Faster Algorithms for Bounded Liveness in Graphs and Game Graphs

Authors Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, Alexander Svozil

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Author Details

Krishnendu Chatterjee
  • IST Austria, Klosterneuburg, Austria
Monika Henzinger
  • Faculty of Computer Science, University of Vienna, Austria
Sagar Sudhir Kale
  • Faculty of Computer Science, University of Vienna, Austria
Alexander Svozil
  • Faculty of Computer Science, University of Vienna, Austria

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Krishnendu Chatterjee, Monika Henzinger, Sagar Sudhir Kale, and Alexander Svozil. Faster Algorithms for Bounded Liveness in Graphs and Game Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 124:1-124:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Graphs and games on graphs are fundamental models for the analysis of reactive systems, in particular, for model-checking and the synthesis of reactive systems. The class of ω-regular languages provides a robust specification formalism for the desired properties of reactive systems. In the classical infinitary formulation of the liveness part of an ω-regular specification, a "good" event must happen eventually without any bound between the good events. A stronger notion of liveness is bounded liveness, which requires that good events happen within d transitions. Given a graph or a game graph with n vertices, m edges, and a bounded liveness objective, the previous best-known algorithmic bounds are as follows: (i) O(dm) for graphs, which in the worst-case is O(n³); and (ii) O(n² d²) for games on graphs. Our main contributions improve these long-standing algorithmic bounds. For graphs we present: (i) a randomized algorithm with one-sided error with running time O(n^{2.5} log n) for the bounded liveness objectives; and (ii) a deterministic linear-time algorithm for the complement of bounded liveness objectives. For games on graphs, we present an O(n² d) time algorithm for the bounded liveness objectives.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Graphs
  • Game Graphs
  • Büchi


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