Quotient Complexity of Regular Languages

TL;DR

This paper introduces quotient complexity as a formal measure of regular language complexity, providing formulas and bounds for the complexity of operations like union and concatenation using derivatives, avoiding automaton constructions.

Contribution

It presents a new approach based on derivatives to analyze quotient complexity, offering formulas and bounds for regular language operations.

Findings

Derived formulas for quotient complexity of regular operations.

Provided upper bounds on the number of quotients for combined languages.

Illustrated advantages of the derivative-based approach with examples.

Abstract

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations.