Abstract
Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic – an extension of modal logic that allows for explicit reference to worlds within the language. The main result is that the systematic search of winning strategies over all models can be finitized and thus reformulated as a proof system.
Research supported by FWF projects P 32684 and W 1255.
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Notes
- 1.
\(\mathcal {P}(\mathsf {W})\) denotes the power set of \(\mathsf {W}\).
- 2.
Our definitions of \(\mathcal {M},\,\mathsf {w}\,\models \,\phi \,\rightarrow \,\psi \) and \(\mathcal {M},\,\mathsf {w}\,\models \,\Box \,\phi \) is equivalent to the usual “if \(\mathcal {M},\,\mathsf {w}\,models\,\phi \), then \(\mathcal {M},\,\mathsf {w}\,models\ psi\)" and “for all \(\mathsf {v}\,\in \,\mathsf {W}\), if \(\mathsf {w}\mathsf {R}\mathsf {v}\), then \(\mathcal {M},\,\mathsf {v}\,\models \,\phi \)". Our formulations are more easily representable as game rules.
- 3.
We often write \(\mathbf {G}(g)\), if \(\mathcal {M}\) is clear from context.
- 4.
For example, for \((R_\rightarrow )\), if \(g=\mathbf {O},\mathsf {w}:\psi _1 \rightarrow \psi _2\), then g is labelled “Y” and its immediate subtrees are \(\mathbf {G}(\mathbf {P}, \mathsf {w}: \psi _1)\) and \(\mathbf {G}(\mathbf {O}, \mathsf {w}: \psi _2)\).
- 5.
This deviates from the standard, more general game-theoretic definition of a winning strategy. It is sufficient for our purposes.
- 6.
Unless noted otherwise, we identify a disjunctive state with the multiset of its game states. This implies that we consider two disjunctive states to be equal if they contain the same game states in the same numbers.
- 7.
A disjunctive state is called elementary if all its game states are elementary, i.e. they involve only elementary formulas.
- 8.
If \(\rho \) is clear from context (or not important), we conveniently write \(\mathbf {DG}(D)\).
- 9.
Due to this rule, the game may continue forever.
- 10.
If one changes the disjunctive game such that infinite runs are considered winning for You, then one can extract the model and the strategies from Your winning strategy in \(\mathbf {DG}(D,\rho )\).
- 11.
In this proof, we conveniently write \(D\bigvee g\) to indicate that \(\rho (D\bigvee g)=g\).
- 12.
We say that a game state g appears along \(\pi \), if it occurs as part of a disjunctive state in \(\pi \).
- 13.
We say that \(\varGamma \vdash \varDelta \) is elementary iff its associated disjunctive state is. Similarly, \(\varGamma \vdash \varDelta \) is called valid, if its associated disjunctive state is game valid.
References
van Benthem, J.: Logic in Games. The MIT Press, Cambridge (2014)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2002). https://books.google.at/books?id=gFEidNVDWVoC
Blackburn, P.: Modal logic as dialogical logic. Synthese 127, 57–93 (2001). https://doi.org/10.1023/A:1010358017657
Blackburn, P., Tzakova, M.: Hybridizing concept languages. Ann. Math. Artif. Intell. 24, 23–49 (2000). https://doi.org/10.1023/A:1018988913388
Blackburn, P., Tzakova, M., Gargov, I.: Hybrid languages and temporal logic. Log. J. IGPL 7, 27–54 (1999). https://doi.org/10.1093/jigpal/7.1.27
Fermüller, C., Lang, T., Pavlova, A.: From truth degree comparison games to sequents-of-relations calculi for Gödel logic. In: Reformat, M.-J., et al. (eds.) IPMU 2020. CCIS, vol. 1237, pp. 257–270. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50146-4_20
Fermüller, C.G., Metcalfe, G.: Giles’s game and the proof theory of Łukasiewicz logic. Stud. Log.: Int. J. Symb. Log. 92(1), 27–61 (2009). http://www.jstor.org/stable/40269050
Hintikka, J.: Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Clarendon Press, Oxford (1973)
Hodges, W., Väänänen, J.: Logic and games. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2019 edn. (2019)
Lorenzen, P., Lorenz, K.: Dialogische Logik. Wissenschaftliche Buchgesellschaft (1978)
Negri, S.: Kripke completeness revisited. In: Primiero, G. (ed.) Acts of Knowledge: History, Philosophy and Logic, pp. 233–266. College Publications (2009)
Väänänen, J.: Models and Games. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2011). https://doi.org/10.1017/CBO9780511974885
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Freiman, R. (2021). Games for Hybrid Logic. In: Silva, A., Wassermann, R., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2021. Lecture Notes in Computer Science(), vol 13038. Springer, Cham. https://doi.org/10.1007/978-3-030-88853-4_9
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