Records Records { x 1 = a 1 ; . . . ; x n = a n } Definition of - PowerPoint PPT Presentation

Records • Records { x 1 = a 1 ; . . . ; x n = a n } Definition of Objects • Record Types in Type Theory { x 1 : A 1 ; . . . ; x n : A n } • Basic Operations – select( r, l ): • Reduction System r . l • Type for “close” objects – update( r, l, a ): • Type for “open” (extendable) objects r . l := a • Basic Reduction { x 1 = a 1 ; . . . ; x n = a n } . x i → a i where x i is not equal to any x j for j > i . 1 2

Basic Operations Basic Operations Another definition: • Objects • Objects } ∆ self . { | x 1 = m 1 [ self ]; . . . ; x n = m n [ self ] | = } ∆ self . { | x 1 = m 1 [ self ]; . . . ; x n = m n [ self ] | = { x 1 = λself.m 1 [ self ]; . . . ; x n = λself.m n [ self ] } λself. { x 1 = m 1 [ self ]; . . . ; x n = m 1 [ self ] } • Method Application apply method( obj ; l ): • Method Application apply method( obj ; l ): obj ◦ x ∆ = ( obj . x ) obj obj ◦ x ∆ = ( obj obj ) . x • Method Update update method( obj ; l ; self.m ): • Method Update update method( obj ; l ; self.m ): obj ◦ x := σ self .m [ self ] ∆ = obj . x := λ self .m [ self ] obj ◦ x := σ self .m [ self ] ∆ = λ self . ( obj self ) . x := m [ self ] 3 4 Basic Reduction Definition of type If o = We want to define a type { | x 1 = m 1 [ self ]; . . . ; x n = m n [ self ] | } Obj = Self . { | x 1 : A 1 [ Self ]; . . . ; x n : A n [ Self ] | } then o ◦ x i → m i [ o/ self ] such that obj ∈ Obj obj ◦ x i ∈ A i [ Obj ] where x i is not equal to any x j for j > i . 5 6

� � � Definition of type Definition of type Definition. Let i X.N ( X ) = � { T : U i | T ⊆ N ( T ) } N ( X ) = { x 1 : X → A 1 [ X ]; . . . ; x n : X → A n [ X ] } Rule. N is a function from U i to U i . obj ∈ X.N ( X ) A i is monotone obj ◦ x i ∈ A i [ X.N ( X )] Lemma. If X ⊆ N ( X ) and o ∈ X then o ◦ x i ∈ A i ( X ). 7 8 Problem 1: How to prove that something is in this type? Problem 2: This type is not extendable Must find X ⊆ N ( X ). Example Idea: • a ∆ = { | move = self | } ∈ { | move : Self | } X 0 = Top, X 1 = N ( X 0 ), X 2 = N ( X 1 ), . . . , X = � X n • b ∆ = { | move = a | } ∈ { | move : Self | } • c ∆ = update method( b ; x ; λ self . 1) / ∈ { | move : Self ; x : Z | } Then X ⊆ N ( X ) because c ◦ move ◦ x is undefined. • d ∆ = update method( a ; x ; λ self . 1) ∈ { | move : Self ; x : Z | } For example, we can already prove So, a is extendable, but b is not. { | x = 1; y = self ◦ x | } ∈ { | x : Z ; y : Z | } because { | x = 1; y = self ◦ x | } ∈ N ( N ( X )). 9 10

� � � � � � � � � � � � Problem 3: We want to update fields Extendable Objects: First Take Example. Naive definition: � obj ∆ X.N ( X ) = { T : U i | ∀ X ⊆ T.X ⊆ N ( X ) } = { | x = 1; y = self ◦ x | | x : Z ; y : N | } ∈ { } obj ◦ x := − 1 / ∈ { | x : Z ; y : N | } We expect the rule: Γ ⊢ obj ∈ X.N 1 ( X ) Γ; X : U i ; X ⊆ N 1 ( X ) ⊢ obj ∈ N 2 ( X ) X. ( N 1 ( X ) � N 2 ( X )) We define type of open objects with guards P X.N ( X ), where a Γ ⊢ obj ∈ guard is a property of types that allows updates: P ( X ) ⇒ ∀ a : Z . ∀ obj ∈ X.obj ◦ x := a ∈ X Unfortunately when we try to prove this we get unprovable subgoal U i ∈ U i . A type X is a “good extension” of P X.N ( X ) if P ( X ) ∧ X ⊆ N ( X ) 11 12 Main Rules Assume that all A i ’s are monotone and continuous functions from Formal definition U i to U i . N 1 ( X ) = { x 1 : X → A 1 [ X ]; . . . ; x n : X → A n [ X ] } N 2 ( X ) = { x n +1 : X → A n +1 [ X ]; . . . ; x m : X → A m [ X ] } P ( X ) : U i → P i +1 Inheritance: ∆ � P N = { T : U i +1 |∀ X : U P . ∃ Y : U P [ X ∩ T ; X ] . Y ⊆ N ( Y ) } Γ ⊢ obj ∈ P X.N 1 ( X ) Γ; X : U i ; P ( X ); X ⊆ N 1 ( X ) ⊢ o ∈ N 2 ( X ) Γ; X : U i ; P ( X ); X ⊆ N 1 ( X ) ⊢ P ( N 2 ( X )) ∆ P X. ( N 1 ( X ) � N 2 ( X )) where U P = { X : U i | P ( X ) } Γ; obj ∈ Closing: and U P [ A ; B ] ∆ = { X : U P | A ⊆ X ⊆ B } . P X.N ( X ) ⊆ X.N ( X ) Method Application: obj ∈ X.N ( X ) obj ◦ x i ∈ A i [ X.N ( X )] 13 14