# Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds

@inproceedings{Gazda2013ZielonkasRA, title={Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds}, author={Maciej Gazda and Tim A. C. Willemse}, booktitle={GandALF}, year={2013} }

Dull, weak and nested solitaire games are important classes of parity games, capturing, among others, alternation-free mu-calculus and ECTL* model checking problems. These classes can be solved in polynomial time using dedicated algorithms. We investigate the complexity of Zielonka's Recursive algorithm for solving these special games, showing that the algorithm runs in O(d (n + m)) on weak games, and, somewhat surprisingly, that it requires exponential time to solve dull games and (nested…

## 14 Citations

BDD-based parity game solving

- Computer Science
- 2018

Zielonka’s BDD-based algorithm beats the B DD-based Priority Promotion algorithm by a small margin for games that are characteristic of practical veriﬁcation problems, and the Fixpoint-Iteration algorithm performs similar to Ziel onka‘s algorithm for games with at most 5 diﬀerent priorities.

Fatal Attractors in Parity Games: Building Blocks for Partial Solvers

- Computer ScienceArXiv
- 2014

New forms of attractors are proposed that are monotone in that they are aware of specific static patterns of colors encountered in reaching a given node set in alternating fashion and can be embedded within greatest fixed-point computations to design solvers of parity games that run in polynomial time but are partial in thatthey may not decide the winning status of all nodes in the input game.

A Comparison of BDD-Based Parity Game Solvers

- Computer ScienceGandALF
- 2018

This paper describes experiments with BDD-based implementations of four parity game solving algorithms, viz.

Oink: an Implementation and Evaluation of Modern Parity Game Solvers

- Computer ScienceTACAS
- 2018

A new and easy to extend tool Oink is implemented, which is a high-performance implementation of modern parity game algorithms and solvers, both on real world benchmarks and randomly generated games.

Improvement in Small Progress Measures

- Computer ScienceGandALF
- 2015

The algorithm is modified so that it derives the winning strategy for both players in one pass, which reduces the upper bound on strategy derivation for SPM to O(dm.(n/floor(d/2))^floor( d/2)).

A Parity Game Tale of Two Counters

- Computer ScienceGandALF
- 2019

This paper presents a parameterized parity game called the Two Counters game, which provides an exponential lower bound for a wide range of parity game solving algorithms and is the first to provide an exponentialLower bound to priority promotion with the delayed promotion policy, and theFirst to provide such a lower bound to tangle learning.

Practical Improvements to Parity Game Solving

- Computer Science
- 2013

The empirical results will support the conclusion that considerable improvements over the state of the art are possible using a combination of careful tool design and implementation, application of powerful preprocessing operations, and the use of advanced heuristics in the implementation of the Small Progress Measures algorithm.

Practical improvements to parity game solving

- Computer Science
- 2013

The empirical results will support the conclusion that considerable improvements over the state of the art are possible using a combination of careful tool design and implementation, application of powerful preprocessing operations, and the use of advanced heuristics in the implementation of the Small Progress Measures algorithm.

Strategy Derivation for Small Progress Measures

- Computer ScienceArXiv
- 2014

This work provides a novel operational interpretation of progress measures, and modify the algorithm so that it derives the winning strategies for both players in one pass, and reduces the upper bound on strategy derivation for SPM.

Evidence extraction from parameterised Boolean equation systems

- Computer Science
- 2018

This paper shows how to extract witnesses and counterexamples from parameterised Boolean equation systems encoding the model checking problem for the first-order modal μ-calculus in the modelling and analysis toolset mCRL2.

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