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Equivalences and Congruences on Infinite Conway Games∗

Published online by Cambridge University Press:  02 March 2012

Furio Honsell ,
Marina Lenisa  and
Rekha Redamalla
Furio Honsell
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Viale delle Scienze 206, 33100 Udine, Italy. [email protected]; [email protected]
Marina Lenisa
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Viale delle Scienze 206, 33100 Udine, Italy. [email protected]; [email protected]
Rekha Redamalla
Affiliation:
Birla Science Centre, Adarsh Nagar, 500063 Hyderabad, India; [email protected]
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Abstract

Taking the view that infinite plays are draws, we study Conwaynon-terminating games and non-losing strategies. These admit asharp coalgebraic presentation, where non-terminating games are seen as afinal coalgebra and game contructors, such as disjunctivesum, as final morphisms. We have shown, in a previous paper,that Conway’s theory of terminating games can be rephrased naturally in terms of game(pre)congruences. Namely, various conceptually independent notions ofequivalence can be defined and shown to coincide on Conway’sterminating games. These are the equivalence induced by the ordering on surrealnumbers, the contextual equivalence determined by observingwhat player has a winning strategy, Joyal’s categoricalequivalence, and, for impartial games, the denotationalequivalence induced by Grundy semantics. In this paper, wediscuss generalizations of such equivalences to non-terminating games andnon-losing strategies. The scenario is even more rich and intriguing inthis case. In particular, we investigate efficient characterizations of the contextualequivalence, and we introduce a category of fair strategies and acategory of fair pairs of strategies, both generalizing Joyal’s categoryof Conway games and winning strategies. Interestingly, the category of fair pairs capturesthe equivalence defined by Berlekamp, Conway, Guy on loopy games.

Keywords

Conway gamesnon-wellfounded gamescoalgebrasequivalencesJoyal’s category
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 2: 12th Italian Conference on Theoretical Computer Science(ICTCS) , April 2012 , pp. 231 - 259
Copyright
© EDP Sciences 2012

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References

Abramsky, S. and Jagadesaan, R., Games and full completeness for multiplicative linear logic, J. Symb. Log. 59 (1994) 543–574. Google Scholar
P. Aczel, Non-wellfounded sets. CSLI Lecture Notes 14 (1988).
J. Barwise and L. Moss, Vicious Circles. CSLI Lecture Notes 60 (1996).
E. Berlekamp, J. Conway and R. Guy, Winning Ways. Academic Press (1982).
J.H. Conway, On Numbers and Games, 2nd edition (1st edition by Academic Press (1976). AK Peters Ltd. (2001).
Forti, M. and Honsell, F., Set-theory with free construction principles. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1983) 493–522. Google Scholar
Grumberg, O., Lange, M., Leucker, M. and Shoham, S., When not losing is better than winning : abstraction and refinement for the full μ-calculus. Inform. Comput. 205 (2007) 1130–1148. Google Scholar
Grundy, P.M., Mathematics and games. Eureka 2 (1939) 6–8. Google Scholar
F. Honsell and M. Lenisa, Conway Games, algebraically and coalgebraically. Log. Meth. Comput. Sci. 7 (2011).
M. Hyland and A. Schalk, Games on Graphs and Sequentially Realizable Functionals, in Proc. of LICS’02. IEEE Computer Science Press (2002) 257–264.
Kissig, C. and Venema, Y., Complementation of Coalgebra Automata, in Proc. of CALCO’09. Lect. Notes Comput. Sci. 5728 (2009) 81–96. Google Scholar
Jacobs, B. and Rutten, J.J.M.M., A Tutorial on (Co)algebras and (Co)induction. Bull. of EATCS 62 (1997) 222–259. Google Scholar
A. Joyal, Remarques sur la théorie des jeux à deux personnes. Gaz. Sci. Math. du Québec 1 (1977).
P.L. Curien, H. Herbelin, J.L. Krivine and P.A. Melliès, Categorical semantics of linear logic, in Interactive models of computation and program behaviour. Panoramas et Synthèses, Société Mathématique de France 27 (2009).
P.A. Melliès, N. Tabareau and C. Tasson, An explicit formula for the free exponential modality of linear logic, in Proc. of ICALP’09. Lect. Notes Comput. Sci. 555 (2009).
M. Pauly, From Programs to Games : Invariance and Safety for Bisimulation, in Proc. of CSL’09 (2009) 485–496.
Santocanale, L., Free μ-lattices. J. Pure Appl. Algebra 168 (2002) 227–264. Google Scholar
Sprague, R., Über mathematische kampfspiele. Tohoku Math. J. 41 (1935) 438–444. Google Scholar
Thomas, W., Infinite games and verification, in Proc. of CAV’02. Lect. Notes Comput. Sci. 2404 (2002) 58–64. Google Scholar
J. van Benthem, Extensive games as process models, J. Log. Lang. Inf. 11 (2002).