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