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. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets

3. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets

4. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets a x b x ax  xbax  xb

5. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets a x b x ax  xbax  xb a c def in a  a  | a  c   a x b x ax  xbax  xb b def in a  b  | c  b  b c a  x b x ax  xbax  xb a def in a  a  | c  b  b c

6. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets a x a  x   def c  y   y  x  in c  c  c y x c y

7. - ICGT 2006 Hernán Melgratti@IMTLucca Join Calculus Join processes can be seen as dynamic and reconfigurable, coloured nets a x a  x   def c  y   y  x  in c  c  a b def c y x c y in a  a  | a  b  a x a  x   def c  y   y  x  in c  c  b def c y x c y in a  b  | c  c  c y c y cy  aycy  ay

8. - ICGT 2006 Hernán Melgratti@IMTLucca DPO Graph Grammar The graph of types The initial T-typed graph The set of productions T-typed Graphs Left-hand-side Interface Right-hand-side Span of injective morphisms

9. - ICGT 2006 Hernán Melgratti@IMTLucca G DPO Rewriting Step m lr p : LKR k b D h d H

10. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions

11. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions p:

12. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n n1n1 n1n1 p:

13. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n n1n1 n1n1 p: GpGp

14. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n n1n1 n1n1 nm f g p: GpGp

15. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n n1n1 n1n1 n1n1 nm f m1m1 f1f1 g p: GpGp

16. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n n1n1 n1n1 n1n1 nm f m1m1 f1f1 g p: q: GpGp

17. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n1n1 n1n1 p: GpGp n n1n1 f m1m1 f1f1 q: … nm g

18. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n1n1 n1n1 p: GpGp n n1n1 f m1m1 f1f1 q: … r p rs t nm g f m g

19. - ICGT 2006 Hernán Melgratti@IMTLucca Towards Dynamic Productions n1n1 n1n1 p: GpGp n n1n1 f m1m1 f1f1 q: … r p rs t f nm g m g p rs t s’ t’ f’ m’ g’ q’: … q: q’:

20. - ICGT 2006 Hernán Melgratti@IMTLucca Dynamic Graph Grammar (DGG) The graph of types The initial T-typed graph The set of productions T-typed Graphs A DGG over the graph of type T  T p Injective Morphism Injective Morphism between T p -typed Graph

21. - ICGT 2006 Hernán Melgratti@IMTLucca G T Dynamic rewriting k b D m r’ d H h l LK T k’ K’ D’

22. - ICGT 2006 Hernán Melgratti@IMTLucca 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

23. - ICGT 2006 Hernán Melgratti@IMTLucca Encoding a Join Process P The graph of types x3x3 x2x2 x1x1 m Where fn (P )  dn(P ) = { x 1, x 2, x 3 }

24. - ICGT 2006 Hernán Melgratti@IMTLucca Encoding a Join Process P A message x  y  m m x y y x m

25. - ICGT 2006 Hernán Melgratti@IMTLucca Encoding a Join Process P A message x  y  y x m m x y

26. - ICGT 2006 Hernán Melgratti@IMTLucca Encoding a Join Process P A definition x 1  u 1  |…| x k  u k   P i m u1u1 x1x1 n1n1 m ukuk xkxk nknk … u1u1 x1x1 n1n1 ukuk xkxk nknk …

27. - ICGT 2006 Hernán Melgratti@IMTLucca Example P = def x  u   def y  v   v  y  in y  u  | x  y  in x  z  x z m z is a free name x is a defined name m nunu x nxnx x z m x z m x nxnx nunu y u y x nxnx m m nunu u m m nvnv y nyny y nyny nvnv m m nvnv y nyny

28. - ICGT 2006 Hernán Melgratti@IMTLucca Example m nunu x nxnx x nxnx nunu y x nxnx m m nunu u m m nvnv y nyny y nyny nvnv m m nvnv y nyny x m z x z u y m m x z u y m m u’ y’ m

29. - ICGT 2006 Hernán Melgratti@IMTLucca Theorem For any Join process P If P  P’ using J i  P i then  Q s.t. and Q  P’ If, then  P’ s.t P  P’ using J i  P i and

30. - ICGT 2006 Hernán Melgratti@IMTLucca DGG as GG We start by defining a graph of types for representing the tree of types created dynamically n f m g TaTa TbTb A chain of types nnfmg TaTa TbTb Refined Type

31. - ICGT 2006 Hernán Melgratti@IMTLucca DGG as GG A typed graph over a refined type n f m g TaTa TbTb nnfmg TaTa TbTb n f m nnfm TaTa TbTb

32. - ICGT 2006 Hernán Melgratti@IMTLucca DGG as GG The refined version of productions n1n1 n1n1 p: n1n1 m1m1 f1f1 nnfm TaTa TbTb nn TaTa nn TaTa nnfmg TaTa TbTb

33. - ICGT 2006 Hernán Melgratti@IMTLucca Theorem

34. - ICGT 2006 Hernán Melgratti@IMTLucca 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)