Descriptive Complexity for Counting Complexity Classes

VCLA hosting a talk by Martin Munoz

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 thesis, 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.

This talk is a presentation of a master thesis, for which Martin Munoz received the VCLA International Student Awards for Outstanding Master Thesis in 2019. More here

Magdalena Ortiz, Kurt Matyas (Vice-Rector for Academic Affairs), Martín Muñoz, Gerti Kappel.