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Dynamic Graph Transformation Systems

Dynamic Graph Transformation Systems. Hernán Melgratti IMT Lucca Institute for Advance Studies Joint Work with Roberto Bruni Dipartimento di Informatica, Università di Pisa. Join Calculus. Join processes can be seen as dynamic and reconfigurable, coloured nets. Join Calculus.

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Dynamic Graph Transformation Systems

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  1. Dynamic Graph Transformation Systems Hernán Melgratti IMT Lucca Institute for Advance Studies Joint Work with Roberto Bruni Dipartimento di Informatica, Università di Pisa

  2. Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets Hernán Melgratti@IMTLucca

  3. Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets Hernán Melgratti@IMTLucca

  4. Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets a x b x ax xb Hernán Melgratti@IMTLucca

  5. a a a c c b b b x x b b x x def ax xb def ax xb   in aa | cb in ab | cb Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets a a c x b x def ax xb in aa | ac Hernán Melgratti@IMTLucca

  6. a x c c y y x ax def cy yx incc Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets Hernán Melgratti@IMTLucca

  7. c a c y y b x c c y y x def ax def cy yx incc cy ay in ab | cc Join Calculus • Join processes can be seen as dynamic and reconfigurable, coloured nets a a b x c c y y x def ax def cy yx incc in aa | ab Hernán Melgratti@IMTLucca

  8. T-typed Graphs DPO Graph Grammar The initial T-typed graph The graph of types The set of productions Left-hand-side Interface Span of injective morphisms Right-hand-side Hernán Melgratti@IMTLucca

  9. l r p : L K R m k h G D H d b DPO Rewriting Step Hernán Melgratti@IMTLucca

  10. Towards Dynamic Productions Hernán Melgratti@IMTLucca

  11. Towards Dynamic Productions p: Hernán Melgratti@IMTLucca

  12. Towards Dynamic Productions p: n1 n1 n Hernán Melgratti@IMTLucca

  13. Towards Dynamic Productions Gp p: n1 n1 n Hernán Melgratti@IMTLucca

  14. Towards Dynamic Productions Gp p: n1 n1 g f n n m Hernán Melgratti@IMTLucca

  15. Towards Dynamic Productions Gp p: n1 n1 n1 m1 f1 g f n n m Hernán Melgratti@IMTLucca

  16. Towards Dynamic Productions Gp p: n1 n1 n1 m1 f1 g f n n m q: Hernán Melgratti@IMTLucca

  17. Towards Dynamic Productions Gp p: n1 n1 n1 m1 f1 g f n n m q: … Hernán Melgratti@IMTLucca

  18. g f m r s t p Towards Dynamic Productions Gp p: n1 n1 n1 m1 f1 g f n n m q: … r Hernán Melgratti@IMTLucca

  19. f’ g’ q: q: q’: m’ r s t p t’ s’ Towards Dynamic Productions Gp p: n1 n1 n1 m1 f1 g g f f n m n m q: … q: … q’: … r s t p r Hernán Melgratti@IMTLucca

  20. The initial T-typed graph The graph of types The set of productions T-typed Graphs Dynamic Graph Grammar (DGG) Injective Morphism A DGG over the graph of type T Tp Injective Morphism between Tp-typed Graph Hernán Melgratti@IMTLucca

  21. L K K’ r’ l T m k k’ h T G H D D’ b d Dynamic rewriting Hernán Melgratti@IMTLucca

  22. Encoding the Join Calculus • A channel (or place) x is encoded as a node n • The actual name of the channel is given by an arc x:n  n • Any firing rule is encoded as a production Hernán Melgratti@IMTLucca

  23. Encoding a Join Process P • The graph of types m Where fn (P )  dn(P ) = { x1, x2, x3 } x3 x1 x2 Hernán Melgratti@IMTLucca

  24. m y x m m y x Encoding a Join Process P • A message xy Hernán Melgratti@IMTLucca

  25. Encoding a Join Process P • A message xy m y x m y x Hernán Melgratti@IMTLucca

  26. u1 uk u1 uk m m … … n1 nk n1 nk x1 xk x1 xk Encoding a Join Process P • A definition x1u1 |…| xkuk Pi Hernán Melgratti@IMTLucca

  27. nu m y u nu nu m m nx x nx nx x x nv nv nv m m m m ny ny ny y y y m m m x x z z z x u y Example x is a defined name • P =defxu defyv vy inyu | xyinxz z is a free name Hernán Melgratti@IMTLucca

  28. nu m y u nu nu m m nx x nx nx x x nv nv nv m m ny ny ny u u u’ y y y m m m m m z z z y y’ y m m m x x x Example Hernán Melgratti@IMTLucca

  29. Theorem • For any Join process P • If P P’ using JiPi then Q s.t. and Q  P’ • If , then P’ s.t P P’ using JiPi and Hernán Melgratti@IMTLucca

  30. Tb g Ta A chain of types Refined Type f m n Ta Tb n n f m g DGG as GG • We start by defining a graph of types for representing the tree of types created dynamically Hernán Melgratti@IMTLucca

  31. DGG as GG • A typed graph over a refined type Tb Ta f n m n n f m Tb Ta Tb Ta g f n m n n f m g Hernán Melgratti@IMTLucca

  32. Tb Ta Ta Ta n n n n n n f m Ta Tb n n f m g DGG as GG • The refined version of productions p: n1 n1 n1 m1 f1 Hernán Melgratti@IMTLucca

  33. Theorem Hernán Melgratti@IMTLucca

  34. Final Remarks • DGG offers a convenient level of abstraction for describing reflexive systems • DGG can be simulated by ordinary GG • Future works: • To study independent derivations, parallelism, process semantics, unfolding semantics and event structure semantics • To show that concurrency is preserved by our encoding • To consider other approaches (like SPO) Hernán Melgratti@IMTLucca

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