Results on the quantitative mu-calculus qMu
<|reference_start|>Results on the quantitative mu-calculus qMu: 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.<|reference_end|>
@article{mciver2003results,
    title={Results on the quantitative mu-calculus qMu},
    author={Annabelle McIver and Carroll Morgan},
    journal={arXiv preprint arXiv:cs/0309024},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0309024},
    primaryClass={cs.LO cs.GT}
}