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CSE-321 Programming Languages Predicative Polymorphic -Calculus POSTECH May 23, 2007 박성우

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2 System F

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3 Identity Function

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4 Pairing Function

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5 Typing Rules

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6 Reduction Rules for System F

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7 Outline System F V Type reconstruction Predicative polymorphic -calculus Let-polymorphism Value restriction

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8 Which is the most readable? fn x => fn y => x + y fn x : int => fn y : int => x + y fn x : int => fn y : int => (x + y) : int

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9 Which is the most readable? fn f => fn g => g o f fn f => fn g => (fn x => g (f x)) fn f : 'a -> 'b => fn g : 'b -> 'c => (fn x : 'a -> g (f x)) fn f : 'a -> 'b => fn g : 'b -> 'c => ((fn x : 'a -> (g (f x)) : 'c) : 'a -> 'c)

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Arguably, type annotations do not always improve code readability.

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11 Type Reconstruction Source code fn x => fn y => x + y After type reconstruction fn x : int => fn y : int => x + y : int -> int -> int

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12 Type Reconstruction in System F type reconstruction type erasure typable

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13 Erasure Function

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14 Type Reconstruction and Erasure type reconstruction type erasure typable

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15 Bad News Not every untyped expression is typable! –e.g. because every expression in System F eventually reduces to a value. Really bad news: –Type reconstruction is undecidable! –There is no algorithm for type reconstruction in System F!

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16 Outline System F V Type reconstruction V Predicative polymorphic -calculus Let-polymorphism Value restriction

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17 Predicative Polymorphic -Calculus Sublanguage of System F –decidable type reconstruction –supports polymorphism Helps us to understand let-polymorphism.

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18 Types in System F

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19 Types in P.P. -Calculus

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20 Predicative Polymorphic -Calculus A variable is assigned a monotype. A type application uses a monotype.

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21 Typing Rules

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Polymorphic types in predicative polymorphic -calculus are USELESS!

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23 Let's apply id to two different types bool and int.

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