CommUnity, Tiles and Connectors

1. Dagstuhl Seminar #04241, 6-11 June 2004 CommUnity, Tiles and Connectors Roberto Bruni Dipartimento di Informatica Università di Pisa Ongoing Work! joint work with José Luiz Fiadeiro Ivan Lanese Antónia Lopes Ugo Montanari

2. Roadmap • Motivation • Background • CommUnity • Tiles • From CommUnity to Tiles • Connectors • Concluding remarks CommUnity, Tiles and Connectors

3. Roadmap • Motivation • Background • CommUnity • Tiles • From CommUnity to Tiles • Connectors • Concluding remarks CommUnity, Tiles and Connectors

4. Motivation I • Categorical Approach • objects are system components (architectural elements) • morphisms express simulation, refinement, … • complex systems are modeled as diagrams • composition via universal construction (colimit) • suited for systems with shared resources • memory, action, channels CommUnity, Tiles and Connectors

5. Motivation II • Categorical Approach • Algebraic Approach • initial algebra where • constants are basic processes • operations compose smaller processes into larger ones • structural axioms collapse structurally equivalent processes • operational semantics (SOS style) • abstract semantics (bisimilarity) • suited for message passing calculi • communication encoded in the labels of the LTS CommUnity, Tiles and Connectors

6. Motivation III • Categorical Approach • Algebraic Approach • Reconcile two selected representatives • CommUnity • categorical approach applied to program design • Tile Model • co-existence of horizontal (space) and vertical (time) dimensions • Advantages • transfer of concepts and techniques CommUnity, Tiles and Connectors

7. Roadmap • Motivation • Background • CommUnity • Tiles • From CommUnity to Tiles • Connectors • Concluding remarks CommUnity, Tiles and Connectors

8. CommUnity • Prototype architectural description language • conceptual distinction between • computation • coordination • (in communicating distributed systems) • System configurations as diagrams • Components compute locally • Interactions as architectural connectors CommUnity, Tiles and Connectors

9. Morphisms Colimit s CommUnity, graphically “Denotational Semantics” Program CommUnity, Tiles and Connectors

10. actions guards concurrent assignments CommUnity Programs design foo is in x, z out v, n doa: true  v:= x+z | n:=v+x [] b: n>MIN  n:=n-x []c: v<MAX  v:=n+z input/output channels a special actions always present: : true  skip CommUnity, Tiles and Connectors

11. Morphsims output channels cannot be merged P1 channels of P1 to channels of P2 we shall focus on actions coordination (actions names are not important) actions of P2 to actions of P1 actions of P1 correspond to disjoint sets of actions of P2 P2 CommUnity, Tiles and Connectors

12. Example design P1is in … out … doa … [] b … design P is in … out … dof|a|p … [] f|a|q … [] f|a|r … [] g|a|p … [] g|a|q … [] g|a|r … [] h|b|s … f,g  a  p,q,r h  b  s design P2is in … out … dof … [] g … [] h … design P3is in … out … dop … [] q … [] r … [] s … CommUnity, Tiles and Connectors

13. Star-Shaped Diagrams glue cables no output channels actions are true skip roles CommUnity, Tiles and Connectors

14. Tile Model • Operational and Abstract Semantics of Open Concurrent Systems • Compositional in Space and Time • Specification Formats • Depend on chosen connectors • Congruence results • Category based CommUnity, Tiles and Connectors

15. parallel composition sequential composition Configurations output interface input interface (interfaces can be typed) CommUnity, Tiles and Connectors

16. Configurations output interface input interface parallel composition functoriality sequential composition CommUnity, Tiles and Connectors

17. Configurations output interface input interface parallel composition functoriality + symmetries = symmetric monoidal cat sequential composition CommUnity, Tiles and Connectors

18. concurrent computation Observations initial interface final interface CommUnity, Tiles and Connectors

19. initial configuration trigger effect final configuration Tiles • Combine horizontal and vertical structures through interfaces CommUnity, Tiles and Connectors

20. Tiles • Compose tiles • horizontally CommUnity, Tiles and Connectors

21. Tiles • Compose tiles • horizontally • (also vertically and in parallel) symmetric monoidal double cat CommUnity, Tiles and Connectors

22. Operational and Abstract Semantics • Structural equivalence • Axioms on configurations (e.g. symmetries) • LTS • states = configurations • transitions = tiles • labels = (trigger,effect) pairs • Abstract semantics • tile bisimilarity • (congruence results for suitable formats) CommUnity, Tiles and Connectors

23. Roadmap • Motivation • Background • CommUnity • Tiles • From CommUnity to Tiles • Connectors • Concluding remarks CommUnity, Tiles and Connectors

24. Notation • Since: • Initial and Final configuration are always equal • We shall consider only two kinds of observations • We use a “flat” notation for tiles 1 0 CommUnity, Tiles and Connectors

25. From CommUnity to Tiles • Aim: • To define the operational and abstract semantics of CommUnity diagrams by translating them to tiles • First step • Standard decomposition of programs and diagrams • identify basic building blocks • make the translation easier • Second step • From standard diagrams to tiles • basic programs to basic boxes • cables, glues and morphisms modeled as connectors CommUnity, Tiles and Connectors

26. Results • The translation of a diagram is tile bisimilar to the translation of its colimit [IFIP-TCS 2004] • Corollary: Diagrams with the same colimit are mapped to tile bisimilar configurations • Colimit axiomatization [Ongoing Work!] • add suitable axioms for connectors • axioms bisimulate • existence of normal forms … hopefully! • the translation of a diagram is equal up-to-axioms to the translation of its colimit CommUnity, Tiles and Connectors

27. Why Standard Decomposition • Decomposition is part of the translation • The translation becomes easier • It is necessary to achieve Colimit Axiomatization • in the colimit, conditional assignments are synchronized according to the diagram • the axiomatization of connectors cannot change the number of computational entities • decomposition is necessary to guarantee that the number of computational entities in the diagram is the same as the number of computational entities in the colimit CommUnity, Tiles and Connectors

28. Standard Decomposition Illustrated • n output channels • m actions P • n channel managers • m guard managers • n+m cables • 1 glue CommUnity, Tiles and Connectors

29. variables needed in assignments input channels actions one action true  x:= … for each assignement channel for x output channel Channel Manager one channel manager for each output variable x of P i channel manager for x i i i i i o CommUnity, Tiles and Connectors

30. variables needed in the guard input channels one action Q  skip Guard Manager one guard manager for each action a of P i guard manager for a i i i i i no output channel CommUnity, Tiles and Connectors

31. all channels of P input channels actions one action true  skip for each action of P Glue one glue i i i i i i no output channel CommUnity, Tiles and Connectors

32. one input channel for each channel of the role input channels actions one action true  skip for each action of the role Cables (morphisms are obvious) one cable for each manager i i i i i i no output channel CommUnity, Tiles and Connectors

33. Scheme of translation state channel fusion channel manager action synchronization through connectors … channel manager guard manager … guard manager CommUnity, Tiles and Connectors

34. i channel manager for x i channel manager for x i channel manager for x i i i i i i i i i i i i … i i i o o o Tiles for components actions of managers are mutually exclusive, but at least one action (e.g. ) must be executed CommUnity, Tiles and Connectors

35. Roadmap • Motivation • Background • CommUnity • Tiles • From CommUnity to Tiles • Connectors • Concluding remarks CommUnity, Tiles and Connectors

36. Connectors • Connectors are used to model diagram morphisms, cables and glues • they express constraints on local choices • Connectors with the same abstract semantics should be identified • different ways of modeling the same constraint • How? • Axioms + reduction to normal form CommUnity, Tiles and Connectors

37. Abstract semantics • Connectors can be seen as black boxes • input interface • output interface • admissible signals on interfaces 1 1 1 1 2 2 2 2 3 3 3 3 4 4 CommUnity, Tiles and Connectors

38. … 001 111 … … 0010  0101  … domain is {input 3, outputs 1,2,3} Abstract semantics • Connectors can be seen as black boxes • input interface • output interface • admissible signals on interfaces • Abstract semantics is just a matrix • n inputs  2n rows • m outputs  2m columns • (not all matrix are interesting) • sequential composition is matrix multiplication • parallel composition is matrix expansion • cells are filled with empty/id copies of matrices 1 1 1 1 2 2 2 2 3 3 3 3 4 4 CommUnity, Tiles and Connectors

39. 00 01 10 11 00  01  = 10  11  An example: Symmetries connectors boxes are immaterial (also identities should be considered) CommUnity, Tiles and Connectors

40. = = AND-Connectors CommUnity, Tiles and Connectors

41. … … AND-Tables andNormal Form • n = m = 0 or 0 < n  m • entry with empty domain is enabled • only one entry with empty input/outputs • entries are closed under (domains) • union • intersection • difference • complementation • at most one entry enabled for every row/column • exactly one entry for every row CommUnity, Tiles and Connectors

42. = = co-AND-Connectors 0 1 (as AND-table, but transposed) 00  01 10 11  CommUnity, Tiles and Connectors

43. = Mixed-AND-Connectors = CommUnity, Tiles and Connectors

44. Mixed-AND-Connectors = = CommUnity, Tiles and Connectors

45. Mixed-AND-Connectors = = = CommUnity, Tiles and Connectors

46. Mixed-AND-Connectors = = = CommUnity, Tiles and Connectors

47. Mixed-AND-Connectors = = = CommUnity, Tiles and Connectors

48. … … … Mixed-AND-Tables andNormal Form • n = m = 0 or 0 < n,m • entry with empty domain is enabled • only one entry with empty input/outputs • entries are closed under (domains) • union • intersection • difference • complementation • at most one entry enabled for every row/column • exactly one entry for every row CommUnity, Tiles and Connectors

49. = ! ! ! = ! ! = . ! ! 0 1 .   (co-)HIDE-Connectors ! ! . 0  1  CommUnity, Tiles and Connectors

50. . . 00  00  01  01 10  10 11  11  !  ! ! A Relevant Difference !  ! ! CommUnity, Tiles and Connectors