1 / 47

Synchronizations with Mobility for Graph Transformations

Dagstuhl seminar, Germany, 7-11 June 2004. Synchronizations with Mobility for Graph Transformations. Ivan Lanese Dipartimento di Informatica Università di Pisa. joint work with Ugo Montanari Dipartimento di Informatica Università di Pisa. Roadmap. Aims of the work

clovis
Download Presentation

Synchronizations with Mobility for Graph Transformations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dagstuhl seminar, Germany, 7-11 June 2004 Synchronizations with Mobility for Graph Transformations Ivan Lanese Dipartimento di Informatica Università di Pisa joint work with Ugo Montanari Dipartimento di Informatica Università di Pisa

  2. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  3. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  4. Aims of the work (1) • To develop an expressive framework for graph transformation • We base on Synchronized Hyperedge Replacement • Rules with local effects • Composition via synchronization with mobility • Different synchronizations mechanisms for different applications • Different applicative scenarios • Modeling network reconfigurations • Modeling software architectures • Giving a graphical semantics to calculi for mobility Synchronizations with Mobility for Graph Transformations

  5. Aims of the work (2) • To define an abstract semantics • For reasoning on behavioural properties • The abstract semantics must be compositional Synchronizations with Mobility for Graph Transformations

  6. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  7. Why graphs? • Are a natural model for distributed systems • Represent the spatial structure • Are a concurrent model • Give a more suggestive representation for process calculi • Easy to understand • Semantically sound Synchronizations with Mobility for Graph Transformations

  8. Which graphs? • We use hypergraphs with labeled (hyper)edges • Subset of nodes (free nodes) as interface • Important for the abstract semantics • Computational interpretation: • Edges are processes or systems • Nodes are links or ports • Synchronization is performed via shared nodes Synchronizations with Mobility for Graph Transformations

  9. Judgements • We use judgements to represent graphs • To simplify the definition of the semantics • ΓG where Γ is a finite set of nodes and G is a term generated by • G ::= nil | s(x1,…,xn) | G|G | νx G • s is an edge label, x and x1,…,xn are nodes • ν is a binder for x • We require that Γ contains at least names in fn(G)  Synchronizations with Mobility for Graph Transformations

  10. w z x,y  z, w. C(x,w) | C(w,y) | C (y,z) | C(z,x)  Example: ring In this case arrows identify the second attachment node Synchronizations with Mobility for Graph Transformations

  11. SHR: a 3 step approach • Productions to describe the behaviour of single hyperedges: • Local effects (easier to implement) • Hyperedges rewritten into generic graphs • Constraints on surrounding nodes • Global constraint-solving algorithm to derive transitions: • To select which productions can be applied • Allows to define complex transformations • Finally transitions are applied Synchronizations with Mobility for Graph Transformations

  12. L R H 3 3 4 4 2 2 1 1 Edge Replacement Systems • A production describes how the hyperedge L is transformed into the graph R Synchronizations with Mobility for Graph Transformations

  13. L R H 3 3 4 4 2 2 1 1 1 1 2 2 3 3 R’ L’ Edge Replacement Systems • A production describes how the hyperedge L is transformed into the graph R Many productions can be applied concurrently Synchronizations with Mobility for Graph Transformations

  14. Synchronized Hyperedge Replacement • We associate actions to nodes attached to edges to be rewritten • A transition is allowed iff the synchronization constraints associated to nodes are satisfied • Many synchronization models are possible (Milner, Hoare, ...) Synchronizations with Mobility for Graph Transformations

  15. a a a B1 A1 3 3 B2 A2 An example: Milner synchronization • Pairs of edges can synchronize by doing complementary actions Synchronizations with Mobility for Graph Transformations

  16. a<x> a<x> a<y> a<y> B1 A1 a<x> ~ a<y> (x) (y) x= y B2 A2 SHR with mobility • We introduce name mobility • Actions carry tuples of references to nodes (new or already existent) • References associated to synchronized actions are matched and corresponding nodes are merged Synchronizations with Mobility for Graph Transformations

  17. Initial Graph Brother: Star Reconfiguration: x C r(w) C C C S (w) r(w) C C C C S Brother Brother Brother Star Rec. x C S S C C C C C C S (1) (2) (3) (4) (5) b ) Example Synchronizations with Mobility for Graph Transformations

  18. ,  G1   G2   Transitions as syntactic judgements • Transitions: • :   (Act x N*) • Associates to each external node its action and its tuple of references to nodes • :  is an idempotent substitution (forces merges on nodes) Synchronizations with Mobility for Graph Transformations

  19. Deriving transitions • Productions • Transitions are generated from productions by applying a suitable set of inference rules • Inference rules are parametric w.r.t. the synchronization model, which is expressed by an algebra , x1,…,xn L(x1,…,xn) G   Synchronizations with Mobility for Graph Transformations

  20. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  21. Synchronization Algebra with Mobility • A quintuple <Act,•,Init,Fin,ε> • Act: ranked set of actions • •: partial function Act x Act->Act • Defines action composition • Undefined if the actions can not synchronize • Returns the composed action otherwise Synchronizations with Mobility for Graph Transformations

  22. Synchronization Algebra with Mobility (2) • Init, Fin: subsets of Act • Init contains trivial actions that can be produced on isolated nodes • Fin contains actions that correspond to completed synchronizations (only actions in Fin are allowed on hidden nodes) • ε: element of Act • ε corresponds to “no synchronization” Synchronizations with Mobility for Graph Transformations

  23. Example: Milner synchronization Synchronizations with Mobility for Graph Transformations

  24. The rule for synchronization Synchronizations with Mobility for Graph Transformations

  25. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  26. Abstract semantics • We use the standard concept of bisimulation Synchronizations with Mobility for Graph Transformations

  27. Properties of abstract semantics • Two bisimilar graphs have the same interface • We can add isolated nodes to the interface to compare graphs with different interfaces • The abstract semantics is compositional • We build a suitable algebra for judgements • We prove that bisimilarity is a congruence w.r.t. the operators of the algebra Synchronizations with Mobility for Graph Transformations

  28. Algebra • We use an algebra in the style of Bauderon & Courcelle • Algebra generated by the operators (modulo axioms) • We consider graphs typed by their interfaces • We have classes of typed operators Synchronizations with Mobility for Graph Transformations

  29. Algebra for judgements • Nil : empty graph • sx1,x2,...,xn : edge s with n attachment nodes • - || - : union of graphs with disjoint interfaces • x,y - : merge of x and y (x representative) • !x - : creation of isolated node x • ?x - : hiding of node x Synchronizations with Mobility for Graph Transformations

  30. The idea of the proof • Inference rules can be written in De Simone format using the algebraic operators • All the axioms bisimulate • Bisimulation is a congruence thanks to standard results on bialgebras • Turi-Plotkin and Buscemi-Montanari Synchronizations with Mobility for Graph Transformations

  31. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  32. Example: routing • We want to use graphs to model dynamically evolving routers • We use two kinds of edges R (router) and L (link) of arity 3 and 2 • Edges can execute complementary actions on two of their attachment nodes and create links among nodes whose names are matched Synchronizations with Mobility for Graph Transformations

  33. A sample production Synchronizations with Mobility for Graph Transformations

  34. Bisimilar routers • A router is characterized by its interface S and its connection relation • Conn is the set of pairs of nodes that are connected by disjoint paths • Bisimilarity captures exactly this notion of equivalence Synchronizations with Mobility for Graph Transformations

  35. Changing the synchronization model • We can use for routers the broadcast synchronization model Synchronizations with Mobility for Graph Transformations

  36. Routers with broadcast synchronization • Multicast is now allowed • Different equivalence on graphs • The left one can send messages from any node to any other • The right one broadcasts messages from any node to the other two • If names are transmitted new broadcast groups are created Synchronizations with Mobility for Graph Transformations

  37. Mapping Fusion Calculus into SHR • SHR is expressive enough to model Fusion Calculus processes • Fusion Calculus can be mapped into Milner SHR • We will not present the mapping in detail • The induced semantics is concurrent • Many actions at the same time on different channels Synchronizations with Mobility for Graph Transformations

  38. Fusion Calculus vs Milner SHR FusionMilner SHR Processes Graphs Sequential processes Hyperedges Names Nodes Parallel comp. Parallel comp. Scope Restriction Sets of conc. transit. Transitions Synchronizations with Mobility for Graph Transformations

  39. Example We can also execute both the steps at the same time Synchronizations with Mobility for Graph Transformations

  40. An apparent mismatch • Bisimulation is a congruence in SHR but not in Fusion Calculus • The reason: SHR semantics is concurrent • In Fusion ux.vy+vy.ux≈ux|vy but the bisimulation relation is not preserved by a context that merges u and v • In SHR the two terms are not bisimilar since the second one can execute both the prefixes at the same time Synchronizations with Mobility for Graph Transformations

  41. Roadmap • Aims of the work • Background: Synchronized Hyperedge Replacement • Parametric inference rules • Abstract semantics • Expressiveness • Conclusions and future work Synchronizations with Mobility for Graph Transformations

  42. Conclusions • We have extended a known approach to graph transformations by making it parametric w.r.t. the synchronization model • We have defined a suitable abstract semantics through bisimilarity • We have proven that bisimilarity is a congruence • We have presented some possible applications Synchronizations with Mobility for Graph Transformations

  43. Future work • Study the properties of our bisimilarity • Which equivalence is induced on Fusion processes? • Allow different kinds of mobility (until now essentially Fusion style moblity) • Consider nodes with different synchronization models in the same graph • Apply synchronization algebras with mobility to other formalisms Synchronizations with Mobility for Graph Transformations

  44. Bibliography (1) • For Synchronized Hyperedge Replacement • P. Degano and U. Montanari. A model for distributed systems based on graph rewriting. Journal of ACM 34(2), 1987 • D. Hirsch and U. Montanari. Synchronized hyperedge replacement with name mobility. Proc. of CONCUR 2001, LNCS, 2001 • G. Ferrari, U. Montanari and E. Tuosto. A lts semantics of ambients via graph synchronization with mobility. Proc. of ICTCS’01, LNCS 2202, 2001 • For abstract semantics for SHR • B. König and U. Montanari. Observational equivalence for synchronized graph rewriting. Proc. of TACS’01, LNCS 2215, 2001 Synchronizations with Mobility for Graph Transformations

  45. Bibliography (2) • For synchronization algebras • G. Winskel. Event structures. LNCS 255, 1986 • For algebras for graphs • M. Bauderon and B. Courcelle. Graph expressions and graph rewriting. Math. Systems Theory 20, 1987 • For results on bialgebras • D. Turi and G. Plotkin. Towards a mathematical operational semantics. Proc. of LICS’97, 1997 • M. Buscemi and U. Montanari. A first order coalgebraic model of pi-calculus early observational equivalence. Proc. of CONCUR’02, LNCS 2421, 2002 • For Fusion Calculus vs SHR • I. Lanese and U. Montanari. A graphical Fusion Calculus. Proc. of CoMeta final workshop, ENTCS, to appear Synchronizations with Mobility for Graph Transformations

  46. Questions? Synchronizations with Mobility for Graph Transformations

  47. Conditions on synchronization algebras Synchronizations with Mobility for Graph Transformations

More Related