Polynomial-time learning of elementary formal systems

Abstract

An elementary formal system (EFS) is a logic program consisting of definite clauses whose arguments have patterns instead of first-order terms. We investigate EFSs for polynomial-time PAC-learnability. A definite clause of an EFS is hereditary if every pattern in the body is a subword of a pattern in the head. With this new notion, we show that H-EFS(m, k, t, r) is polynomial-time learnable, which is the class of languages definable by EFSs consisting of at mostm hereditary definite clauses with predicate symbols of arity at mostr, wherek andt bound the number of variable occurrences in the head and the number of atoms in the body, respectively. The class defined by all finite unions of EFSs in H-EFS(m, k, t, r) is also polynomial-time learnable. We also show an interesting series ofNC-learnable classes of EFSs. As hardness results, the class of regular pattern languages is shown not polynomial-time learnable unlessRP=NP. Furthermore, the related problem of deciding whether there is a common subsequence which is consistent with given positive and negative examples is shownNP-complete.