Principal type-schemes for functional programs

Transcript

1. Principal type-schemes for functional programs Luis Damas and Robin Milner

   (POPL `82)

2. Agenda • Slides • Code

3. ML • Meta Language for LCF • Type inference •

   Influence on Haskell, Rust, F#, OCaml, ... • “Sweet spot” in type system design

4. ML letrec f xs = if null xs then nil

   else snoc (f (tl xs)) (hd xs) What type does this function have? null : ∀ ( list → bool) snoc : ∀ ( list → → list) hd, tl : ∀ ( list → ) nil : ∀ ( list)

5. ML What about: let s x y z = x

   z (y z) ?

6. Type Inference f : ∀ ( list → list) •

   Given f, how can we infer this type? • What does it even mean for a value to have a type? • How can we be sure we have the most general type?

7. Lambda Calculus Expressions e: • Identifiers: , , … •

   Applications: e e’ • Abstractions: . e • Let bindings: let = e in e’

8. Lambda Calculus For example: . . . . let =

   . . in

9. Types Monotypes : • Variables: • Primitives: • Functions: →

10. Type Schemes Type schemes : • Monomorphic: • Polymorphic: ∀

    . Type schemes are types with identifiers bound by ∀ at the front.

11. Substitutions Mappings from variables to types • Can instantiate type

    schemes using substitutions • Gives a simple subtyping relation on type schemes

12. Semantics Construct a semantic domain (CPO) V containing • Primitives

    • Functions • An error element and a semantic function : e → (Id → V) → V

13. Semantics Identify types with subsets of V Define the judgment

    A ╞ e : when (∀ ( : ’) ∈ A. ∈ ’) ⇒ e ∈

14. Declarative System Variable rule:

15. Declarative System Application rule:

16. Declarative System Abstraction rule:

17. Declarative System Let rule:

18. Declarative System Instantiation rule:

19. Declarative System Generalization rule:

20. Soundness If A e : then A ╞ e :

    “Static behavior determines dynamic behavior”

21. Example Prove: . : ∀ . ( → → )

    → →

22. Algorithm W • The inference rules do not translate easily

    into an algorithm (why not?) • Introduce w : Exp → Env → (Env, )

23. Algorithm W • W attempts to build a substitution, bottom-up

    • W can fail with an error if there is no valid typing • Intuition: ◦ Collect constraints ◦ Then solve constraints • Reality: W is the fusion of these two steps • See the code!

24. Unification • Unification gives local information about types • We

    assemble a global solution from local information

25. Unification Example: ( → ) ~ (( → ) →

    ) ~ ( → ) ~ ~ ( → )

26. Occurs Check Prevents inference of infinite types w( . ,

    nil) = error! Can’t unify ~ if occurs in the body of . E.g. ~ → ~ ((… → ) → ) →

27. Soundness If w(A, e) = (S, ) then A e

    : “Algorithm W constructs typing judgments”

28. Completeness If A e : then w(A, e) constructs a

    typing judgment for e which generalises the above. “Algorithm W constructs principal types”

29. Further Reading More type systems • System F, F⍵ •

    Rank-N types • Type Classes • Dependent Types • Refinement Types Other approaches • Constraints • Bidirectional typechecking • SMT See TAPL & ATAPL!

30. Acknowledgments DHM axioms reproduced from Wikipedia under the CC-3.0 Attribution/ShareAlike

    license