Learning of Elementary Formal Systems with Two Clauses Using Queries

Abstract

An elementary formal system, EFS for short, is a kind of logic program over strings, and regarded as a set of rules to generate a language. For an EFS Γ, the language L(Γ) denotes the set of all strings generated by Γ. Many researchers studied the learnability of EFSs in various learning models. In this paper, we introduce a subclass of EFSs, denoted by $r\cal E\!F\!S$, and study the learnability of $r\cal E\!F\!S$ in the exact learning model. The class $r\cal E\!F\!S$ contains the class of regular patterns, which is extensively studied in Learning Theory.

Let Γ_∗ be a target EFS of learning in $r\cal E\!F\!S$. In the exact learning model, an oracle for superset queries answers “yes” for an input EFS Γ in $r\cal E\!F\!S$ if L(Γ) is a superset of L(Γ_∗), and outputs a string in L(Γ_∗) – L(Γ), otherwise. An oracle for membership queries answers “yes” for an input string w if w is included in L(Γ_∗), and answers “no”, otherwise.

We show that any EFS in $r\cal E\!F\!S$ is exactly identifiable in polynomial time using membership and superset queries. Moreover, for other types of queries, we show that there exists no polynomial time learning algorithm for $r\cal E\!F\!S$ by using the queries. This result indicates the hardness of learning the class $r\cal E\!F\!S$ in the exact learning model, in general.