Inductive Inference of Context-free Languages - Context-free Expression Method

An inductive inference problem of context-free languages is investigated. There have been many attempts to this problem, and most of them are based on a problem setting in which a representation space for hypotheses is a class of context-free grammars. An inference algorithm given in this paper , on the contrary, employs a kind of extensions of regular expressions called context-free expressions as a representation space for context-free languages. The algorithm, based on the notion of an identification in the l imit, is significantly concise when compared with existing algorithms. 1. I n t r o d u c t i o n We consider the following model of inductive inference problem: Given an object L of inference, an inductive inference device (IID) tries to infer a representation H for the object from examples. It is assumed that IID has an enumeration mechanism by which any possible hypothesis from the representation space can be eventually enumerated at least once. It is also assumed that we can utilize an oracle for presenting examples concerning the object. IID asks the oracle for an example, and computes hypothesis and outputs i t , and again asks another example for the next step, and this process is cycled. In a sequence of hypotheses H1, H2 , . . . I ID is said to identify L in the l imit if there exists a positive integer n such that Hn represents L and Hn + i equals to Hn for all i >0. A simple algorithm for identification in the l imit is the one based on the notion of identification by enumeration. Let H1, H2,... be an effective enumeration of the possible hypotheses, and suppose a set of examples e1,e2,...,ek are presented. Then, ED provides as its next output the first hypothesis which is compatible with all these examples. Under the assumption of a perfect oracle, the sequence of hypotheses converges in the limit.([Gold 1967]) In this paper, we deal with the inductive inference problem for context-free languages, and employ a representation space for hypotheses different from the ones in the existing methods. This enables us to make an elegant discussion on the problem and to obtain a simple algorithm for solving the problem.