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!

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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 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