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ACM Transactions on Programming Languages and Systems (TOPLAS)
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Competitive Markov decision processes
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TACAS 2001 Proceedings of the 7th International Conference on Tools and Algorithms for the Construction and Analysis of Systems
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STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
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STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
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APSEC '01 Proceedings of the Eighth Asia-Pacific on Software Engineering Conference
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A survey of stochastic ω-regular games
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Information and Computation
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ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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The µ-calculus is a powerful tool for specifying and verifying transition systems, including those with demonic (universal) and angelic (existential) choice; its quantitative generalisation qMµ extends that to probabilistic choice. We show for a finite-state system that the straightforward denotational interpretation of the quantitative µ-calculus is equivalent to an operational interpretation given as a turn-based gambling game between two players. Kozen defined the standard Boolean-typed calculus denotationally; later Stirling gave it an operational interpretation as a turn-based game between two players, and showed the two interpretations equivalent. By doing the same for the quantitative real-typed calculus, we set it on a par with the standard calculus, in that it too can benefit from a solid interface linking the logical and operational frameworks. Stirling's game analogy, as an aid to intuition, continues in the more general context to provide a surprisingly practical specification tool, meeting for example Vardi's challenge to "figure out the meaning of AFAXp" as a branching-time formula. We also show that memoriless strategies suffice for achieving the minimax value of a quantitative game, when the state space is finite.