# Improved set-based symbolic algorithms for parity games

K. Chatterjee, W. Dvorák, M. Henzinger, V. Loitzenbauer, in:, Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2017. 2017_LIPIcs-Chatterjee.pdf 710.18 KB

Conference Paper | Published | English
Department
Abstract
Graph games with omega-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify omega-regular winning conditions. Graph games with parity conditions are equivalent to mu-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators. We consider graph games with n vertices and parity conditions with c priorities (equivalently, a mu-calculus formula with c alternations of least and greatest fixed points). While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires O(n^c) symbolic operations and linear space; and (b) an improved algorithm that requires O(n^{c/2+1}) symbolic operations but also O(n^{c/2+1}) space (i.e., exponential space). In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires O(n^{c/2+1}) symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires O(n^{c/3+1}) symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations.
Publishing Year
Date Published
2017-08-01
Volume
82
Article Number
18
Conference
CSL: Conference on Computer Science Logic
Conference Location
Stockholm, Sweden
Conference Date
2017-08-20 – 2017-08-24
IST-REx-ID

### Cite this

Chatterjee K, Dvorák W, Henzinger M, Loitzenbauer V. Improved set-based symbolic algorithms for parity games. In: Vol 82. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik; 2017. doi:10.4230/LIPICS.CSL.2017.18
Chatterjee, K., Dvorák, W., Henzinger, M., & Loitzenbauer, V. (2017). Improved set-based symbolic algorithms for parity games (Vol. 82). Presented at the CSL: Conference on Computer Science Logic, Stockholm, Sweden: Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPICS.CSL.2017.18
Chatterjee, Krishnendu, Wolfgang Dvorák, Monika Henzinger, and Veronika Loitzenbauer. “Improved Set-Based Symbolic Algorithms for Parity Games,” Vol. 82. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2017. https://doi.org/10.4230/LIPICS.CSL.2017.18.
K. Chatterjee, W. Dvorák, M. Henzinger, and V. Loitzenbauer, “Improved set-based symbolic algorithms for parity games,” presented at the CSL: Conference on Computer Science Logic, Stockholm, Sweden, 2017, vol. 82.
Chatterjee K, Dvorák W, Henzinger M, Loitzenbauer V. 2017. Improved set-based symbolic algorithms for parity games. CSL: Conference on Computer Science Logic vol. 82.
Chatterjee, Krishnendu, et al. Improved Set-Based Symbolic Algorithms for Parity Games. Vol. 82, 18, Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2017, doi:10.4230/LIPICS.CSL.2017.18.
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