Learning pattern languages using queries

A pattern is a finite string of constant and variable symbols. For k≥1, we denote by kμΠ the set of all patterns in which each variable symbol occurs at most ktimes. In particular, we abbreviate μΠ for k=1. The language L(π) of a pattern πis the set of all strings obtained by substituting any non-null constant string for each variable symbol in π. In this paper, we show that any pattern π ∈ kμΠ is exactly identifiable in O(¦ω¦^k+2) time from one positive example w ∈ L(π) using ¦ω¦^k+1+¦π¦^k membership queries. Moreover, we introduce the notion of critical pattern, and show that the number of membership queries can be reduced to ¦ω¦+¦π¦ if the target pattern π∈μΠ is not critical. For instance, any patternπ∈μΠ whose constant parts are of length at most 3 is not critical. Finally, we show a nontrivial subclass of μΠ that is identified using membership queries only, without any initial positive example.