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Mathematical Operational Semantics and Finitary System Behaviour

This seminar presents an exploration of mathematical operational semantics with a focus on generalizing Aceto's Theorem within the framework of abstract GSOS rules formulated by Turi and Plotkin. We investigate the interplay between syntax and semantics in system behaviour, examining operations on rational behaviour in various contexts, including streams and labelled transition systems. Applications and examples such as Milner's CCS combinators and rational formal power series will be discussed, highlighting the advancements in operational models and the unique features of finitely presentable objects.

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Mathematical Operational Semantics and Finitary System Behaviour

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  1. Mathematical Operational SemanticsandFinitary System Behaviour Stefan Milius, Marcello Bonsangue, Robert Myers, Jurriaan Rot

  2. Motivation Processalgebra: SOS rulesspecifyalgebraicoperations on systembehaviour. GSOS format well-behavedoperations(bisimilarityis a congruence) Aceto‘s Theorem: The termmodelof a simple GSOSspecificationisregular. Turi& Plotkin(Power et al., Bartels, Klin, …): Mathematicaloperational semanticsInterplay betweensyntaxandsemantics (sosrules) capturedbydistributive laws Main question: Can Aceto‘s Theorem begeneralizedtomathematical operational semantics? Ourresults: GeneralizationofAceto‘s Theorem Abstract ruleformatspecifyingoperations on rational behaviourApplications:concreteformatsfor: streams, (weighted) LTS‘s, (non-)determ. automata B. Bloom, S. Istrail & A. Meyer: Bisimlationcan‘tbetraced. JACM 42, 1995. Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  3. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  4. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  5. Operations on behaviour SOS rulesspecifyalgebraicoperations on systembehaviour. Example:Milner‘sCCS combinators Example: thezip-operation on streams: Example: theshuffle-operation on languages: Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  6. Abstract GSOS rules behaviourfunctor Turi‘s and Plotkin‘s abstract GSOS rules: free (term) monad signaturefunctor operationalmodel initial ¸-bialgebra final denotationalmodel Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  7. Example: labelledtransitionsystems B. Bloom, S. Istrail & A. Meyer: Bisimulationcan‘tbetraced. J. ACM 42, 1995. GSOS format • Classicaltransitionsystemspecificationswithrulesofthe form operationsymbolfromgivensignatureΣ Example:Milner‘sCCS combinators Rational Operational Models | MFPS XXIX | June 23, 2013

  8. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  9. Example: labelledtransitionsystems L. Aceto: GSOS and Finite Labelled Transition Systems, TCS 131, 1994. Aceto‘s Simple GSOS • Classicaltransitionsystemspecificationswithrulesofthe form Rational Operational Models | MFPS XXIX | June 23, 2013

  10. Aceto‘s Theorem • Examples: infinitedependency finitedependency • Theorem (L. Aceto). • For a boundedtransitionsystemspecificationhavingfinite dependencythe operational modelisregular. L. Aceto: GSOS and finite labelledtransitionsystems. TCS 131, 1994. Howtogeneralizethisto distributive laws? Rational Operational Models | MFPS XXIX | June 23, 2013

  11. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  12. BipointedSpecifications Given: Definition: Bipointedspecificationsarenaturaltransformations Example: bipointedspecifications = simple GSOS specificationwithboundedopns Whatabout finite dependency? Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  13. Finite dependency Definition: Preservingfinitelypresentableobjects Example: Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  14. Aceto‘s Theorem generalized Operational model unique Theorem. Example. Definition. finitelypresentableobjects Rational Operational Models | MFPS XXIX | June 23, 2013

  15. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  16. (Rational) DenotationalModel Denotationalmodel unique Nowconsider: finallocally finite F-coalgebra finitelypresentableobjects Proposition. J. Adamek, S. Milius, J. Velebil: Iterative Algebrasat Work, MSCS 2006 Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  17. Examples More examples: rational formal power series,rational Ʃ-trees, rational ¸-trees, … Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  18. Operations on the rational fixpoint Extends: M. Bonsangue, S. Milius, J. Rot: On the specification of operations on the rational behaviour of systems, EXPRESS/SOS 2012. Theorem. „rational denotationalmodel“ unique F-coalgebahomomorphism Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  19. Whataboutmoregeneralruleformats? Counterexample: rational behaviourisnot closedunderoperationsspecifiedbyabstract GSOS rules Conjecture: all results still hold trueforKlin‘s „coGSOS“ laws: cofreecomonad on F

  20. Overview • Abstract GSOS rulesofTuri & Plotkin • Simple GSOS andAceto‘s Theorem • GeneralizationofAceto‘s Theorem • Operations on rational behaviour • Applications Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  21. Application 1: Labelled Transition Systems Corollary. Operationsdefinedby simple GSOS rulesrestricttotherational fixpointof F. Corollary. Aceto's Theorem. coproductof all finitelabelledtransitionsystemsmodulobisimilarity • Examples. All CCS combinators, e.g. Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  22. Application 2: Streams Corollary. Operationsdefinedbybipointedstream SOS rulesrestricttoeventuallyperiodicstreams. • Examples. coGSOSrule Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  23. Application 3: Non-deterministicautomata • Bipointed NDA SOS specifications acceptingstates • Remark. This formatis not complete w.r.t. tobipointedspecifications. • Corollary. • Example. Shuffle operator Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  24. Application 4: DeterministicAutomata join-semilatticeswithbottom • Bipointed DA SOS specifications • Remark. Not complete w.r.t. tobipointedspecifications. • Corollary. Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  25. Application4: DeterministicAutomata • Shuffle operator • Sequentialcomposition • Other examples: regularexpressionopns incl. Kleene star, … • Corollary. (togeneralizationofAceto‘stheorem) Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  26. Application 5: weightedtransitionsystems • Bipointed WTS SOS specifications • Remark. This formatis not complete w.r.t. tobipointedspecifications. • Corollary. • Example. Priorityoperator Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

  27. Conclusions • Mathematicaloperational semanticsmeetsfiniteness: • bipointedspecificationscaptureAceto‘s simple GSOS • GeneralizationofAceto‘sresultthattheoperational modelisregular • rational fixpointisclosedunderoperationsspecifiedbybipointedspecifications • Manyinterestingapplications: • labelledtransitionsystems, streams, (non-)deterministicautomata, weightedtransitionsystems, deterministicautomata on join-semilattices, etc. Future work • More on bipointedspecificationsin algebraiccategories(e.g. completeformats, othercategories: locally finite varieties, …) • Rational andcontextfree power series • Localfinitenessof operational modelsand rational fixpoints: decidabilityofbisimilarity, algorithms, tooldevelopment Mathematical Operational SemanticsandFinitary System Behaviour | Oberseminar Erlangen | October 29, 2013

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