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Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2026.191

Infinite-State Games with Energy Objectives Beyond Counters

Authors: Irmak Sağlam and Georg Zetzsche

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

In the theory of games on infinite-state arenas, there is a stark contrast between (i) recursion-based models such as pushdown systems and extensions on one hand, and (ii) counter-based models like vector addition systems with states (VASS) on the other. For pushdown systems and extensions, there is a rich variety of decidable and well-understood games, whereas on VASS arenas, even extremely simple games are undecidable. Here, a VASS is an automaton with counters that can be incremented and decremented, but not tested for zero. Crucially, the counters can only assume non-negative values. However, certain VASS games become decidable when using energy semantics: An energy game is played on a system with counters, but the arena includes configurations with negative counters. The requirement that the counters stay non-negative is, instead, part of the winning condition of the existential player. We study an analogue of energy semantics - legality of instructions as part of the winning condition rather than arena - on a broad class of infinite-state systems, where we call them viability games. Specifically, we study viability games in the framework of valence systems over graph monoids, where (undirected, loops allowed) graphs specify various infinite-state systems, such as pushdowns, VASS counters, integer counters, and combinations thereof. In our main results, we provide a complete description of the decidability and complexity landscape of viability games across valence systems over graph monoids. Our results reveal encouraging decidability properties. For example, in certain combinations of pushdowns and counters, viability games are decidable, despite non-termination games being undecidable there. Moreover, viability games are even decidable for certain systems where (single-player) control-state reachability is undecidable.

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Irmak Sağlam and Georg Zetzsche. Infinite-State Games with Energy Objectives Beyond Counters. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 191:1-191:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{saglam_et_al:LIPIcs.ICALP.2026.191,
  author =	{Sa\u{g}lam, Irmak and Zetzsche, Georg},
  title =	{{Infinite-State Games with Energy Objectives Beyond Counters}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{191:1--191:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.191},
  URN =		{urn:nbn:de:0030-drops-265803},
  doi =		{10.4230/LIPIcs.ICALP.2026.191},
  annote =	{Keywords: Games on Graphs, Decidability, Complexity, Energy Games, Vector addition systems, Pushdown, Groups, Valence Systems}
}

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DOI: 10.4230/LIPIcs.STACS.2024.36

Directed Regular and Context-Free Languages

Authors: Moses Ganardi, Irmak Sağlam, and Georg Zetzsche

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract

We study the problem of deciding whether a given language is directed. A language L is directed if every pair of words in L have a common (scattered) superword in L. Deciding directedness is a fundamental problem in connection with ideal decompositions of downward closed sets. Another motivation is that deciding whether two directed context-free languages have the same downward closures can be decided in polynomial time, whereas for general context-free languages, this problem is known to be coNEXP-complete. We show that the directedness problem for regular languages, given as NFAs, belongs to AC¹, and thus polynomial time. Moreover, it is NL-complete for fixed alphabet sizes. Furthermore, we show that for context-free languages, the directedness problem is PSPACE-complete.

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Moses Ganardi, Irmak Sağlam, and Georg Zetzsche. Directed Regular and Context-Free Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ganardi_et_al:LIPIcs.STACS.2024.36,
  author =	{Ganardi, Moses and Sa\u{g}lam, Irmak and Zetzsche, Georg},
  title =	{{Directed Regular and Context-Free Languages}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.36},
  URN =		{urn:nbn:de:0030-drops-197465},
  doi =		{10.4230/LIPIcs.STACS.2024.36},
  annote =	{Keywords: Subword, ideal, language, regular, context-free, equivalence, downward closure, compression}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2023.34

Solving Odd-Fair Parity Games

Authors: Irmak Sağlam and Anne-Kathrin Schmuck

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract

This paper discusses the problem of efficiently solving parity games where player Odd has to obey an additional strong transition fairness constraint on its vertices - given that a player Odd vertex v is visited infinitely often, a particular subset of the outgoing edges (called live edges) of v has to be taken infinitely often. Such games, which we call Odd-fair parity games, naturally arise from abstractions of cyber-physical systems for planning and control. In this paper, we present a new Zielonka-type algorithm for solving Odd-fair parity games. This algorithm not only shares the same worst-case time complexity as Zielonka’s algorithm for (normal) parity games but also preserves the algorithmic advantage Zielonka’s algorithm possesses over other parity solvers with exponential time complexity. We additionally introduce a formalization of Odd player winning strategies in such games, which were unexplored previous to this work. This formalization serves dual purposes: firstly, it enables us to prove our Zielonka-type algorithm; secondly, it stands as a noteworthy contribution in its own right, augmenting our understanding of additional fairness assumptions in two-player games.

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Irmak Sağlam and Anne-Kathrin Schmuck. Solving Odd-Fair Parity Games. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 34:1-34:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{saglam_et_al:LIPIcs.FSTTCS.2023.34,
  author =	{Sa\u{g}lam, Irmak and Schmuck, Anne-Kathrin},
  title =	{{Solving Odd-Fair Parity Games}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{34:1--34:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.34},
  URN =		{urn:nbn:de:0030-drops-194077},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.34},
  annote =	{Keywords: parity games, strong transition fairness, algorithmic game theory}
}

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Documents authored by Sağlam, Irmak

Infinite-State Games with Energy Objectives Beyond Counters

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Directed Regular and Context-Free Languages

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Solving Odd-Fair Parity Games

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