CSE-321 Programming Languages

Extensions to the Simply Typed -Calculus

POSTECH

April 10, 2006

박성우

2

Abstract Syntax

3

Operational Semantics

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Type System

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Outline• Product types

– pairs– generalized product types– unit type

• Sum types• Fixed point construct

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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

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Abstract Syntax and Typing Rules

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Eager Reduction Rules

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Lazy Reduction Rules

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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

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Generalized Product Types

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Generalized Product Types

• We have n components.– n = 2

• pair types– n > 2

• tuple types– n = 1

• seems irrelevant– n = 0?

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n = 0

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Outline• Product types

– pairs V– generalized product types V– unit type

• Sum types• Fixed point construct

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Unit in SML- ();

val it = () : unit

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• We need n elements to build a tuple:

• I want to extract the i-th element where 1 · i :

n = 0: Nullary Product Type

No elimination rule for the unit type!

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Type unit: No Elim; Only Intro

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Outline• Product types V

– pairs– generalized product types– unit type

• Sum types• Fixed point construct

21

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

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Datatypes in SML 2007- 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

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Sum Types: Two Alternatives

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Reduction Rules

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bool = unit + unit

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n = 0: Nullary Sum Type

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What about Elimination Rule?

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Type void: No Intro; Only Elim

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Why do we need the Elim rule then?

• void type is not found in SML.

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Outline• Product types V

– pairs– generalized product types– unit type

• Sum types V • Fixed point construct

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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.

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Fixed Point Construct

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Recursive Functions in SML

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Mutually Recursive Functions in SML