Second Order Types Giuseppe Castagna: Foundation of OOP Tutorial - PDF document

Second Order Types Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 68 0 0

Loss of information Consider the function I = λx T .x : T → T By the rule for application I : T → T M : U < T I ( M ) : T Therefore ( λx � � a : int � � .x ) � a = 1 � b = 2 � : � � a : int � � Second order I : ∀ X ≤ T.X → X Two ways: 1. Implicit polymorphism 2. Explicit polymorphism Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 69 62 0

Implicit polymorphism No types in terms λx.x : ∀ α.α → α ( λx.x )3 : int x : α � x : α [ α = β ] � λx.x : α → β � 3 : int [ α = int ] � ( λx.x )3 : β Subtyping λx. (( λy.x )( x.� + 3)) : ∀ α ≤� � � : int � � .α → α Therefore λx. (( λy.x )( x.� +3))( � � = 1 � m = true � ) : � � � : int� m : bool � � Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 70 0 0

Giuseppe Castagna: Foundation of OOP �Tutorial Slides) Inference with subtyping x : α � x : α α ≤ � � � : � � � x : α� y : γ � x : α x : α � 3 : int x : α � x.� : � α = β δ = int� � ≤ int x : α � λy.x : γ → β x : α � x.� + 3 : δ δ ≤ γ x : α � ( λy.x )( x.� + 3) : β � λx. (( λy.x )( x.� + 3)) : α → β Resulting type ∀ � ≤ int . ∀ α ≤� � � : � � � . α → α Simplified ∀ α ≤� � � : int � � . α → α 71 0 0

Explicit polymorphism Λ X.λx X .x : ∀ X.X → X The programmer specifies the type (Λ X.λx X .x )( int )(3) ( λx int .x )(3) � Subtyping � .λx X .x Λ X ≤� � a : int � The application � .λx X .x )( � (Λ X ≤� � a : int � � a : int� b : int � � ) has type � � a : int� b : int � � → � � a : int� b : int � � thus � .λx X (Λ X ≤� � a : int � .x )( � � a : int� b : int � � )( � a = 1 � b = 3 � ) has type � a : int� b : int � � � Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 72 0 0

F ≤ Types ::= X | Top | T → T | ∀ ( X ≤ T ) T T Terms x | ( λx T .a ) | a ( a ) ::= a | top | Λ X ≤ T.a | a ( T ) Reduction ( λx T .a )( b ) � a [ x T := b ] ( β ) (Λ X ≤ T.a )( T � ) � a [ X := T � ] ( β ∀ ) Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 73 0 0

Subtyping (refl) C � T ≤ T C � T 1 ≤ T 2 C � T 2 ≤ T 3 (trans) C � T 1 ≤ T 3 (taut) C � X ≤ C ( X ) (Top) C � T ≤ Top C � T 1 ≤ S 1 C � S 2 ≤ T 2 ( → ) C � S 1 → S 2 ≤ T 1 → T 2 C � T 1 ≤ S 1 C� ( X ≤ T 1 ) � S 2 ≤ T 2 ( ∀ ) C � ∀ ( X ≤ S 1 ) S 2 ≤ ∀ ( X ≤ T 1 ) T 2 Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 74 0 0

Type system [Vars] C ; Γ � x : Γ( x ) C ; Γ � ( x : T ) � a : T � [ → Intro] C ; Γ � ( λx T .a ): T → T � C ; Γ � a : S → T C ; Γ � b : S [ → Elim] C ; Γ � a ( b ): T [Top] C ; Γ � top : Top C� ( X ≤ T ) ; Γ � a : T � [ ∀ Intro] C ; Γ � Λ X ≤ T.a : ∀ ( X ≤ T ) T � C ; Γ � a : ∀ ( X ≤ S ) T [ ∀ Elim] C ; Γ � a ( S ): T [ X := S ] C � T � ≤ T C ; Γ � a : T � [Subsump] C ; Γ � a : T Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 75 0 0

Transitivity elimination Id A | X T | Top T | c → c � | ∀ ( X ≤ c ) c � | c c � c : : = C � Id A : A ≤ A (refl) C � c � : T 2 ≤ T 3 C � c : T 1 ≤ T 2 (trans) C � c � c : T 1 ≤ T 3 C ∪ � X ≤ T } � X T : X ≤ T (taut) C � Top T : T ≤ Top (Top) C � c 1 : T � C � c 2 : T 2 ≤ T � 1 ≤ T 1 2 ( → ) C � c 1 → c 2 : T 1 → T 2 ≤ T � 1 → T � 2 C � c 1 : T � C ∪ � X ≤ T � 1 } � c 2 : T 2 ≤ T � 1 ≤ T 1 2 ( ∀ ) C � ∀ ( X ≤ c 1 ) c 2 : ∀ ( X ≤ T 1 ) T 2 ≤ ∀ ( X ≤ T � 1 ) T � 2 Theorem 5 There is a 1-1 correspondence be- tween well-typed coerce expressions and sub- typing derivations. Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 76 0 0

Giuseppe Castagna: Foundation of OOP �Tutorial Slides) The rewriting system ( Asc ) ( c d ) e c ( d e ) � ( c → d ) ( c � → d � ) ( c � c ) → ( d d � ) ( → � ) � ( → �� ) ( c → d ) (( c � → d � ) e ) (( c � c ) → ( d d � )) e � ∀ ( X ≤ c � c )( d d � [ X T : = c X S ]) ( ∀ � ) ( ∀ ( X ≤ c ) d ) ( ∀ ( X ≤ c � ) d � ) � ( ∀ ( X ≤ c � c )( d d � [ X T := c X S ])) e ( ∀ �� ) ( ∀ ( X ≤ c ) d ) (( ∀ ( X ≤ c � ) d � ) e ) � Normal forms are subterms of ( c → d ) e 1 . . . e n or of ( ∀ ( X ≤ c ) d ) e 1 . . . e n where c� c i � d� d i are in normal form and e 1 � . . . � e n are either X t or Top T . They normal forms correspond to derivations in which every left premise of a (trans) rule is a leaf. Thus, the rewriting system pushes the transitivity up to the leaves. 76 0 0

Example ( c → d ) (( c � → d � ) e ) � (( c � c ) → ( d d � )) e ∗ Theorem 6 (Soundness) If c � d and C � c : Δ then C � d : Δ Theorem 7 (Weak normalization) Every in- nermost strategy for � terminates. Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 76 0 0

Coherence Let c : S ≤ T (id l ) Id T c c � (id r ) c Id S c � (top) Top T c � Top S (varTop) X Top � Top X Consider the composition of the rewriting sys- tems: Theorem 8 (normal forms) Every well-typed coerce expression in normal form has the form c 0 c 1 ... c n with n ≥ 0 , where c 0 can be any co- erce expression different from a composition �of other coerce expressions) whose subformu- lae are in normal form, and c 1 . . . c n are vari- ables. Theorem 9 For every provable subtyping judg- ment, there exists only one coerce expression in normal form proving it. Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 76 0 0

Coherence Theorem 10 (coherence) Let Π 1 and Π 2 be two proofs of the same judgment C � Δ . If c 1 and c 2 are the corresponding coerce ex- pressions then c 1 and c 2 are equal modulo the rewriting system. Shape of NFs and the subtyping algorithm The normal forms of Theorem 8 correspond to derivations in which every application of a (trans) rule has as left premise an application of the rule (taut). Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 76 0 0

Subtyping algorithm (AlgRefl) C � X ≤ X C � C ( X ) ≤ T (AlgTrans) C � X ≤ T (Top) C � T ≤ Top C � T 1 ≤ S 1 C � S 2 ≤ T 2 ( → ) C � S 1 → S 2 ≤ T 1 → T 2 C� ( X ≤ T 1 ) � S 2 ≤ T 2 C � T 1 ≤ S 1 ( ∀ ) C � ∀ ( X ≤ S 1 ) S 2 ≤ ∀ ( X ≤ T 1 ) T 2 Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 76 0 0

Typing algorithm [Vars] C ; Γ � x : Γ( x ) C ; Γ � ( x : T ) � a : T � [ → I] C ; Γ � ( λx T .a ): T → T � [ → E] C ; Γ � a : U C ; Γ � b : S � C � S � ≤ S B C ( U ) = S → T C ; Γ � a ( b ): T [Top] C ; Γ � top : Top C� ( X ≤ T ) ; Γ � a : T � [ ∀ I] C ; Γ � Λ X ≤ T.a : ∀ ( X ≤ T ) T � C � S � ≤ S C ; Γ � a : U [ ∀ E] B C ( U ) = ∀ ( X ≤ S ) T C ; Γ � a ( S � ): T [ X := S � ] Definition 2 � B C ( C ( X )) if T ≡ X B C ( T ) = otherwise T Giuseppe Castagna: Foundation of OOP �Tutorial Slides) 77 0 0

Typing and subtyping algorithms are sound and complete Giuseppe Castagna: Foundation of OOP �Tutorial Slides) Sound and complete does not mean decidable�� let ¬ T and ∀ ( X ) T denote T → Top and ∀ ( X ≤ Top ) T : X 0 ≤ ∀ ( Y ) ¬ ( ∀ ( Z ≤ Y ) ¬ Y ) ∀ ( X 1 ≤ X 0 ) ¬ X 0 � X 0 ≤ by applying AlgTrans: X 0 ≤ ∀ ( Y ) ¬ ( ∀ ( Z ≤ Y ) ¬ Y ) � ∀ ( X 1 ) ¬ ( ∀ ( X 2 ≤ X 1 ) ¬ X 1 ) ∀ ( X 1 ≤ X 0 ) ¬ X 0 ≤ by applying ( ∀ ): X 0 ≤ ∀ ( Y ) ¬ ( ∀ ( Z ≤ Y ) ¬ Y ) � X 1 ≤ X 0 � ¬ ( ∀ ( X 2 ≤ X 1 ) ¬ X 1 ) ≤ ¬ X 0 by the contravariance of ( → ): X 0 ≤ ∀ ( Y ) ¬ ( ∀ ( Z ≤ Y ) ¬ Y ) � X 1 ≤ X 0 � X 0 ≤ ∀ ( X 2 ≤ X 1 ) ¬ X 1 the same judgement as the one we started from. Just semi-decidability holds Kernel-Fun: compare quantifications with equal bounds. 78 0 0