The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distribution-free sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) Boolean threshold functions, and (c) Boolean formulas in which each variable occurs at most once. Relationships between learning of heuristics and finding approximate solutions to NP-hard optimization problems are given.

E cient distribution-free learning of Boolean formulae from positive and negative examples is considered. It is shown that classes of formulae that are e ciently learnable from only positive examples or only negative examples have certain closure properties. A new substitution technique is used to show that in the distribution-free case learning DNF disjunctive normal form formulae is no harder than learning monotone DNF. We prove that monomials cannot be e ciently learned from negative examples alone, even if the negative examples are uniformly distributed. It is also shown that if the examples are drawn from uniform distributions then the class of DNF in which each v ariable occurs at most once is e ciently learnable, while the class of all monotone Boolean functions is e ciently weakly learnable i.e., individual examples are correctly classi ed with a probability larger than 1 2 + 1 p , where p is a polynomial in the relevant parameters of the learning problem. We then show an equivalence between the notion of weak learning and the notion of group learning, where a group of examples of polynomial size, either all positive or all negative, must be correctly classi ed with high probability.

Abstract. An algorithm is presented for learning the class of Boolean formulas that are expressible as conjunctions of Horn clauses. (A Horn clause is a disjunction of literals, all but at most one of which is a negated variable.) The algorithm uses equivalence queries and membership queries to produce a formula that is logically equivalent to the unknown formula to be learned. The amount of time used by the algorithm is polynomial in the number of variables and the number of clauses in the unknown formula.

This paper surveys recent results concerning the inference of deterministic finite automata (DFAs). The results discussed determine the extent to which DFAs can be feasibly inferred, and highlight a number of interesting approaches in computational learning theory.

No abstract