The Fixpoint-Iteration Algorithm for Parity Games

TL;DR

This paper investigates the use of fixpoint iteration, a straightforward mu-calculus model checking algorithm, for solving parity games, revealing its exponential complexity and limitations in computing winning strategies.

Contribution

It demonstrates the exponential complexity of fixpoint iteration for parity games and introduces the concept of eventually-positional strategies derived from this method.

Findings

Fixpoint iteration is exponential on certain families of parity games.

It does not inherently compute positional winning strategies.

Eventually-positional strategies can be extracted from the fixpoint iteration results.

Abstract

It is known that the model checking problem for the modal mu-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal mu-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.