Solving Parity Games via Priority Promotion

Abstract

We consider parity games, a special form of two-player infiniteduration games on numerically labelled graphs, whose winning condition requires that the maximal value of a label occurring infinitely often during a play be of some specific parity. The problem has a rather intriguing status from a complexity theoretic viewpoint, since it belongs to the class UPTime ∩ CoUPTime, and still open is the question whether it can be solved in polynomial time. Parity games also have great practical interest, as they arise in many fields of theoretical computer science, most notably logic, automata theory, and formal verification. In this paper, we propose a new algorithm for the solution of the problem, based on the idea of promoting vertices to higher priorities during the search for winning regions. The proposed approach has nice computational properties, exhibiting the best space complexity among the currently known solutions. Experimental results on both random games and benchmark families show that the technique is also very effective in practice.

Simple Fixpoint Iteration To Solve Parity Games

“…Selecting a random strategy that stays in the won region is therefore incorrect. A smarter but still incorrect method is to try to play to the highest priority vertex of the winner's parity, by repeatedly attracting to the highest vertices like in attractor-based algorithms [1,4,25]. This however does not always produce a correct result.…”

A Delayed Promotion Policy for Parity Games

“…The priority promotion approach proposed in [2] attacks the problem of solving a parity game by computing one of its dominions D, for some player α ∈ {0, 1}, at a time. Indeed, once the α-attractor D ⋆ of D is removed from , the smaller game \ D ⋆ is obtained, whose positions are winning for one player iff they are winning for the same player in the original game.…”

“…A closed α-region in a game is clearly an α-dominion in that game. These observations give us an easy and efficient way to extract a quasi dominion from every subgame: collect the α-attractor of the positions with maximal priority p in the subgame, where p ≡ 2 α, and assign p as priority of the resulting region R. This priority, called measure of R, intuitively corresponds to an under-approximation of the best priority player α can force the opponent α to visit along any play exiting from R. Proposition 3.1 (Region Extension [2]). Let ∈ PG be a game and R ⊆ Ps an α-region in .…”

“…). An open quasi dominion pair (R, α) ∈ QD − is compatible with a state s ((r, p), _, _) ∈ S, in symbols s (R, α), iff (1) (R, α) ∈ Rg s and (2) if R is α-open in s or it is α-locked w.r.t. s then R = atr α s (r − (p)).…”

“…PGSOLVER was compiled with OCaml version 2.12.1 2. The version of PP used in the experiments is actually an improved implementation of the one described in[2].…”