On the Complexity of Parity Games

Abstract

Parity games underlie the model checking problem for the modal μ-calculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem – which is known to be both in NP and coNP – has a polynomial-time solution (without the assumption that P = NP) and those who believe that it does not. (A third, pessimistic, faction believes that the answer to this question will remain unknown in their lifetime.)

In this paper we explore the possibility of employing Bounded Arithmetic to resolve this question, motivated by the fact that problems which are both NP and coNP, and where the equivalence between their NP and coNP description can be formulated and proved within a certain fragment of Bounded Arithmetic, necessarily admit a polynomial-time solution. While the problem remains unresolved by this paper, we do proposed another approach, and at the very least provide a modest refinement to the complexity of parity games (and in turn the μ-calculus model checking problem): that they lie in the class PLS of Polynomial Local Search problems. This result is based on a new proof of memoryless determinacy which can be formalised in Bounded Arithmetic.

The approach we propose may offer a route to a polynomial-time solution. Alternatively, there may be scope in devising a reduction between the problem and some other problem which is hard with respect to PLS, thus making the discovery of a polynomial-time solution unlikely according to current wisdom.

Contributors

Arnold Beckmann

Faron Moller

Conference

Publication date: September 2008

Publication date (Print): September 2008

Pages: 237-247

Affiliations

Department of Computer Science

Swansea University

Article

DOI: 10.14236/ewic/VOCS2008.20

Copyright statement: © Arnold Beckmann et al. Published by BCS Learning and Development Ltd. Visions of Computer Science - BCS International Academic Conference

License:

This work is licensed under a Creative Commons Attribution 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Conference name: Visions of Computer Science - BCS International Academic Conference

Conference acronym: VOCS

Conference number:

Conference location: Imperial College, London, UK

Conference date: 22 - 24 September 2008

Conference sponsor: Electronic Workshops in Computing (eWiC)

Conference theme: BCS International Academic Conference

Product

Product Information: 1477-9358BCS Learning & Development

Self URI (article page): https://www.scienceopen.com/hosted-document?doi=10.14236/ewic/VOCS2008.20

Self URI (journal page): https://ewic.bcs.org/

Categories

Series title: Electronic Workshops in Computing