Results on the quantitative mu-calculus qMu

Details Results on the quantitative mu-calculus qMu Results on the quantitative mu-calculus qMu

2003-09-15

arxiv.org/abs/cs/0309024v1

Computer Science

Logic in Computer Science

45 pages, 4 figures

Scientific paper

The mu-calculus is a powerful tool for specifying and verifying transition systems, including those with both demonic and angelic choice; its quantitative generalisation qMu extends that to probabilistic choice. We show that for a finite-state system the logical interpretation of qMu, 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 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. 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 qMu games/formulae, of arbitrary (including alternating) nesting structure.

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