Abstract
We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.
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References
Abramsky S. (2006) Socially responsive, environmentally friendly logic. In: Tuomo A., Pietarinen A.-V. (eds) Truth and games: Essays in honour of Gabriel Sandu. Acta Philosophica Fennica, Helsinki
Abramsky S., Jagadeesan R. (1994) Games and full completeness for multiplicative linear logic. Journal of Symbolic Logic 59: 543–574
Alur R., Henzinger T.A., Kupferman O. (2002) Alternating-time temporal logic. Journal of the ACM 49: 672–713
Baeten, J. C. M., & Verhoef, C. (1995). Concrete process algebra. In S. Abramsky, D. M. Gabbay, & T. S. E. Maibaum (Eds.), Handbook of logic in computer science (Vol. 4, pp. 149–268). London: Semantic Modelling, Clarendon Press.
Bergstra, J.A., Ponse, A., Smolka , S.A. (eds) (2001) Handbook of process algebra. North-Holland, Amsterdam
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. CUP.
Blass A. (1972) Degrees of indeterminacy of games. Fundamenta Mathematicae 77: 151–166
Ghosh, S. (2008). Strategies made explicit in dynamic game logic. In J. van Benthem & E. Pacuit (Eds.), Proceedings of the Workshop on Logic and Intelligent Interaction, ESSLLI 2008, pp. 74–81.
Goldblatt R. (1992) Parallel action: Concurrent dynamic logic with independent modalities. Studia Logica 51: 551–578
Goranko V. (2003) Basic algebra of game equivalences. Studia Logica 75: 221–238
Hansen, H., Kupke, C., & Pacuit, E. (2007). Bisimulation for neighbourhood structures. In M. Haveraaen, U. Montanari, & T. Mossakowski (Eds.), Proceedings of 2nd Conference on Algebra and Coalgebra in Computer Science (CALCO 2007) (pp. 279–293). LNCS 4624.
Harel, D., Kozen, D., & Tiuryn, J. (2000). Dynamic logic. The MIT Press.
Hintikka, J., & Sandu, G. (1997). Game-theoretical semantics. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 361–410). MIT Press.
Hodges W. (1997) Compositional semantics for a language of imperfect information. Journal of the Interest Group in Pure and Applied Logics 5: 539–563
Netchitailov, I. (2000). An extension of game logic with parallel operators. Master’s thesis, University of Amsterdam.
Parikh R. (1985) The logic of games and its applications. Annals of Discrete Mathematics 24: 111–140
Pauly, M. (2001). Logics for social software. Ph.D. Dissertation, University of Amsterdam.
Pauly M., Parikh R. (2003) Game logic—An overview. Studia Logica 75: 165–182
Peleg D. (1987) Concurrent dynamic logic. Journal of the ACM 34: 450–479
Sevenster, M. (2006). Branching of imperfect information: Logic, games and computation. Ph.D. dissertation, University of Amsterdam.
van Benthem, J. (1999). Logic in games. Lecture Notes, Amsterdam and Stanford.
van Benthem J. (2001) Games in dynamic-epistemic logic. Bulletin of Economic Research 53: 219–248
van Benthem J. (2002) Extensive games as process models. Journal of Logic, Language and Information 11: 289–313
van Benthem J. (2003) Logic games are complete for game logics. Studia Logica 75: 183–203
van Benthem, J. (2006). The epistemic logic of IF games. In R. E. Auxier & L. E. Hahn (Eds.), Philosophy of Jaakko Hintikka. Open Court.
van Benthem, J. (2007). In praise of strategies, 2007. In J. van Eijck & R. Verbrugge (Eds.), Games, actions and social software (to appear).
van Benthem J., Ghosh S., Liu F. (2007) Modelling simutaneous games with concurrent dynamic logic. In: van Benthem J., Ju S., Veltman F. (eds) A meeting of the minds—Proceedings of the workshop on logic, rationality and interaction. King’s College Publications, London, pp 243–258
van Benthem, J., & Sarenac, D. (2004). The geometry of knowledge. In J.-Y. Béziau, A. Costa Leite, & A. Facchini (Eds.), Aspects of universal logic (pp. 1–31). Centre de Recherches Sémiologiques, Université de Neuchatel.
van Benthem J., van Eijck J., Stebletsova V. (1994) Modal logic, transition systems and processes. Journal of Logic Computation 4: 1–50
van Eijck, J., & Verbrugge, R. (2008). A fresh look at propositional dynamic logic and game logic. CWI Amsterdam and Department of AI, University of Groningen.
Venema Y. (2003) Representation of game algebras. Studia Logica 75: 239–257
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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van Benthem, J., Ghosh, S. & Liu, F. Modelling simultaneous games in dynamic logic. Synthese 165, 247–268 (2008). https://doi.org/10.1007/s11229-008-9390-y
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DOI: https://doi.org/10.1007/s11229-008-9390-y