Quotient complexity of ideal languages

Abstract

A language L over an alphabet Σ is a right (left) ideal if it satisfies L=LΣ∗ (L=Σ∗L). It is a two-sided ideal if L=Σ∗LΣ∗, and an all-sided ideal if L=Σ∗▪L, the shuffle of Σ∗ with L. Ideal languages are not only of interest from the theoretical point of view, but also have applications to pattern matching. We study the state complexity of common operations in the class of regular ideal languages, but prefer to use the equivalent term “quotient complexity”, which is the number of distinct left quotients of a language. We find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator, and also on the complexity of the minimal generator in terms of the complexity of the language. Moreover, tight upper bounds on the complexity of union, intersection, set difference, symmetric difference, concatenation, star, and reversal of ideal languages are derived.