Quotient Complexity of Star-Free Languages

TL;DR

This paper investigates the quotient complexity of star-free languages, establishing tight bounds for various operations and highlighting differences from regular languages, especially for reversal.

Contribution

It extends known complexity bounds to star-free languages, showing they largely match regular languages with specific exceptions.

Findings

Tight bounds for union, intersection, difference, symmetric difference, concatenation, and star are similar to regular languages.

The quotient complexity of reversal for star-free languages is 2^n - 1.

Some small exceptions to the bounds are identified for star-free languages.

Abstract

The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star-free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation, and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas the bound for reversal is 2^n-1.