Principal type-schemes for functional programs

Transcript

1. Principal type-schemes
   for functional programs
   Luis Damas and Robin Milner (POPL `82)
   

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2. Agenda
   ● Slides
   ● Code
   

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3. ML
   ● Meta Language for LCF
   ● Type inference
   ● Influence on Haskell, Rust, F#, OCaml, ...
   ● “Sweet spot” in type system design
   

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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)
   

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

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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?
   

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7. Lambda Calculus
   Expressions e:
   ● Identifiers: , , …
   ● Applications: e e’
   ● Abstractions: . e
   ● Let bindings: let = e in e’
   

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

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

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10. Type Schemes
    Type schemes :
    ● Monomorphic:
    ● Polymorphic: ∀ .
    Type schemes are types with identifiers bound by ∀ at the
    front.
    

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11. Substitutions
    Mappings from variables to types
    ● Can instantiate type schemes using substitutions
    ● Gives a simple subtyping relation on type schemes
    

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12. Semantics
    Construct a semantic domain (CPO) V containing
    ● Primitives
    ● Functions
    ● An error element
    and a semantic function : e → (Id → V) → V
    

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13. Semantics
    Identify types with subsets of V
    Define the judgment
    A ╞ e :
    when (∀ ( : ’) ∈ A. ∈ ’) ⇒ e ∈
    

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14. Declarative System
    Variable rule:
    

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15. Declarative System
    Application rule:
    

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16. Declarative System
    Abstraction rule:
    

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17. Declarative System
    Let rule:
    

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18. Declarative System
    Instantiation rule:
    

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19. Declarative System
    Generalization rule:
    

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20. Soundness
    If
    A e :
    then
    A ╞ e :
    “Static behavior determines dynamic behavior”
    

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21. Example
    Prove:
    . : ∀ . ( → → ) → →
    

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22. Algorithm W
    ● The inference rules do not translate easily into an
    algorithm (why not?)
    ● Introduce
    w : Exp → Env → (Env, )
    

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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!
    

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24. Unification
    ● Unification gives local information about types
    ● We assemble a global solution from local information
    

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

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26. Occurs Check
    Prevents inference of infinite types
    w( . , nil) = error!
    Can’t unify ~ if occurs in the body of .
    E.g. ~ →
    ~ ((… → ) → ) →
    

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27. Soundness
    If
    w(A, e) = (S, )
    then
    A e :
    “Algorithm W constructs typing judgments”
    

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28. Completeness
    If
    A e :
    then w(A, e) constructs a typing judgment for e which
    generalises the above.
    “Algorithm W constructs principal types”
    

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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!
    

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

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