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The Memory of ω-Regular and BC(Σ⁰₂) Objectives

Authors Antonio Casares , Pierre Ohlmann

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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Infinite duration games
  • Strategy complexity
  • Omega-regular

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Abstract

In the context of 2-player zero-sum infinite duration games played on (potentially infinite) graphs, the memory of an objective is the smallest integer k such that in any game won by Eve, she has a strategy with ≤ k states of memory. For ω-regular objectives, checking whether the memory equals a given number k was not known to be decidable. In this work, we focus on objectives in BC(Σ⁰₂), i.e. recognised by a potentially infinite deterministic parity automaton. We provide a class of automata that recognise objectives with memory ≤ k, leading to the following results:  
- for ω-regular objectives, the memory can be computed in NP; 
- given two objectives W₁ and W₂ in BC(Σ⁰₂) and assuming W₁ is prefix-independent, the memory of W₁ ∪ W₂ is at most the product of the memories of W₁ and W₂.  Our results also apply to chromatic memory, the variant where strategies can update their memory state only depending on which colour is seen.

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Antonio Casares and Pierre Ohlmann. The Memory of ω-Regular and BC(Σ⁰₂) Objectives. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 149:1-149:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.149

Author Details

Antonio Casares
  • University of Warsaw, Poland
Pierre Ohlmann
  • CNRS, Laboratoire d'Informatique et des Systèmes (LIS), Marseille, France

Funding

  • Casares, Antonio: Supported by the Polish National Science Centre (NCN) grant "Polynomial finite state computation" (2022/46/A/ST6/00072).

Acknowledgements

We thank Nathan Lhote for his participation in the scientific discussions which led to this paper. We also thank Pierre Vandenhove for pointing us to references concerning lifting results for memory.

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