Descriptive Complexity for Counting Complexity Classes

TL;DR

This paper develops a logical framework called Quantitative Second Order Logics (QSO) to characterize and analyze counting complexity classes like FP, #P, and FPSPACE, extending descriptive complexity to counting problems.

Contribution

It introduces QSO, a new logic framework for counting complexity classes, and demonstrates its ability to characterize classes, define hierarchies, and capture lower complexity classes with recursion.

Findings

QSO captures fundamental counting classes such as FP, #P, and FPSPACE.

A hierarchy inside #P is defined with classes having good closure and approximation properties.

Recursion in QSO captures lower counting classes like #L.

Abstract

Descriptive Complexity has been very successful in characterizing complexity classes of decision problems in terms of the properties definable in some logics. However, descriptive complexity for counting complexity classes, such as FP and #P, has not been systematically studied, and it is not as developed as its decision counterpart. In this paper, we propose a framework based on Weighted Logics to address this issue. Specifically, by focusing on the natural numbers we obtain a logic called Quantitative Second Order Logics (QSO), and show how some of its fragments can be used to capture fundamental counting complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to define a hierarchy inside #P, identifying counting complexity classes with good closure and approximation properties, and which admit natural complete problems. Finally, we add recursion to QSO, and show how this extension naturally captures lower counting complexity classes such as #L.