAC example oracle, and