CSE-321 Programming Languages

Simply Typed -Calculus

POSTECHApril 3, 2006

박성우

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-Calculus• Abstract syntax

• Operational semantics (call-by-value)

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Simulating Base Types• Booleans

• Natural numbers

• Fixed point combinator

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• The -calculus is equally expressive as Turing machines.– booleans, integers, lists, recursive functions, ...

• But it is highly inefficient to program in the -calculus.– Why not just use

instead of ?

• So we introduce a type system!

Motivation

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Simply Typed -Calculus• An extension of the untyped -calculus with types

• Assumes a fixed set of base types– E.g.

• base type • primitive constructs

• A subset of Standard ML

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Outline• Introduction V• The simply typed -calculus

– Abstract syntax– Operational semantics– Type system

• Type safety

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Abstract Syntax

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What if there is no base type?

No interesting expression!

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Outline• Introduction V• The simply typed -calculus

– Abstract syntax V– Operational semantics– Type system

• Type safety

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Simply Untyped -Calculus

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Reduction Rules for Booleans

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Capture-Avoiding Substitutions• Completely analogous

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Free Variables• Completely analogous

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Outline• Introduction V• The simply typed -calculus

– Abstract syntax V– Operational semantics V– Type system

• Type safety

What is the type of ?

Answer:

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How to find the type of 1. Assume that the type of x is A.

• okay

2. Find the type of x.• A

3. Build a function type• A ! A

Need to make assumptions on types of variables!

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Type System• Typing context

• Typing judgment

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Typing Rules --- Top-down

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Typing Rules --- Bottom-up

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Typing Rules for Booleans

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Typing Derivation

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Typing Derivation

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Outline• Introduction V• The simply typed -calculus V

– Abstract syntax V– Operational semantics V– Type system V

• Type safety

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Unsafe Operations in C• Adding two pointers

• Subtracting an integer from a string– which is okay, but likely to be unintended

• Null-pointer dereferencing– Argh... segmentation fault!

• Using an integer as a destination address in a function call

• ...

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Huffman Coding in SML• Suppose that you spent 10 hours before getting

your program to typecheck.– 0am: You start.– 10am: Your program compiles with no type

errors.

• How many more hours did you spend after that?

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Type Safety• Slogan

"well-typed expressions never go wrong"

• Two theorems– Type preservation:

"A well-typed expression reduces to another expression of the same type."

– Progress:"A well-typed expression is not stuck: either it is a value or

reduces to another expression."

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Type Preservation + Progress• A well-typed expression e:

– If it is a value, we are finished.– If it is not,

• It reduces to another e' [Progress]• e' has the same type as e. [Type

preservation]

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Type Safety• Type preservation

• Progress

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Proof of Type Safety• Use the rule induction.• In the next lecture!