Small Progress Measures for Solving Parity Games

@inproceedings{Jurdzinski2000SmallPM,
  title={Small Progress Measures for Solving Parity Games},
  author={Marcin Jurdzinski},
  booktitle={STACS},
  year={2000}
}
In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal µ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to… 
An ordered approach to solving parity games in quasi polynomial time and quasi linear space
TLDR
A first implementation for a quasi-polynomial algorithm is provided, and a number of side results are provided, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al.
An ordered approach to solving parity games in quasi-polynomial time and quasi-linear space
TLDR
This work provides a first implementation for a quasi-polynomial algorithm, test it on small examples, and provides a number of side results, including minor algorithmic improvements, and a complexity index associated to the approach, which is compared to the recently proposed register index.
An Optimal Value Iteration Algorithm for Parity Games
TLDR
The technical result of this paper is to show that the latter construction is asymptotically tight: universal trees have at least quasipolynomial size, suggesting that the succinct progress measure algorithm of Jurdzi\'nski and Lazi\'c is in this framework optimal, and that the polynomial time algorithm for parity games is hiding someplace else.
A Discrete Strategy Improvement Algorithm for Solving Parity Games
A discrete strategy improvement algorithm is given for constructing winning strategies in parity games, thereby providing also a new solution of the model-checking problem for the modal μ-calculus.
A Super-Polynomial Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it
TLDR
A new lower bound for the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdziski is presented, answering in the negative the long-standing question whether this algorithm runs in polynomial time.
The Fixpoint-Iteration Algorithm for Parity Games
TLDR
This paper studies the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration, and shows that particular exponential-space strategies which are eventually-positional can be extracted from them, and are shown to be positional winning strategies.
An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it
  • Oliver Friedmann
  • Computer Science
    2009 24th Annual IEEE Symposium on Logic In Computer Science
  • 2009
TLDR
A family of games on which the discrete strategy improvement algorithm for solving parity games due to Voege and Jurdzinski requires exponentially many strategy iterations is outlined, answering in the negative the long-standing question whether this algorithm runs in polynomial time.
Local Strategy Improvement for Parity Game Solving
TLDR
A local strategy improvement algorithm which explores the game graph on-the-fly whilst performing the improvement steps and can outperform existing global strategy improvement algorithms by several orders of magnitude.
Two Local Strategy Iteration Schemes for Parity Game Solving
TLDR
It turns out that local strategy iteration can outperform these others significantly and be compared empirically with existing global strategy iteration algorithms and the currently only other local algorithm for solving parity games.
TWO LOCAL STRATEGY IMPROVEMENT SCHEMES FOR PARITY GAME SOLVING
TLDR
Two local strategy improvement algorithms which explore the game graph on-the-fly whilst performing the improvement steps and can outperform existing global strategy improvement algorithm for solving parity games by several orders of magnitude.
...
...

References

SHOWING 1-10 OF 25 REFERENCES
Deciding the Winner in Parity Games is in UP \cap co-Up
The Complexity of Mean Payoff Games on Graphs
Infinite Games Played on Finite Graphs
Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees
  • W. Zielonka
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1998
Progress measures, immediate determinacy, and a subset construction for tree automata
  • Nils Klarlund
  • Mathematics
    [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
  • 1992
TLDR
Using the concept of a progress measure, a simplified proof is given of M.O. Rabin's (1969) fundamental result that the languages defined by tree automata are closed under complementation and a graph-theoretic duality theorem for such acceptance conditions is shown.
An Improved Algorithm for the Evaluation of Fixpoint Expressions
Pushdown Processes: Games and Model-Checking
TLDR
It is shown that the model checking problem for push-down automata and the propositional μ-calculus is DEXPTIME-complete and there is a winning strategy which is realized by a pushdown process.
The complexity of tree automata and logics of programs
  • E. Emerson, C. Jutla
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
TLDR
It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)/sup 3n/), even though the problem is in general NP-complete, and it follows that satisfiability for propositional dynamic logic with a repetition construct and for the propositional mu-calculus can be tests in deterministic single exponential time.
Progress measures for complementation omega -automata with applications to temporal logic
  • Nils Klarlund
  • Computer Science
    [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
  • 1991
TLDR
It is shown that a graph-theoretic approach based on the notion of progress measures is a potent tool for complementing omega -automata, and that the powerful temporal logic ETLs is much more tractable than previously thought.
Tree automata, mu-calculus and determinacy
  • E. Emerson, C. Jutla
  • Mathematics, Computer Science
    [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
  • 1991
TLDR
It is shown that the propositional mu-calculus is equivalent in expressive power to finite automata on infinite trees, which provides a radically simplified, alternative proof of M.O. Rabin's (1989) complementation lemma for tree automata, which is the heart of one of the deepest decidability results.
...
...