On the Complexity of Parity Games

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.