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CSE-321 Programming Languages Extensions to the Simply Typed  -Calculus

CSE-321 Programming Languages Extensions to the Simply Typed  -Calculus. 박성우. POSTECH April 4, 2007. Abstract Syntax. Operational Semantics. Type System. Outline. Product types pairs generalized product types unit type Sum types Fixed point construct. Pairs in SML.

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CSE-321 Programming Languages Extensions to the Simply Typed  -Calculus

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  1. CSE-321 Programming LanguagesExtensions to the Simply Typed -Calculus 박성우 POSTECH April 4, 2007

  2. Abstract Syntax

  3. Operational Semantics

  4. Type System

  5. Outline • Product types • pairs • generalized product types • unit type • Sum types • Fixed point construct

  6. Pairs in SML - (1, true); val it = (1,true) : int * bool - #1 (1, true) ; val it = 1 : int - #2 (1, true) ; val it = true : bool

  7. Abstract Syntax and Typing Rules

  8. Eager Reduction Rules

  9. Lazy Reduction Rules

  10. Outline • Product types • pairs V • generalized product types • unit type • Sum types • Fixed point construct

  11. Tuples in SML - (1, true, "hello", fn x : int => x); val it = (1,true,"hello",fn) : int * bool * string * (int -> int) - #1 (1, true, "hello", fn x : int => x); val it = 1 : int - #2 (1, true, "hello", fn x : int => x); val it = true : bool - #3 (1, true, "hello", fn x : int => x); val it = "hello" : string

  12. Generalized Product Types

  13. Generalized Product Types • We have n components. • n = 2 • pair types • n > 2 • tuple types • n = 1 • seems irrelevant • n = 0?

  14. n = 0

  15. Outline • Product types • pairs V • generalized product types V • unit type • Sum types • Fixed point construct

  16. Unit in SML - (); val it = () : unit

  17. n = 0: Nullary Product Type • We need n elements to build a tuple: • I want to extract the i-th element where 1 ·i : No elimination rule for the unit type!

  18. Type unit: No Elim; Only Intro

  19. Outline • Product types V • pairs • generalized product types • unit type • Sum types • Fixed point construct

  20. Datatypes in SML - datatype t = Inl of int | Inr of bool; datatype t = Inl of int | Inr of bool - val x = Inl 1; val x = Inl 1 : t - val y = Inr true; val y = Inr true : t - case (Inl 1) of Inl x => x | Inr _ => 0; val it = 1 : int - case (Inr true) of Inl x => x | Inr _ => 0; val it = 0 : int

  21. Datatypes in PostechML - datatype 'int + bool' = Inl of int | Inr of bool; datatype 'int + bool' = Inl of int | Inr of bool - val x = Inl 1; val x = Inl 1 : 'int + bool' - val y = Inr true; val y = Inr true : 'int + bool' - case (Inl 1) of Inl x => x | Inr _ => 0; val it = 1 : int - case (Inr true) of Inl x => x | Inr _ => 0; val it = 0 : int

  22. Sum Types: Two Alternatives

  23. Reduction Rules

  24. bool = unit + unit

  25. n = 0: Nullary Sum Type

  26. What about Elimination Rule?

  27. Type void: No Intro; Only Elim

  28. Why do we need the Elim rule then? • void type is not found in SML.

  29. Outline • Product types V • pairs • generalized product types • unit type • Sum types V • Fixed point construct

  30. Recursion? • Why not use the fixed point combinator? • It does not typecheck in the simply typed -calculus. • f has type A and also A ! A.

  31. Fixed Point Construct

  32. Recursive Functions in SML

  33. Mutually Recursive Functions in SML

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