Just Enough Type Theory

or,

Featherweight JavaA Simple Formal Model of Objects

Jonathan Aldrich

15-819

Why Formal Models?

• Make precise what a language means– What can you say in the language?– How does a program execute?

• Allow us to prove formal properties– Typically, lack of certain run-time errors

• This course– Type theory not required for much of the reading– However, some papers use formal notation

• A brief introduction will help us get more out of it

Example: Featherweight Java

• A minimal core calculus for Java– Classes, methods, fields, inheritance– Any FJ program is a Java program

• Purpose of a core language– Leaves out unnecessary details– Focuses attention on issues of interest– Makes proving formal properties easier

• Citation– Atsushi Igarashi, Benjamin Pierce, and Philip Wadler.

Featherweight Java: A Minimal Core Calculus for Java and GJ. OOPSLA ’99.

FJ Syntax

• Standard BNF definition• Overbar represents a sequence

Subtyping Judgments

If we know this

Then we can conclude this

Base case: each class subtypes itself

Transitivity ruleBoth exprs on top must hold

Dynamic Semantics

• Computation expressed as rewriting rules• [d/x] e – substitute d for x in e

Evaluation Examples

Type System

• Conceptually:– Annotates an object or expression– Describes operations that are applicable

• Prevents run-time errors from undefined operations– X = “hello” – 2– snail.fly()

• Type soundness– A well-typed program will not halt with an undefined operation

error• Java’s type system does a dynamic check at casts, and so

programs can halt with a cast error.• FJ’s type system, however prevents all other run time errors.

– Real languages have additional error cases; however, the type soundness guarantee is still useful

FJ Types

maps var -> class

• Read ├ e C as, “in the context of type environment , expression e has type C

Class/Method Typing

Other definitions

Type Soundness

A well-typed program remains well-typed after a reduction step

A well-typed program can take a step

Well-typed FJ programs eventually reduce to either a value or an expression with an embedded cast error

Proofs are by induction, beyond the scope of this course