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An advanced course @ ESSLLI 2026
We focus on an emerging direction regarding the well-studied interaction between logic and games — a modal logic approach to graph games. We present different types of games having deep roots in computer science and show how to design modal logics to reason about some core game-theoretic concepts. We provide a general background on the topic and a detailed exploration on two levels. First, we explore specific games and develop formal frameworks to reason about such games. Then, we study the general characteristics underlying these games and offer a unified logical framework to reason about winning strategies. From a technical perspective, we discuss expressiveness, frame definability, computational complexity, axiomatization and so on. By the end of this course, participants are expected to have a grasp on the development and frontier results in this area, to obtain insights into some open problems, and to get familiar with the relevant frameworks including dynamic logic, dynamic-epistemic logic, many-dimensional logic and fixed-point logic.
LECTURERS
TENTATIVE OUTLINE OF THE COURSE
esslli-2026-lecture-1.pdf1st lecture: We introduce a general background on the interaction between games and logic. Then, we motivate the study of graph games with modal logic and show the existing broad research program on the topic. In particular, we will highlight the basic ideas of sabotage games, poison games and hide-and-seek games, and show their applications in relevant fields.
esslli-2026-lecture2.pdf2nd lecture: We dive into the details of sabotage games, poison games and the corresponding modal logics. We present the applications of logic to games and discuss the expressive power of language, computational complexity of decision problems, and the axiomatization of logic. Moreover, we will compare the logic with other important paradigms, including dynamic-epistemic logic, logic of definable link deletion and logic of point deletion.
esslli-2026-lecture3.pdf3rd lecture: We turn to another setting, that of hide-and-seek (i.e., cops and robber) and explore a corresponding modal logic. We study expressiveness, frame definability, axiomatization and computational complexity. We compare the proposed logic with standard many-dimensional logic, provide its hybrid extension, and show how our approach provides significant improvements in terms of distinct properties.
esslli-2026-lecture4.pdf4th lecture: We move to the imperfect information setting and introduce the detailed design of the corresponding hide-and-seek games. We then explore a logical framework that can define the key features of the games and automatically capture the information dynamics. For the resulting logic, we study its axiomatization and show that it has a decidable satisfiability problem. Moreover, we compare our approach to that of dynamic-epistemic logic and show the advantages of our proposal. Finally, we discuss how to apply the ideas to other graph games, including sabotage games.
esslli-2026-lecture5.pdf5th lecture: For the last lecture of the course, we move to a broader setting. First, we show how the techniques developed for LHS can be used to improve the properties of the product logic with the diagonal constant, and this also provides a new way to understand the logic for the hide-and-seek games. Next, for the games discussed in the course, we study some general features and focus on how to express the existence of winning positions of players in an arbitrary setting. To this end, we use substitution as an important tool, discuss different approaches to the notion, and develop a logic of substitution.
PRE-REQUISITES
Participants for the course are expected to be masters and/or PhD students, who are familiar with the basics of modal logic and game-theoretic notions, and may be looking for relevant research topics to explore. However, we will make our material self-contained so that other interested persons can also participate.
REFERENCES
J. van Benthem (2011): Logical Dynamics of Information and Interaction. Cambridge University Press.
J. van Benthem (2014): Logic in Games. The MIT Press.
J. van Benthem & F. Liu, editors (2026): Graph Games and Logic Design: Recent Developments and Future Directions. Springer.
P. Blackburn & B. ten Cate (2006): Pure extensions, proof rules, and hybrid axiomatics. Studia Logica 84(3), pp. 277–322.
P. Blackburn, M. de Rijke & Y. Venema (2001): Modal Logic. Cambridge University Press.
F. Z. Blando, K. Mierzewski & C. Areces (2020): The modal logics of the poison game. In F. Liu, H. Ono & J. Yu, editors: Knowledge, Proof and Dynamics, Logic in Asia: Studia Logica Library, Springer, pp. 3–23.
H. van Ditmarsch, W. van der Hoek & B. Kooi (2008): Dynamic Epistemic Logic. Springer.
P. Du, F. Liu & D. Li (2025): Modal logics for the poison game: Axiomatization and undecidability. The Review of Analytic Philosophy 5, pp. 43–71.
P. Duchet & H. Meyniel (1993): Kernels in directed graph: A poison game. Discrete Mathematics 115, p. 273–276.
R. Fagin, J.Y. Halpern, Y. Moses & M.Y. Vardi (1995): Reasoning about Knowledge. The MIT Press.
D. Grossi & S. Rey (2019): Credulous acceptability, poison games and modal logic. In N. Agmon, M. E. Taylor, E. Elkind & M. Veloso, editors: Proceedings of AAMAS 2019, pp. 1994–1996.
Y. Tu, S. Ghosh, F. Liu & D. Li (2026): A modal approach towards substitutions. Annals of Pure and Applied Logic 117, 103742.
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