CS 4110 Programming Languages & Logics. Lecture 25 Records and Subtyping

Transcription

1 CS 4110 Programming Languages & Logics Lecture 25 Records and Subtyping 31 October 2016

2 Announcements 2 Homework 6 returned: x = 34 of 37, σ = 3.8 Preliminary Exam II in class on Wednesday, November 16 New date! Please me as soon as you can if you have a conflict. Topics: λ-calculus through subtyping (today) Not cumulative (unlike the final) Practice problems available on CMS now

3 Records 3 We ve seen binary products (pairs), and they generalize to n-ary products (tuples). Records are a generalization of tuples where we mark each field with a label.

4 Records 3 We ve seen binary products (pairs), and they generalize to n-ary products (tuples). Records are a generalization of tuples where we mark each field with a label. Example: {foo = 32, bar = true} is a record value with an integer field foo and a boolean field bar.

5 Records 3 We ve seen binary products (pairs), and they generalize to n-ary products (tuples). Records are a generalization of tuples where we mark each field with a label. Example: {foo = 32, bar = true} is a record value with an integer field foo and a boolean field bar. Its type is: {foo:int, bar:bool}

6 Syntax 4 l L e ::= {l 1 = e 1,..., l n = e n } e.l v ::= {l 1 = v 1,..., l n = v n } τ ::= {l 1 :τ 1,..., l n :τ n }

7 Dynamic Semantics 5 E ::=... {l 1 = v 1,..., l i 1 = v i 1, l i = E, l i+1 = e i+1,..., l n = e n } E.l {l 1 = v 1,..., l n = v n }.l i v i

8 Static Semantics 6 i 1..n. Γ e i :τ i Γ {l 1 = e 1,..., l n = e n }:{l 1 :τ 1,..., l n :τ n } Γ e:{l 1 :τ 1,..., l n :τ n } Γ e.l i :τ i

9 Example 7 GETX λp:{x : int, y : int}. p.x

10 Example 7 GETX λp:{x : int, y : int}. p.x GETX {x = 4, y = 2}

11 Example 7 GETX λp:{x : int, y : int}. p.x GETX {x = 4, y = 2} GETX {x = 4, y = 2, z = 42}

12 Example 7 GETX λp:{x : int, y : int}. p.x GETX {x = 4, y = 2} GETX {x = 4, y = 2, z = 42} GETX {y = 2, x = 4}

13 Subtyping 8 Definition (Subtype) τ 1 is a subtype of τ 2, written τ 1 τ 2, if a program can use a value of type τ 1 whenever it would use a value of type τ 2. If τ 1 τ 2, we also say τ 2 is the supertype of τ 1.

14 Subtyping 8 Definition (Subtype) τ 1 is a subtype of τ 2, written τ 1 τ 2, if a program can use a value of type τ 1 whenever it would use a value of type τ 2. If τ 1 τ 2, we also say τ 2 is the supertype of τ 1. Γ e:τ τ τ Γ e:τ SUBSUMPTION This typing rule says that if e has type τ and τ is a subtype of τ, then e also has type τ.

15 Record Subtyping 9 We ll define a new subtyping relation that works together with the subsumption rule. τ 1 τ 2

16 Record Subtyping 10 This program isn t well-typed (yet): (λp:{x : int}. p.x) {x = 4, y = 2}

17 Record Subtyping 10 This program isn t well-typed (yet): (λp:{x : int}. p.x) {x = 4, y = 2} So let s add width subtyping: k 0 {l 1 :τ 1,..., l n+k :τ n+k } {l 1 :τ 1,..., l n :τ n }

18 Record Subtyping 11 This program also doesn t get stuck: (λp:{x : int, y : int}. p.x + p.y) {y = 37, x = 5}

19 Record Subtyping 11 This program also doesn t get stuck: (λp:{x : int, y : int}. p.x + p.y) {y = 37, x = 5} So we can make it well-typed by adding permutation subtyping: π is a permutation on 1..n {l 1 :τ 1,..., l n :τ n } {l π(1) :τ π(1),..., l π(n) :τ π(n) }

20 Record Subtyping 12 Does this program get stuck? Is it well-typed? (λp:{x : {y : int}}. p.x.y) {x = {y = 4, z = 2}}

21 Record Subtyping 12 Does this program get stuck? Is it well-typed? (λp:{x : {y : int}}. p.x.y) {x = {y = 4, z = 2}} Let s add depth subtyping: i 1..n. τ i τ i {l 1 :τ 1,..., l n :τ n } {l 1 :τ 1,..., l n :τ n }

22 Record Subtyping 13 Putting all three forms of record subtyping together: i 1..n. j 1..m. l i = l j τ j τ i {l 1 :τ 1,..., l m :τ m } {l 1 :τ 1,..., l n :τ n} S-RECORD

23 Standard Subtyping Rules 14 We always make the subtyping relation both reflexive and transitive. τ τ S-REFL τ 1 τ 2 τ 2 τ 3 τ 1 τ 3 S-TRANS Think of every type describing a set of values. Then τ 1 τ 2 when τ 1 s values are a subset of τ 2 s.

24 Top Type 15 It s sometimes useful to define a maximal type with respect to subtyping: τ ::= τ S-TOP Everything is a subtype of, as in Java s Object or Go s interface{}.

25 Subtype All the Things! 16 We can also write subtyping rules for sums and products: τ 1 τ 1 τ 2 τ 2 τ 1 + τ 2 τ 1 + τ 2 S-SUM

26 Subtype All the Things! 16 We can also write subtyping rules for sums and products: τ 1 τ 1 τ 2 τ 2 τ 1 + τ 2 τ 1 + τ 2 S-SUM τ 1 τ 1 τ 2 τ 2 τ 1 τ 2 τ 1 τ 2 S-PRODUCT

27 Function Types 17 How should we decide whether one function type is a subtype of another???? τ 1 τ 2 τ 1 τ 2 S-FUNCTION

28 Desiderata 18 We d like to have: int {x:int, y:int} int {x:int}

29 Desiderata 18 We d like to have: int {x:int, y:int} int {x:int} And: {x:int} int {x:int, y:int} int

30 Desiderata We d like to have: And: In general, to prove: int {x:int, y:int} int {x:int} {x:int} int {x:int, y:int} int τ 1 τ 2 τ 1 τ 2 we ll require: Argument types are contravariant: τ 1 τ 1 Return types are covariant: τ 2 τ 2 18

31 Function Subtyping 19 Putting these two pieces together, we get the subtyping rule for function types: τ 1 τ 1 τ 2 τ 2 τ 1 τ 2 τ 1 τ 2 S-FUNCTION

32 Reference Subtyping 20 What should the relationship be between τ and τ in order to have τ ref τ ref?

33 Example 21 If r has type τ ref, then!r has type τ. Imagine we replace r with r, where r has a type τ ref that we ve somehow decided is a subtype of τ ref.

34 Example 21 If r has type τ ref, then!r has type τ. Imagine we replace r with r, where r has a type τ ref that we ve somehow decided is a subtype of τ ref. Then!r should still produce something can be treated as a τ. In other words, it should have a type that is a subtype of τ. So the referent type should be covariant: τ τ τ ref τ ref

35 Example 22 If v has type τ, then r := v should be legal. If we replace r with r, then it must still be legal to assign r := v. So!r would then produce a value of type τ.

36 Example 22 If v has type τ, then r := v should be legal. If we replace r with r, then it must still be legal to assign r := v. So!r would then produce a value of type τ. So the referent type should be contravariant! τ τ τ ref τ ref

37 Reference Subtyping 23 In fact, subtyping for reference types must be invariant: a reference type τ ref is a subtype of τ ref if and only if τ τ and τ τ. τ τ τ τ τ ref τ ref S-REF

38 Java Arrays 24 Tragically, Java s mutable arrays use covariant subtyping!

39 Java Arrays 24 Tragically, Java s mutable arrays use covariant subtyping! Suppose that Cow is a subtype of Animal. Code that only reads from arrays typechecks: Animal[ ] arr = new Cow[ ] { new Cow( Alfonso ) }; Animal a = arr[0];

40 Java Arrays 24 Tragically, Java s mutable arrays use covariant subtyping! Suppose that Cow is a subtype of Animal. Code that only reads from arrays typechecks: Animal[ ] arr = new Cow[ ] { new Cow( Alfonso ) }; Animal a = arr[0]; but writing to the array can get into trouble: arr[0] = new Animal( Brunhilda ); Specifically, this generates an ArrayStoreException.