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Two-Player Boundedness Counter Games

Authors Emmanuel Filiot, Edwin Hamel-de le Court

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Author Details

Emmanuel Filiot
  • Université libre de Bruxelles, Belgium
Edwin Hamel-de le Court
  • Université libre de Bruxelles, Belgium

Funding

This work was supported by the Fonds de la Recherche Scientifique – F.R.S.-FNRS under the MIS project F451019F. Emmanuel Filiot is a senior research associate at F.R.S.-FNRS.

Acknowledgements

We warmly thank Jean-François Raskin for his valuable inputs.

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Emmanuel Filiot and Edwin Hamel-de le Court. Two-Player Boundedness Counter Games. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CONCUR.2022.21

Abstract

We consider two-player zero-sum games with winning objectives beyond regular languages, expressed as a parity condition in conjunction with a Boolean combination of boundedness conditions on a finite set of counters which can be incremented, reset to 0, but not tested. A boundedness condition requires that a given counter is bounded along the play. Such games are decidable, though with non-optimal complexity, by an encoding into the logic WMSO with the unbounded and path quantifiers, which is known to be decidable over infinite trees. Our objective is to give tight or tighter complexity results for particular classes of counter games with boundedness conditions, and study their strategy complexity. In particular, counter games with conjunction of boundedness conditions are easily seen to be equivalent to Streett games, so, they are CoNP-c. Moreover, finite-memory strategies suffice for Eve and memoryless strategies suffice for Adam. For counter games with a disjunction of boundedness conditions, we prove that they are in solvable in NP∩CoNP, and in PTime if the parity condition is fixed. In that case memoryless strategies suffice for Eve while infinite memory strategies might be necessary for Adam. Finally, we consider an extension of those games with a max operation. In that case, the complexity increases: for conjunctions of boundedness conditions, counter games are EXPTIME-c.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Automata over infinite objects
Keywords
  • Controller synthesis
  • Game theory
  • Counter Games
  • Boundedness objectives

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