Just Enough Type Theory or, Featherweight Java A Simple Formal Model of Objects

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Just Enough Type Theoryor,Featherweight
JavaA Simple Formal Model of Objects

• Jonathan Aldrich
• 15-819

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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

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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.

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FJ Syntax

• Standard BNF definition
• Overbar represents a sequence

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Subtyping Judgments
Base case each class subtypes itself
Transitivity rule Both exprs on top must hold
If we know this
Then we can conclude this
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Dynamic Semantics

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

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Evaluation Examples
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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
• Javas type system does a dynamic check at casts,
  and so programs can halt with a cast error.
• FJs type system, however prevents all other run
  time errors.
• Real languages have additional error cases
  however, the type soundness guarantee is still
  useful

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FJ Types

• ? maps var -gt class
• Read ? e ? C as, in the context of type
  environment ?, expression e has type C

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Class/Method Typing
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Other definitions
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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