1. Focusing in Proof-search and Concurrent Synchronization Deepak Garg Carnegie Mellon University (Based on joint work with Frank Pfenning)

2. Focusing in Proof-search and Concurrent Synchronization2 Objectives Direct translation of asynchronous pi- calculus to intuitionistic linear logic formulas Committed Forward chaining in linear logic to simulate pi-calculus reductions Combine with focusing to obtain atomicity in synchronization

3. Focusing in Proof-search and Concurrent Synchronization3 Direct translation Connectives of pi-calculus map to connectives of linear logic

4. First attempt at translation (No focusing, unsound!)

5. Focusing in Proof-search and Concurrent Synchronization5 Translation

6. Focusing in Proof-search and Concurrent Synchronization6 Pi-calculus, Linear Logic Asynchronous pi-calculus (without replication): Intuitionistic linear logic fragment:

7. Focusing in Proof-search and Concurrent Synchronization7 Pi-calculus: semantics Modified CHAM semantics

8. Focusing in Proof-search and Concurrent Synchronization8 Linear Logic: Forward Chaining Judgment: Term variablesFormulas (Linear)

9. Focusing in Proof-search and Concurrent Synchronization9 Linear Logic: Forward Chaining

10. Focusing in Proof-search and Concurrent Synchronization10 Forward Chaining as Rewriting

11. Focusing in Proof-search and Concurrent Synchronization11 Forward Chaining as Rewriting

12. Focusing in Proof-search and Concurrent Synchronization12 Forward Chaining as Rewriting

13. Focusing in Proof-search and Concurrent Synchronization13 Translation

14. Focusing in Proof-search and Concurrent Synchronization14 Simulation example

15. Focusing in Proof-search and Concurrent Synchronization15 Unsound! Another possible reduction sequence:

16. Focusing in Proof-search and Concurrent Synchronization16 Unsoundness analysis Forward chaining may get stuck incorrectly Universal quantifier and implication in inputs must be eliminated simultaneously Can be done using focusing –Each communication is exactly one focusing

17. Focusing in Forward Chaining

18. Focusing in Proof-search and Concurrent Synchronization18 Synchronous and Asynchronous Divide formulas into (right) synchronous S and (right) asynchronous A Coerce S to A via a monad (CLF, LolliMon)

19. Focusing in Proof-search and Concurrent Synchronization19 Focused Forward Chaining Judgments Different from CLF focusing –Decomposition of ­, 9 on left is not in focus –No right rules

20. Focusing in Proof-search and Concurrent Synchronization20

21. Focusing in Proof-search and Concurrent Synchronization21

22. Focusing in Proof-search and Concurrent Synchronization22

23. Focusing in Proof-search and Concurrent Synchronization23 Focused Forward Chaining as Rewriting We read rules bottom up, ignore synchronous goals and get a conditional rewrite system. Judgments: Unchanged

24. Focusing in Proof-search and Concurrent Synchronization24

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28. Focusing in Proof-search and Concurrent Synchronization28 Complete Rewrite System

29. Focusing in Proof-search and Concurrent Synchronization29 Translation of the pi-calculus

30. Focusing in Proof-search and Concurrent Synchronization30 Simulation example

31. Focusing in Proof-search and Concurrent Synchronization31 Simulation example

32. Focusing in Proof-search and Concurrent Synchronization32 Correctness of translation

33. Focusing in Proof-search and Concurrent Synchronization33 A Strange External Choice… What does A&B correspond to in the pi- calculus? It corresponds to external choice between input/output actions

34. Focusing in Proof-search and Concurrent Synchronization34 Extension to the logic Extension of asynchronous formulas New focusing rules

35. Focusing in Proof-search and Concurrent Synchronization35 Translation of choice Correctness results remain the same

36. Focusing in Proof-search and Concurrent Synchronization36 Expressiveness How expressive is this extension of the asynchronous pi-calculus? Conjecture: It is as expressive as the synchronous pi-calculus (with choice, without replication) - Translation follows from Boudol’s encoding

37. Focusing in Proof-search and Concurrent Synchronization37 3-way synchronization Can encode receivers with 2 simultaneous inputs

38. Focusing in Proof-search and Concurrent Synchronization38 Encoding 3-way synchronization Works because both implications must be eliminated in one focusing step Generalizes to n-way inputs

39. Focusing in Proof-search and Concurrent Synchronization39 Other connectives

40. Focusing in Proof-search and Concurrent Synchronization40 Summary Dynamic semantics of the asynchronous pi-calculus can be simulated using focusing and forward chaining Focusing – atomicity in synchronization Correspondence between connectives of the pi-calculus and intuitionistic linear logic

41. Focusing in Proof-search and Concurrent Synchronization41 Related Work Translation idea is not new –Miller92 – pi-calculus as a theory in linear logic –Cervesato03 – similar idea, no focusing Abramsky93 – classical logic and concurrency Concurrent Logic Programming