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 graph games, and logic as an inspiration for designing new graph games. Our aim is modest: we propose a perspective that complements existing game-theoretic 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.
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- 1.
The travel game matches the usual evaluation game for the \(\mu \)-calculus formula \(\nu p \cdot \, \Box \, \Diamond \, p\).
- 2.
However, van Benthem, Mierzewski and Zaffora (2019) has new axiomatization techniques combining hybrid logic and dynamic-epistemic logic.
- 3.
For a general study of graph change encompassing sabotage, but also other destructive mechanisms, see the systematic approach in Areces et al. (2015), which studies more general hybrid logics of graph change.
- 4.
If there is such a rich subtree, Traveler can obviously stay inside it, no matter where Demon cuts, and eventually, as the game moves down the tree, Traveler ends up in an endpoint in the goal region. If there is no rich subtree, then each point t in the subtree has at most one successor u whose subtree is rich. For, if there were two such successors, then there is a rich subtree at s after all. But then, Demon can cut the connection to this single rich subtree (if one exists at all) and force Traveler to a situation without a rich subtree. In the end, this forces Traveler into an endpoint outside of the goal region.—Note that, if Traveler can move back and forth in the tree, this outcome changes. Suppose s is not a winning node for Traveler, but there are three sibling nodes that are winning. Then, no matter how Demon cuts, Traveler can move back one step, and then still reach a winning node. For a more precise version of this argument, see the fixed-point analysis in Sect. 2.2. The existence of a rich subtree is not definable in basic modal logic since one can always create rich subtrees modulo bisimulation by duplicating parts of trees. It is not even first-order definable, Valentin Goranko (p.c.).
- 5.
This classical geometric setting is also a good antidote to a hidden bias in the literature toward using only simple two-dimensional ‘planar graphs’ in sabotage scenarios.
- 6.
This is not to say that the connection between graph games and ‘matching’ logics is entirely unproblematic, and we will also raise some conceptual problems in Sect. 4.
- 7.
Our examples in this paper are mainly cast as sequential games: but it would make sense to also consider versions that allow simultaneous moves for the players.
- 8.
However, there appears to be no single generic formula in the basic modal language, or even in the modal \(\mu \)-calculus, that expresses the property of reaching the goal first.
- 9.
There are much more complex modal ‘product logics’ (Gabbay and Shethman 1998), but these are not needed here.
- 10.
The crux is that working with just variables x, y for our pairs, bounded existential quantifiers for travel modalities at a point can reuse the variable marking the other point.
- 11.
There is an interesting analogy between the meeting game and modal bisimulation. A bisimulation is a relation Z between points s, t in two models M, N such that each step from s to an accessible point \(s'\) in M can be matched by a step from t to an accessible \(t'\) in N. Here ‘matched’ means that \(s' Z t'\) holds. The game version of bisimulation is much like our meeting/avoiding game. Now it is known that the notion of bisimulation cannot be expressed with two first-order variables (van Benthem et al. 2009), whence its meta-theory is somewhat complex.
- 12.
In other widely played parlor games, such as Halma or Chinese Checkers, barriers of occupied points can be jumped over, but we will not consider such variations here.
- 13.
Interestingly, Grossi and Rey (2019) shows how to view graph-changing games as a model for argumentation.
- 14.
This variety of relevant logics, whose mutual relations are sometimes not so easy to establish, also shows that our matching of games with even just modal logics is not an automatic process, and that there may even be significantly different logics for the same game. For a recent example concerning modal poison logics, cf. Areces, Mierzewski and Zaffora (2018).
- 15.
The number 2 here reflects the out-degree 2 of the ‘rich subtrees’ of Footnote 3, where local sabotage had the same power for Demon as global sabotage.
- 16.
There are other natural ways of changing the sabotage game. Say, instead of deleting links, we might also let players delete points and all links leading to them. This makes Demon much more powerful than in the original game. Even so, the logical theory of point deletion shows many similarities with that of link deletion, cf. Chen (2018), and for a broader study connecting up with hybrid logic and first-order logic (van Benthem, Mierzewski and Zaffora 2019). Another natural variation would allow cutting off all links according to a definable rule in a graph. In a uniform global case, this is like link cutting in dynamic-epistemic logic. But the theory of a more realistic local variant shares many features with other complex logics discussed here (Li 2018).
- 17.
Such variations need not be a priori, they are just as well suggested by actual board games that are reminiscent of the sabotage game, such as ‘Quoridor’ (Glendenning 2005).
- 18.
There are many further possible moves in graph games than those discussed here. A quick survey of common parlor games reveals of the host of further types, having to do with how players can move counters, jump across barriers: in single steps or in iterated sequences, and so on.
- 19.
There are precedents for this restriction in other logical literature on game design (Ågotnes and van Ditmarsch 2011).
- 20.
In its turn, this meta-perspective might suggest a generalized ‘protocol semantics’ where models may constrain available graph changes (van Benthem et al. 2009). This widening of the standard models has been used to lower the complexity of dynamic-epistemic logics, and it might also improve our earlier treatment by lowering the current high complexity of logics of graph change.
- 21.
Even so, this is just one line, and the emphasis on modal logics in this paper may be a bit biased. If our focus had been on infinite graph games, then an alternative attractive formalism would have been that of temporal logics. These, too, can be very useful for defining complex behavior and strategies over time.
- 22.
For an up to date survey, see the Stanford Encyclopedia entry (van Benthem 2019).
- 23.
A simple illustration is that a position in a graph does not tell us how the game will go until we know which player has the turn (a game-internal notion) and needs to move there.
- 24.
Slightly richer invariants would arise on annotated graphs that encode some procedural game-internal information by means of additional unary properties of nodes.
- 25.
To see how remarkable this is, it suffices to make some obvious variations on well-known evaluation games, and observe how an immediate match with standard logics vanishes.
- 26.
After the first version of this paper started circulating, a solution to the question posed here has been found using the countable random graph, where truth coincides with truth almost everywhere in finite models. Mierzewski (2018) shows that, if the goal region has at least two points, all positions in the random graph are winning for Traveler—making the sabotage game massively favorable to Traveler. For special model classes, however, the result remains open.
- 27.
In a broader perspective, the logic and games connection pursued here would also need a third component, namely, the study of games in computer science. These games go back a long time, cf. Aigner and Fromme (1984), Grädel et al. (2002), and they would form a natural complement to our discussion here.
- 28.
There is much more to the match between graph games and evaluation games than what we have probed here, especially, when one tries to correlate equivalent formulations of the same graph game with equivalent formulas in matching fixed-point logics, cf. Grossi (2013).
- 29.
In some cases, the ballast may even be misleading. For instance, it is emphasized in Aucher et al. (2017) how the valid equivalence \((\Diamond \top \wedge \, \blacksquare \, \Diamond \, p) \leftrightarrow \langle {\geqslant } 2 \rangle \, p\) of the modal logic for local sabotage is a fluke, in being not schematically valid. But as we saw in Sect. 2, for just defining the winning positions of Traveler, this equivalence was the right observation to make.
- 30.
One concrete way in which players might have limited observational powers in graph games is through ‘short sight’ in the sense of Grossi and Turrini (2012). We might assume that localized players only see the graph up to a certain reachability depth around them. Again, these dynamics can be described in dynamic-epistemic terms (Liu et al. 2015).
- 31.
Other scenarios with imperfect information that may be connected to graph games are the Boolean Network Games of Seligman and Thompson (2015), with their connections with the epistemic friendship logic of Girard et al. (2014). The social network dynamics in this work is not the same as the graph dynamics in the present paper, since agents do not move, but can only change local properties triggered by those of their neighbors—but a serious comparison seems in order.
- 32.
These approaches can also be combined in felicitous ways, witness the joint methodology of logical analysis of agent types, simulations, and cognitive experiments pursued in Ghosh (2018).
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This research is supported by Tsinghua University Initiative Scientific Research Program (2017THZWYX08).
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van Benthem, J., Liu, F. (2020). Graph Games and Logic Design. In: Liu, F., Ono, H., Yu, J. (eds) Knowledge, Proof and Dynamics. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-2221-5_7
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