A Survey on Operational State Complexity

TL;DR

This survey comprehensively reviews the state complexities of regular language operations, their combinations, and methods for estimating complexities, providing a detailed overview of operational state complexity in formal language theory.

Contribution

It offers a systematic review of the current knowledge on operational state complexity, including individual and combined operations, and discusses estimation and approximation techniques.

Findings

Summarizes known state complexities for various regular language operations.

Analyzes complexity of combined operations and their estimation methods.

Highlights open problems and future research directions in operational state complexity.

Abstract

Descriptional complexity is the study of the conciseness of the various models representing formal languages. The state complexity of a regular language is the size, measured by the number of states of the smallest, either deterministic or nondeterministic, finite automaton that recognises it. Operational state complexity is the study of the state complexity of operations over languages. In this survey, we review the state complexities of individual regularity preserving language operations on regular and some subregular languages. Then we revisit the state complexities of the combination of individual operations. We also review methods of estimation and approximation of state complexity of more complex combined operations.