Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming

1. Dagstuhl Seminar #05081, 20-25 February 2005 Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming Ivan Lanese Dipartimento di Informatica Università di Pisa joint work with Ugo Montanari To be published on a special issue of Theory and Practice of Logic Programming

2. Roadmap • Lot of background • Fusion Calculus • Synchronized Hyperedge Replacement • Logic programming • From Fusion Calculus to Hoare SHR • From Hoare SHR to logic programming • Conclusions

3. Motivations • Many models proposed for global computing systems • Each model has its strengths and its weaknesses • Comparing different models • To understand the relationships among them • To devise new (hybrid) models • Cannot analyze all the models, naturally…

4. Roadmap • Lot of background • Fusion Calculus • Synchronized Hyperedge Replacement • Logic programming • From Fusion Calculus to Hoare SHR • From Hoare SHR to logic programming • Conclusions

5. Fusion Calculus • Process calculus that is an evolution of -calculus • Simpler and more symmetric but also more expressive • Introduces fusions of names

6. Syntax for Fusion Calculus • Agents: S::=i i.Pi P::=0 | S | P1|P2 | (x)P | rec X. P | X Processes are agents up to a standard structural congruence

7. Reduction semantics

8. Synchronized Hyperedge Replacement • Follows the approach of graph transformation • (Hyper)edges are systems connected through common nodes • Productions describe the evolution of single edges • Local effect, easy to implement • Productions are synchronized via constraintson nodes • Global constraint solving algorithm to find allowed transitions • Productions applied indipendently • Allows to define complex transformations

9. Hyperedge Replacement Systems • A production describes how the hyperedge L is transformed into the graph R L R H H 3 3 4 4 2 2 1 1

10. L R H 3 3 4 4 2 2 1 1 1 1 2 2 3 3 R’ L’ Hyperedge Replacement Systems • A production describes how the hyperedge L is transformed into the graph R H Many concurrent rewritings are allowed

11. Synchronizing productions • Productions associate actions to nodes • A transition is allowed iff thesynchronization constraints imposed on actions are satisfied • Many synchronization models are possible (Hoare, Milner, ...)

12. a a a a B1 A1 3 3 B2 A2 An example: Hoare SHR • Hoare synchronization: all the edges must perform the same action • Milner synchronization:pairs of edges do complementary actions

13. We use 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 a<x> a<x> a<y> B1 A1 a<y> (x) (y) x=y B2 A2 SHR with mobility

14. Logic programming Very quickly… • Syntax • Semantics

15. Roadmap • Lot of background • Fusion Calculus • Synchronized Hyperedge Replacement • Logic programming • From Fusion Calculus to Hoare SHR • From Hoare SHR to logic programming • Conclusions

16. From Fusion to HSHR • We separate the topological structure (graph) from the behaviour (productions) • We give a visual representation to processes • Processes translated into graphs • Sequential processes become hyperedges • Names become structures called amoeboids • Our approach deals only with closed processes

17. Translation by example • We take agents in standard form • restrictions with inside parallel composition of sequential agents • We transform it into a linear agent + substitution

18. Translation by example • We translate the process into a graph • Each linear agent becomes an edge labelled with a copy of the agent with standard names • Q(x1,y1,z1) becomes an edge labelled by Q(x1,x2,x3) attached to x1,y1,z1 • σ transformed into amoeboids • Each amoeboid connects a group of names merged by σ

19. x1x2x3.R(x4,x5) Translation by example x x1 z1 Q(x1,x2,x3) y1 z x2 x3 y y2 z2 z3 u2 u1 w1 u3 x1x2x3.S(x4,x5) u w2 w

20. Dynamics • We have actions for input and output prefixes • Productions for process edges • Correspond to executions of prefixes of normalized linear agents • Prefixes modeled by corresponding SHR actions • RHS contains the translation of the resulting sequential process • Fusions implemented by connecting nodes via amoeboids

21. x2 x3 x1x2x3.R(x4,x5) y2 R(x1,x2) u2 u1 A sample production x2 x3  y2 u2 u1 out2x2y2

22. in x out y P Q y x What is an amoeboid? • Amoeboids must allow two complementary actions on the interface and create new amoeboids connecting corresponding names • Connected amoeboids are merged P Q =

23. Implementing amoeboids in HSHR • Amoeboids implemented as networks of edges with a particular structure • composed essentially by edges that act as routers • each internal node is shared by two edges • some technical conditions • Satisfy the desired properties but… • interleaving must be imposed from the outside • produce some garbage (disconnected from the system)

24. Correspondence theorem • Reductions of fusion processes correspond to (interleaving) HSHR transitions • up to garbage • up to equivalence of amoeboids that does not change the behaviour • The correspondence can be extended to computations

25. x1x2x3.R(x4,x5) Translation by example x Q(x1,x2,x3) z y out x2 y2 x1x2x3.S(x4,x5) u in z2 w1 w

26. Translation by example x Q(x1,x2,x3) z y R(x1,x2) S(x1,x2) u w

27. Summary : Fusion Calculus vs HSHR FusionHoare SHR Closed process Graph Sequential process Hyperedge Name Amoeboid Prefix Action Prefix execution Production Reduction Transition

28. Roadmap • Lot of background • Fusion Calculus • Synchronized Hyperedge Replacement • Logic programming • From Fusion Calculus to Hoare SHR • From Hoare SHR to logic programming • Conclusions

29. From Hoare SHR to logic programming • Useful for implementation purposes • Logic programming as goal rewriting engine • Very similar syntax (with the textual representation for HSHR) • Logic programming allows for many execution strategies and data structures  we need some restrictions • limited function nesting • synchronized execution • We define Synchronized Logic Programming (SLP)

30. Synchronized Logic Programming • A transactional version of logic programming (in the zero-safe nets style) • Safe states are goals without function symbols (goal-graphs) • Transactions are sequences of SLD steps • During a transaction each atom can be rewritten at most once • Transactions begin and end in safe states • Transactions are called big-steps • A computation is a sequence of big-steps

31. Synchronized clauses • Clauses with syntactic restrictions • bodies are goal-graphs • heads are A(t1,…,tn) where ti is either a variable or a single function symbol applied to variables

32. HSHR vs logic programming • Graphs translated to goal-graphs • edges modeled by predicates applied to the attachment nodes • Productions are synchronized clauses • Transitions are matched by big-steps • Actions are implemented by function symbols • the constraint that all function symbols have to be removed corresponds to the condition for Hoare synchronization • names are the arguments of the function symbol • we choose the first one to represent the new name for the node where the interaction is performed (needed since substitutions are idempotent) • fusions performed by unification

33. Correspondence theorem • Correspondence between HSHR transitions and big-steps • An injective (at each step) substitution keeps track of the correspondence between HSHR nodes and logic programming variables

34. y y C C x z C x An example (simpler than Fusion…) C(x,y)←C(x,z),C(z,y) r <w> y y C(r(x,w),r(y,w))←S(y,w) C S (w) x x r <w>

35. S S S C C C x C S C C C C C C Dynamics

36. S S S C C C x C S C C C C C C Dynamics

37. Summary : HSHR vs SLP Hoare SHR SLP Graph Goal Hyperedge Atom Node Variable Parallel comp. AND comp. Action Function sym. Production Clause Transition Big-step

38. Roadmap • Lot of background • Fusion Calculus • Synchronized Hyperedge Replacement • Logic programming • From Fusion Calculus to Hoare SHR • From Hoare SHR to logic programming • Conclusions

39. Conclusions • Many relations among the three models • similar underlying structure (e.g. parallel composition) • name generations ability • fusions • Distinctive features • Fusion: same structure for system and elementary actions, interleaving semantics, Milner synchronization, restriction • HSHR: distributed parallel computations, Hoare synchronization, synchronous execution • SLP: Hoare synchronization, asynchronous execution engine

40. Future work • Analyzing different name-handling mechanisms • In π calculus bound names are guarenteed distinct • Useful for analyzing protocols (nonces, key generation) • Hybrid models • Different synchronizations for Fusion or logic programming • Process calculi with unification (of terms) • Logic programming with restriction • Logic programming for implementation purposes • For HSHR systems • For Fusion Calculus

41. End of talk Thanks Questions?

42. w z A textual notation for graphs Ring Example x,y,w,z C(x,w) | C(w,y) | C (y,z) | C(z,x) 

43. Transitions as judgements • Transitions    G1  ,,I G2  :  (A x N* ) (x, a , y)   if (x) = (a , y)  is the set of new names that are used in synchronization  = {z |  x. (x) = (a , y), z  , z set(y)} I contains new internal names

44. Computations 1 2 n    0 G0  1 G1  …  n Gn Transitions as judgements • Productions  x1,…,xn L(x1,…,xn)  x1,…,xn , , IG   Names can be merged (and new names can be added) Identity productions are always available • Transitions are generated from the productions by applying the transition rules for the chosen synchronization model

45. e <> y y C C x z e <> C x Textual representation for productions (x,ε,<>)(y,ε,<>)  x, y  C(x,y) x, y, z .C(x,z) | C(z,y) r <w> y y C S (w) x x r <w> , (x,r,<w>)(y,r,<w>)   x, y x, y S(w,y) C(x,y)

46. From SHR to SLP

47. Structural congruence Process: agent up to the following laws: • | and + are associative, commutative and with 0 as unit • -conversion • (x)0 = 0, (x)(y)P=(y)(x) P • P|(x)Q=(x)(P|Q) if x not free in P • rec X.P=P[rec X.P/X]