Graph Games and Logic Design

Abstract

Graph games are interactive scenarios with a wide range of applications. This position paper discusses old and new graph games in tandem with matching logics, and identifies general questions behind this match. Throughout, we pursue two strands: logic as a way of analyzing existing games, and logic as an inspiration for designing new games. Our aim is modest: we propose a style of thinking that complements existing gametheoretic and computational ones, we raise questions, make observations, and suggest research directions-technical results are left to future work. But frankly, our main aim with this survey paper is to show that graph games are concrete, fun, easy to grasp, and yet challenging to study. 1 Two graph games Graph games are played on graphs, directed or undirected, with one or many relations, perhaps with annotations at vertices. This setting may seem specialized-but the scope of these games is broad, with applications to computation, argumentation, communication, social networks, warfare, and other scenarios. To introduce our perspective, we start with two concrete examples. For general background on logic and games presupposed in this paper, we refer to [10], [13].

On the Right Path: A Modal Logic for Supervised Learning

“…We are inspired by the work on SG [19], SML [5] and their application to formal learning theory [12]. This article is also relevant to other work studying graph games with modal logics, such as [9,14,18,20]. Technically, the logic SLL has resemblances to several recent logics with model modifiers, such as [2,3,4].…”

Losing connection: the modal logic of definable link deletion

“…It remains to note that this article fits with the general program recently proposed in [12] for a much broader study of analysis and design for graph games in tandem with matching modal logics. In particular, it proposed various meaningful new games, and identified general questions behind the match between logic and game.…”

The Modal Logic of Stepwise Removal

“…The winning positions of the Demon and the Traveler can be analyzed using standard modalities, together with additional modalities describing what holds in a pointed model after one link has been removed from the current accessibility relation. However, validity in modal logics for various graph games of this sort can be undecidable, and the resulting model theory is quite complex (see [Aucher et al, 2018] and [van Benthem and Liu, 2020]). This difference in complexity calls for an explanation.…”

“…However, validity in modal logics for various graph games of this sort can be undecidable, and the resulting model theory is quite complex (see (Aucher et al. , 2018) and (van Benthem & Liu, 2020)).…”

“…Note on game perspectives. While not strictly necessary for the proof of Theorem 4.1, the above traveling game with point removal over the initial structure is independently appealing, and it suggests links with the graph games that motivate sabotage logics and related logics for graph change mentioned in §1 (van Benthem & Liu, 2020). As a further perspective, the above traveling game is virtually identical to the standard logical evaluation game for the crucial quantified Boolean formula in the above proof.…”