Quotient Complexity of Closed Languages

TL;DR

This paper investigates the quotient complexity of various closed language classes, establishing tight bounds for operations like closure, boolean operations, concatenation, star, and reversal, and analyzing the effects of closure and complement.

Contribution

It provides the first comprehensive analysis of quotient complexity bounds for prefix-, suffix-, factor-, and subword-closed languages, including effects of closure and complement operations.

Findings

Tight upper bounds for complexity of closures and operations in closed languages.

Repeated positive closure and complement produce at most four distinct languages.

Kleene closure and complement produce at most eight distinct languages.

Abstract

A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in the same way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the prefix-, suffix-, factor-, and subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star and reversal in each of the four classes of closed languages. We show that repeated application of positive closure and complement to a closed language results in at most four distinct languages, while Kleene closure and complement gives at most eight languages.