Logical characterizations of computational complexity classes

TL;DR

This paper reviews the development of descriptive complexity theory, which characterizes computational complexity classes using logical descriptions rather than algorithms, highlighting its historical evolution since 1974.

Contribution

It provides a comprehensive overview of the methods and progress in descriptive complexity theory, emphasizing its role in characterizing complexity classes logically.

Findings

Logical methods can describe combinatorial problems without algorithms

Descriptive complexity links logic and computational complexity classes

Historical development of the theory since 1974

Abstract

Descriptive complexity theory is an important area in the study of computational complexity. In this direction, it is possible to describe combinatorial problems exclusively by logical methods, without resorting to the use of complicated algorithms. The first work in this direction was written in 1974 by the American mathematician Fagin. The article describes the development of methods of the theory of descriptive complexity.