parity game

Note that as opposed to parity games, this game is no longer symmetric with respect to players 0 and 1.

Parity games lie in the third level of the borel hierarchy, and are consequently determined.

Solving a parity game played on a finite graph means deciding, for a given starting position, which of the two players has a winning strategy.

Let be parity game, where resp.

Also, decision problems like validity or satisfiability for modal logics can be reduced to parity game solving.

Moreover, parity games are history-free determined.

Zielonka outlined a recursive algorithm that solves parity games.

The game semantics of μ-calculus is related to two-player games with perfect information, particularly infinite parity games.

The model-checking problem for the modal μ-calculus for instance is known to be equivalent to parity game solving.

The Knaster-Tarski theorem leads to a relatively simple proof of determinacy of parity games.

A key component of the proof requires showing determinacy of parity games, which lie in the third level of the Borel hierarchy.

Parity Game Solvers:

Despite its interesting complexity theoretic status, parity game solving can be seen as the algorithmic backend to problems in automated verification and controller synthesis.

Similar to other graph measures, such as cycle rank, some algorithmic problems, e.g. parity game, can be efficiently solved on graphs of bounded entanglement.

A parity game is played on a colored directed graph, where each node has been colored by a priority - one of (usually) finitely many natural numbers.

Given that parity games are history-free determined, solving a given parity game is equivalent to solving the following simple looking graph-theoretic problem.

Games related to parity games were implicitly used in Rabin's proof of decidability of second order theory of n successors, where determinacy of such games was proven.