Quotient complexity of ideal languages

TL;DR

This paper investigates the quotient complexity of various ideal languages and their operations, providing tight upper bounds using derivatives and analyzing the complexity based on minimal generators.

Contribution

It introduces the concept of quotient complexity for ideal languages and derives tight upper bounds for their complexity under various operations.

Findings

Tight upper bounds for quotient complexity of ideal languages are established.

Complexity bounds are expressed in terms of minimal generators and their derivatives.

The study covers union, intersection, difference, symmetric difference, concatenation, star, and reversal operations.

Abstract

We study the state complexity of regular operations in the class of ideal languages. A language L over an alphabet Sigma is a right (left) ideal if it satisfies L = L Sigma* (L = Sigma* L). It is a two-sided ideal if L = Sigma* L Sigma *, and an all-sided ideal if it is the shuffle of Sigma* with L. We prefer the term "quotient complexity" instead of "state complexity", and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.