1. Strict Bidirectional Type Checking Adam Chlipala, Leaf Petersen, and Robert Harper

2. 1/10/052 Type Systems Less Type Annotation More Type Annotation ML Concise syntax. Compact representation. Complex (global) type reconstruction problem. Undecidable past a certain point. Verbose syntax. Representation issues. Simple or no type reconstruction. Powerful type systems are still decidable. Haskell Java TILT IL

3. 1/10/053 Type Systems Less Annotation More Annotation ML Concise (enough) syntax. Compact representation. Simple syntax directed typechecking. Powerful and decidable type systems. Haskell Java TILT IL Pierce, Turner (POPL ‘98)

4. 1/10/054 Simply Typed Calculus

5. 1/10/055 TermsTypes 10

6. 1/10/056 TermsTypes 10 23

7. 1/10/057 TermsTypes 10 23 38

8. 1/10/058 TermsTypes 10 23 38 415

9. 1/10/059 TermsTypes 10 23 38 415 …… ii 2 -1 Quadratic! 

10. 1/10/0510 Algorithmic view Inputs are black. Outputs are red.

11. 1/10/0511 Algorithmic view Inputs are black. Outputs are red. Redundant

12. 1/10/0512 Algorithmic view Inputs are black. Outputs are red.

13. 1/10/0513 Algorithmic view Inputs are black. Outputs are red. Redundant

14. 1/10/0514 Bidirectional Typechecking Synthesis: Typed terms. Analysis: Checkable terms.

15. 1/10/0515 TermsTypes 11

16. 1/10/0516 TermsTypes 11 23

17. 1/10/0517 TermsTypes 11 23 35

18. 1/10/0518 TermsTypes 11 23 35 47

19. 1/10/0519 TermsTypes 11 23 35 47 …… i2i-1 Linear!

20. 1/10/0520 Unnecessary! Much simpler technique works fine. Good asymptotic behavior. Syntax directed. Extends directly to sum types.

21. 1/10/0521 Recursive types.

22. 1/10/0522 Simple term

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24. 1/10/0524 Fold Annotations Only

25. 1/10/0525 With Sum Annotations

26. 1/10/0526 Bidirectional version

27. 1/10/0527 Let Binding Surprisingly, bidirectional typechecking interacts badly with let binding. –Quadratic increase in annotation size in bad cases. Bad - programmers use let. Bad - compilers use let. –Put code in “named form” (sequentialization)

28. 1/10/0528 Binding forms Analysis binding: Synthesis binding:

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32. 1/10/0532 Sequentializing bidirectional terms may lead to quadratic increase in type size! 

33. 1/10/0533 Binding forms (revisited) Analysis binding: – Synthesis binding: – Undecorated analysis binding? – –Might solve our problem.

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37. 1/10/0537 Can we typecheck it?

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42. 1/10/0542 Key points: Typecheck body first. Context of occurrence gives a unique type to z 0 because it occurs in an analysis position.

43. 1/10/0543 Algorithmically Speaking Divide context into input and output zones. Input zone variables must declare types. –Explicitly: –Implicitly: Output zone variables get types from occurrences: – –Check A after getting type for u from E. –Must ensure variables actually occur!!

44. 1/10/0544 Formally Look to strict logic –Substructural logic in which hypotheses must be used at least once. Normal variables in unrestricted context (  may occur zero or more times. Strict variables in restricted context (  must occur at least once. Contrast with linear logic: –Linear variables in restricted context (  ) must occur exactly once.

45. 1/10/0545 Strict Bidirectional Typechecking Judgments: –Analysis: –Synthesis: Unrestricted context (  is input mapping variables to their types. Restricted context (  is output mapping variables to their types. Strict variables must occur in analysis positions.

46. 1/10/0546 Cool Fact #1 Theorem: The strict bidirectional typechecking rules constitute an algorithm. –Syntax directed. –Uniquely determined outputs. –Proof is in paper.

47. 1/10/0547 Cool Fact #2 Theorem: Sequentialization into the strict language never increases type size. –Size of sequentialized form is bounded above by the size of the original. –Proof is in paper.

48. 1/10/0548 Recap Bidirectional representation yields asymptotic improvements in type size. Let binding destroys these benefits. Recognizing strictness allows benefits to be recaptured without unification.

49. 1/10/0549 The End