Subtying Kangwon National University 임현승 Programming Languages These slides were originally created by Prof. Sungwoo Park at POSTECH.

2 Key Features of Object-Oriented Languages Multiple representation –different methods of the same name –dynamic dispatch Encapsulation –Only an object's own methods can see private data. Inheritance –classes, subclasses,... Subtyping –interface, interface extension

3 Outline Overview V Principle of subtyping Subtyping relations Coercion semantics

4 Principle of Subtyping

5 Examples

6 Basic Rules Reflexivity and transitivity Rule of subsumption

7 Subset Semantics for Subtyping Example –32-bit word for nat –32-bit word for int –64-bit word for float

8 Outline Overview V Principle of subtyping V Subtyping relations Coercion semantics

9 Setup

10 Subtyping for Product Types

11 Subtyping for Sum Types

12 Subtyping for Function Types

13 Subtyping for Reference Types

You are all so gullible!

15 Wrong Subtyping Rules!

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18 Subtyping for Function Types Covariant in return types B and B' Contravariant in argument types A and A'

19 Dereferencing ref A

20 Assigning to ref A

21 Subtyping for Reference Types

22 Subtyping for Array Types Array types = extension of reference types Subtyping for array types in Java –runtime overhead of dynamic tag-checks.

Subtyping for Java Array Types class Animal {} class Dog extends Animal {} class Cat extends Animal {} class Main { public static void main(String[] args) { Dog[] dogs = new Dog[1]; dogs[0] = new Dog(); Animal[] animals = dogs; animals[0] = new Cat(); } 23 Exception in thread "main" java.lang.ArrayStoreException: Cat at Main.main

24 Outline Overview V Principle of subtyping V Subtyping relations V Coercion semantics

25 Coercion Semantics for Subtyping Example –32-bit word for int –64-bit word for float

26 Coercion Subtyping Judgment

27 Coherence