Learning Regular Languages from Counterexamples

Abstract

Abstract We study the problem of learning an unknown language given a teacher which can only answer equivalence queries. The teacher presenting a language L can test (in unit time) whether a conjectured language L ′ is equal to L and, if L ′ ≠ L , provide a counterexample (i.e., a string in the symmetric difference of L and L ′). It has recently been shown that the family of regular languages and the family of pattern languages are not learnable in polynomial time under this protocol. We consider the learnability of subfamilies of regular languages. It is shown that the problem of learning a subfamily of regular languages can be reduced to the problem of learning its finite members. Using this reduction, we show that the family of κ-bounded regular languages is learnable in polynomial time. We investigate how a partial ordering on counterexamples affects the learnability of the family of regular languages and the family of pattern languages. Two partial orderings are considered: ordering by length and lexicographical ordering. We show that the first ordering on counterexamples does not reduce the complexity of learning the family of regular languages. In contrast, the family of pattern languages is learnable in polynomial time if the teacher always provides counterexamples of minimal length and the family of regular languages is learnable in polynomial time if the teacher always provides the lexicographically first counterexamples.