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Complete Axioms for Stateless Connectors

CALCO 2005, Swansea, Wales, UK, 3-6 September 2005. Complete Axioms for Stateless Connectors. Ivan Lanese Dipartimento di Informatica Università di Pisa. joint work with Roberto Bruni and Ugo Montanari Dipartimento di Informatica Università di Pisa. Roadmap. Why connectors?

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Complete Axioms for Stateless Connectors

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  1. CALCO 2005, Swansea, Wales, UK, 3-6 September 2005 Complete Axioms for Stateless Connectors Ivan Lanese Dipartimento di Informatica Università di Pisa joint work with Roberto Bruni and Ugo Montanari Dipartimento di Informatica Università di Pisa

  2. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  3. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  4. Interaction and connectors • Modern systems are huge • composed by different entities that collaborate to reach a common goal • interactions are performed at some well-specified interfaces… • …and are managed by connectors • Connectors allow separation between computation and coordination CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  5. Coordination via connectors • Connectors useful to • ensure compatibility among independently developed components • allow to reuse them • allow run-time reconfiguration • Connectors exist at different levels of abstraction (architecture, applications, …) CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  6. Which connectors? • We follow the algebraic approach • system as term in an algebra • We propose an algebra of simple stateless connectors for synchronization and mutual exclusion • expressive enough to model the architectural connectors of CommUnity [IFIP TCS 04] • build on symmetric monoidal categories and P-monoidal categories • related to Stefanescu’s flow algebras and REO connectors CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  7. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  8. The tile model • Operational and observational semantics of open concurrent systems • compositional in space and time • Category based CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  9. parallel composition sequential composition Configurations output interface input interface CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  10. Configurations output interface input interface parallel composition functoriality sequential composition CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  11. Configurations output interface input interface parallel composition functoriality + symmetries = symmetric monoidal cat sequential composition CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  12. concurrent computation Observations initial interface final interface CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  13. initial configuration trigger effect final configuration Tiles • Combine horizontal and vertical structures through interfaces CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  14. Tiles • Compose tiles • horizontally CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  15. Tiles • Compose tiles • horizontally • (also vertically and in parallel) symmetric monoidal double cat CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  16. Tiles as LTS • Structural equivalence • axioms on configurations (e.g. symmetries) • LTS • states = configurations • transitions = tiles • labels = (trigger,effect) pairs • Observational semantics • tile trace equivalence/bisimilarity • congruence results for suitable formats CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  17. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  18. Connectors • Connectors to express synchronization and mutual exclusion constraints on local choices • Possible outcomes: tick (1, action performed) or untick (0, action forbidden) • Operational semantics via tiles and observational semantics via tile bisimilarity • Denotational semantics via tick-tables (boolean matrices) • Complete axiomatization of connectors and reduction to normal form CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  19. ! ! 0 0 Basic connectors Symmetry Duplicator Bang Mex Zero CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  20. 0 1 Notation • Only two kinds of allowed observations • Initial and final states always coincide (since connectors are stateless) • Thus we can use a “flat” notation for tiles CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  21. Operational semantics • Tiles specify the behaviours of basic connectors • When composed, connectors must agree on the observation at the interfaces CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  22. Basic tiles (I) Dual connectors have dual tiles CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  23. ! 0 ! Basic tiles (II) CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  24. … 001 111 … … 0010  0101  … domain is {input 3, outputs 1,2,3} Denotational semantics • Connectors can be seen as black boxes • input interface • output interface • admissible observations on interfaces • Denotations are just matrixes • n inputs  2n rows • m outputs  2m columns • dual is transposition • sequential composition is matrix multiplication • parallel composition is matrix expansion • cells are filled with empty/copies of matrices 1 1 1 1 2 2 2 2 3 3 3 3 4 4 CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  25. 0 1 . . 00 01 10 11 0  0 0   0  1  1 1  1  00 01 10 11 0 00  01  ! 00 01 10 11 10  0  11  1   Denotational semantics CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  26. Semantic correspondance • Tile bisimilarity coincides with tile trace equivalence (stateless property) • Two connectors are tile bisimilar iff they have the same associated tick-tables • Tile bisimilarity is a congruence CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  27. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  28. Axiomatization • We want to find a complete axiomatization for the bisimilarity of connectors • Synch-connectors (without mex and zero) • symmetries, duplicators and bangs form a gs-monoidal category • adding dual connectors we get a P-monoidal category • No simple known axiomatization works for mex, but we show an axiomatization for the full class of connectors CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  29. = = ! = = Gs-monoidal axioms CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  30. = = = . ! ! Additional P-monoidal axioms CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  31. Synch-tables • Entry with empty domain is enabled • Entries are closed under (domains) • union • intersection • difference • complementation • Base: set of minimal (non empty) entries w.r.t. domain intersection • Each synch-table is determined by its base CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  32. ! ! … … … … Normal form • Sort connectors Central points (correspond to cells of the base) Hiding connectors directly connected to central points Central points are connected to at least one external interface CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  33. Properties • All the axioms bisimulate (correctness) • Each connector can be transformed in normal form using the axioms • Bijective correspondance between synch-tables and connectors in normal form CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  34. Roadmap • Why connectors? • The tile model • Stateless connectors • Axiomatization of synch-connectors • Adding mutual exclusion • Concluding remarks CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  35. Adding mex and zero • Synch-connectors are not expressive enough (only synchronization) • Adding mex and zero to express mutual exclusion constraints and enforce inactivity • Just mex has to be inserted: zero and dual connectors can be derived • Mex and zero form a gs-monoidal category CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  36. . 0 1 0  00 01 10 11 00  0 1 = 1 x 0  01  0  1  1 10  11 0 0 Obtaining zero connector = def = ! CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  37. Obtaining comex connector • Hiding and synchronization allow to flip wires ! = ! ! CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  38. Looking for axiomatization of mex = CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  39. Looking for axiomatization of mex =  CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  40. Looking for axiomatization of mex =   CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  41. Looking for axiomatization of mex =   CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  42. Looking for axiomatization of mex =   CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  43. Key axioms  CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  44. ! = ! ! ! Key axioms = ! CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  45. = Some axioms about mex-dup = CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  46. 0 0 = . = 0 ! ! 0 0 = = 0 0 0 = = 0 Some axioms about zero 0 = CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  47. = ! 0 0 = ! 0 = ! 0 = . A sample proof 0 0 CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  48. ! Additional axioms = CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  49. An axiom scheme ! … … ! ! CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

  50. An axiom scheme ! … ! CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari

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