Abstract. The µ-calculus is a powerful tool for specifying and verifying transition systems, including those with both demonic (universal) and angelic (existential) choice; its quantitative generalisation qMµ [17, 29, 9] extends that to probabilistic choice. We show here that for a finite-state system the logical interpretation of qMµ, via fixed-points in a domain of real-valued functions into [0, 1] , is equivalent to an operational interpretation given as a turn-based gambling game between two players. The equivalence sets qMµ on a par with the standard µ-calculus, in that it too can benefit from a solid interface linking the logical and operational frameworks. The logical interpretation provides direct access to axioms, laws and meta-theorems. The operational, game-based interpretation aids the intuition and 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 AF AX p" as a branching-time formula. A corollary of our proofs is an extension of Everett's singly-nested games result in the finite turn-based case: we prove well-definedness of the minimax value, and existence of fixed memoriless strategies, for all qMµ games/formulae, of arbitrary (including alternating) nesting structure.
In REF , a quantitative generalisation of the µ-calculus is proposed, and shown to be able to encode stochastic parity games.
Results on the quantitative mu-calculus qMu
ArXiv
ArXiv
Mathematics Computer Science