{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T10:53:46Z","timestamp":1772794426118,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2008,2,1]],"date-time":"2008-02-01T00:00:00Z","timestamp":1201824000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2008,2]]},"abstract":"<jats:p>Behavioural equivalences are a means of establishing whether computing systems possess the same properties. The specific set of properties that are preserved by a specific behavioural equivalence clearly depends on how the system behaviour is observed and can usually be characterised by means of a modal logic. In this paper we consider three different approaches to the definition of behavioural equivalences \u2013 bisimulation, testing and trace \u2013 applied to three different classes of systems \u2013 non-deterministic, probabilistic and Markovian \u2013 and we survey the nine resulting modal logic characterisations, each of which stems from the Hennessy\u2013Milner logic. We then compare the nine characterisations with respect to the logical operators, in order to emphasise the differences between the three approaches in the definition of behavioural equivalences and the regularities within each of them. In the probabilistic and Markovian cases we also address the issue of whether the probabilistic and temporal aspects should be treated in a local or global way and consequently whether the modal logic interpretation should be qualitative or quantitative.<\/jats:p>","DOI":"10.1017\/s0960129507006408","type":"journal-article","created":{"date-parts":[[2008,3,5]],"date-time":"2008-03-05T14:42:14Z","timestamp":1204728134000},"page":"29-55","source":"Crossref","is-referenced-by-count":8,"title":["A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems"],"prefix":"10.1017","volume":"18","author":[{"given":"MARCO","family":"BERNARDO","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"STEFANIA","family":"BOTTA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2008,2,1]]},"reference":[{"key":"S0960129507006408_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2005.10.042"},{"key":"S0960129507006408_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0039071"},{"key":"S0960129507006408_ref12","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1006\/inco.1995.1123","article-title":"Reactive, Generative and Stratified Models of Probabilistic Processes","volume":"121","author":"van","year":"1995","journal-title":"Information and Computation"},{"key":"S0960129507006408_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/S1571-0661(04)00121-5"},{"key":"S0960129507006408_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00090-1"},{"key":"S0960129507006408_ref17","volume-title":"Communicating Sequential Processes","author":"Hoare","year":"1985"},{"key":"S0960129507006408_ref14","doi-asserted-by":"publisher","DOI":"10.1145\/2455.2460"},{"key":"S0960129507006408_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90065-X"},{"key":"S0960129507006408_ref13","doi-asserted-by":"publisher","DOI":"10.1145\/4221.4249"},{"key":"S0960129507006408_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0039056"},{"key":"S0960129507006408_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/B978-044482830-9\/50019-9"},{"key":"S0960129507006408_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/0890-5401(91)90030-6"},{"key":"S0960129507006408_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(84)90113-0"},{"key":"S0960129507006408_ref15","doi-asserted-by":"crossref","unstructured":"Hermanns H. (2002) Interactive Markov Chains. Springer-Verlag Lecture Notes in Computer Science 2428.","DOI":"10.1007\/3-540-45804-2"},{"key":"S0960129507006408_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00344-9"},{"key":"S0960129507006408_ref9","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1999.2808"},{"key":"S0960129507006408_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-44618-4_23"},{"key":"S0960129507006408_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-48778-6_13"},{"key":"S0960129507006408_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511569951"},{"key":"S0960129507006408_ref3","doi-asserted-by":"crossref","unstructured":"Bernardo M. and Botta S. (2006) Modal Logic Characterization of Markovian Testing and Trace Equivalences. In: Proc. of the First Int. Workshop on Logic, Models and Computer Science (LMCS 2006), Camerino (Italy). Electronic Notes in Theoretical Computer Science.","DOI":"10.1016\/j.entcs.2006.07.024"},{"key":"S0960129507006408_ref2","doi-asserted-by":"crossref","unstructured":"Bernardo M. (2007) Non-Bisimulation-Based Markovian Behavioral Equivalences. To appear in Journal of Logic and Algebraic Programming.","DOI":"10.1016\/j.jlap.2007.02.002"},{"key":"S0960129507006408_ref21","volume-title":"Communication and Concurrency","author":"Milner","year":"1989"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129507006408","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T17:45:06Z","timestamp":1557337506000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129507006408\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,2]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,2]]}},"alternative-id":["S0960129507006408"],"URL":"https:\/\/doi.org\/10.1017\/s0960129507006408","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,2]]}}}