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Mathematical Operational Semantics and Finitary System Behaviour

Mathematical Operational Semantics and Finitary System Behaviour. Stefan Milius , Marcello Bonsangue , Robert Myers, Jurriaan Rot. Motivation. Process algebra : SOS rules specify algebraic operations on system behaviour .

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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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