{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T17:11:17Z","timestamp":1760202677485,"version":"3.41.2"},"reference-count":1,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2012,11,27]],"date-time":"2012-11-27T00:00:00Z","timestamp":1353974400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>The probabilistic modal {\\mu}-calculus is a fixed-point logic designed for\nexpressing properties of probabilistic labeled transition systems (PLTS's). Two\nequivalent semantics have been studied for this logic, both assigning to each\nstate a value in the interval [0,1] representing the probability that the\nproperty expressed by the formula holds at the state. One semantics is\ndenotational and the other is a game semantics, specified in terms of\ntwo-player stochastic parity games. A shortcoming of the probabilistic modal\n{\\mu}-calculus is the lack of expressiveness required to encode other important\ntemporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL).\nTo address this limitation we extend the logic with a new pair of operators:\nindependent product and coproduct. The resulting logic, called probabilistic\nmodal {\\mu}-calculus with independent product, can encode many properties of\ninterest and subsumes the qualitative fragment of PCTL. The main contribution\nof this paper is the definition of an appropriate game semantics for this\nextended probabilistic {\\mu}-calculus. This relies on the definition of a new\nclass of games which generalize standard two-player stochastic (parity) games\nby allowing a play to be split into concurrent subplays, each continuing their\nevolution independently. Our main technical result is the equivalence of the\ntwo semantics. The proof is carried out in ZFC set theory extended with\nMartin's Axiom at an uncountable cardinal.<\/jats:p>","DOI":"10.2168\/lmcs-8(4:18)2012","type":"journal-article","created":{"date-parts":[[2013,11,29]],"date-time":"2013-11-29T13:21:33Z","timestamp":1385731293000},"source":"Crossref","is-referenced-by-count":13,"title":["Probabilistic modal {\\mu}-calculus with independent product"],"prefix":"10.46298","volume":"Volume 8, Issue 4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4050-3617","authenticated-orcid":false,"given":"Matteo","family":"Mio","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2012,11,27]]},"reference":[{"key":"761:not-found"}],"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/789\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/789\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T19:56:06Z","timestamp":1681242966000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/789"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,27]]},"references-count":1,"URL":"https:\/\/doi.org\/10.2168\/lmcs-8(4:18)2012","relation":{"is-same-as":[{"id-type":"arxiv","id":"1211.1511","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1211.1511","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2012,11,27]]},"article-number":"789"}}