Logical Properties of CPS Transforms

1. Logical Properties of CPS Transforms Deepak Garg Fall, 2004

2. Introduction • CPS = Continuation Passing Style • Studied to simulate cbv with cbn • We look at CPS transforms as proof-transformations • CPS transforms embed classical logic into intuitionistic logic

3. Simply Typed -calculus • Terms: • Types (Logical Propositions):

4. Evaluation and Typing Rules • Standard Typing Rules • Small step, call by value evaluation

5. Abort (A) Operator

6. What we have so far … • We have INTUITIONISTIC logic with ?

7. Control (C) Operator

8. What we have now … • We have CLASSICAL logic! • Why? Rule of double negation elimination.

9. -calculus + A + C $ Classical logic Summary -calculus + A $ Intuitionistic logic with ?

10. CPS Transforms

11. CPS Transform: History • [1975] Plotkin. CPS studied formally. • [1986] Felleisen et al. Extended to control operators (call/cc, C and A). • [1993] Griffin. Typed CPS transforms. • [2003] Wadler. Duality with CPS transforms.

12. CPS Transform: Properties • Translate terms, types and proofs • On terms: • No control (C) operator in transformed terms • On types: • No double negation elimination in transformed proofs • Translates classical into intuitionistic logic!

13. CPS: Term Translation • -calculus • CPS translation • Each translated term expects a continuation

14. CPS: Operational Interpretation • Explicitly formalize evaluation in terms of continuations To evaluate (M N) in the continuation k, evaluate M in the continuation that binds its input to m and evaluates N in the continuation that binds its input to n and evaluates (m n) in the continuation k.

15. CPS: Type Translation • Translation for types: • Types:

16. CPS: Logical Interpretation

17. CPS: A-Operator • k is thrown away like E[ ]

18. Soundness: A-Operator

19. CPS: C-Operator • d is thrown away like E’[ ].

20. Soundness: C-Operator

21. CPS: Logical Interpretation

22. Summary of CPS

23. CPS as an Embedding

24. CPS as an Embedding

25. CPS as an Embedding

26. CPS as an Embedding Two more theorems (unrelated to proof-terms):

27. Doing it all in Twelf

28. Representing Terms • Use HOAS

29. Classical and Intuitionistic Terms • The previous definition is not enough. • We have to distinguish classical and intuitionistic logic • Introduce two types of terms: • Classical: termc • Intuitionistic: termi

30. Classical and Intuitionistic Terms

31. Representing the CPS Transform • Terms represented with HOAS • No direct representation for variables • How do we represent the following? • Create a judgment:

32. More Trouble …

33. The solution • We make the translation a hypothetical judgment • Recall: • We get:

34. The solution

35. Soundness Theorem

36. Input Coverage Problem: Problems: Soundness Theorem

37. Problem: Soundness Theorem Output External From worlds (soundnessblock) Make this an output? Can’t be changed (Needed for Induction)

38. Soundness Theorem Solution • New Theorem:

39. Soundness Theorem? • What have we shown? • What we need …

40. Soundness Theorem? • Are these theorems the same? • To us they are • Why? • We know that given A, there is exactly one A’ such that A* = A’. • We never told Twelf this fact • So, in Twelf these are different theorems!

41. Telling Twelf about Uniqueness • Can we tell Twelf that A* is unique? • Not directly! There is no uniqueness check • We can make an equality judgment

42. Telling Twelf about Uniqueness • Now we prove a theorem

43. A Typing-soundness Theorem • We also need the following theorem

44. True Soundness • Using all our previous theorems, we can now prove the soundness with correct modes.

45. Extensions • Can be extended to include conjunction and disjunction. • We can also use a call-by-name transform • Gives a translation from classical to minimal logic • Very similar to Kolmogrov’s double negation translation