CSE-321 Programming Languages

Subtyping

POSTECH

May 14, 2007

박성우

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!

16

17

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

• A · B corresponds to dereferencing.• B · A corresponds to assigning a new value.• Nonvariant (neither covariant nor contravariant)

22

Subtyping for Array Types• Array types = extension of reference types

• Subtyping for array types in Java

– runtime overhead of dynamic tag-checks.

23

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

24

Coercion Semantics for Subtyping

• Example

– 32-bit word for int– 64-bit word for float

25

Coercion Subtyping Judgment

26

• All coercion functions for A · B) exhibit the same behavior.

Coherence