Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory
Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory
CONCUR '95 Proceedings of the 6th International Conference on Concurrency Theory
Quantitative Analysis and Model Checking
LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science
Quantitative stochastic parity games
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Quantitative solution of omega-regular games
Journal of Computer and System Sciences - STOC 2001
Abstraction, Refinement And Proof For Probabilistic Systems (Monographs in Computer Science)
Abstraction, Refinement And Proof For Probabilistic Systems (Monographs in Computer Science)
Automatic verification of probabilistic concurrent finite state programs
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Perfect information stochastic priority games
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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The quantitative @m-calculus qM@m extends the applicability of Kozen's standard @m-calculus [D. Kozen, Results on the propositional @m-calculus, Theoretical Computer Science 27 (1983) 333-354] to probabilistic systems. Subsequent to its introduction [C. Morgan, and A. McIver, A probabilistic temporal calculus based on expectations, in: L. Groves and S. Reeves, editors, Proc. Formal Methods Pacific '97 (1997), available at [PSG, Probabilistic Systems Group: Collected reports, http://web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html]; also appears at [A. McIver, and C. Morgan, ''Abstraction, Refinement and Proof for Probabilistic Systems,'' Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 9], M. Huth, and M. Kwiatkowska, Quantitative analysis and model checking, in: Proceedings of 12th annual IEEE Symposium on Logic in Computer Science, 1997] it has been developed by us [A. McIver, and C. Morgan, Games, probability and the quantitative @m-calculus qMu, in: Proc. LPAR, LNAI 2514 (2002), pp. 292-310, revised and expanded at [A. McIver, and C. Morgan, Results on the quantitative @m-calculus qM@m (2005), to appear in ACM TOCL]; also appears at [A. McIver, and C. Morgan, ''Abstraction, Refinement and Proof for Probabilistic Systems,'' Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 11], A. McIver, and C. Morgan, ''Abstraction, Refinement and Proof for Probabilistic Systems,'' Technical Monographs in Computer Science, Springer, New York, 2005, A. McIver, and C. Morgan, Results on the quantitative @m-calculus qM@m (2005), to appear in ACM TOCL] and by others [L. de Alfaro, and R. Majumdar, Quantitative solution of omega-regular games, Journal of Computer and System Sciences 68 (2004) 374-397]. Beyond its natural application to define probabilistic temporal logic [C. Morgan, and A. McIver, An expectation-based model for probabilistic temporal logic, Logic Journal of the IGPL 7 (1999), pp. 779-804, also appears at [A. McIver, and C. Morgan, ''Abstraction, Refinement and Proof for Probabilistic Systems,'' Technical Monographs in Computer Science, Springer, New York, 2005, Chap.10]], there are a number of other areas that benefit from its use. One application is stochastic two-player games, and the contribution of this paper is to depart from the usual notion of ''absolute winning conditions'' and to introduce a novel game in which players can ''draw''. The extension is motivated by examples based on economic games: we propose an extension to qM@m so that they can be specified; we show that the extension can be expressed via a reduction to the original logic; and, via that reduction, we prove that the players can play optimally in the extended game using memoryless strategies.