CSE-321 Programming Languages Typechecking

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CSE-321 Programming LanguagesTypechecking
???

• POSTECH
• May 3, 2006

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Declarative Interpretation of ? e A

• Input
• ?
• e
• A
• Output
• Yes e has type A under ?.
• No e does not have type A under ?.

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Algorithmic Interpretation of ? e A

• Input
• ?
• e
• Output
• A Yes, A is the type of e.
• TypeError No, e has no type.

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Outline

• Purely synthetic typechecking
• algorithmic interpretation only
• essentially Assignment 5, Part I
• Bidirectional typechecking

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Implementation of the Type System

• A new function typing

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Function typing (1/2)
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Function typing (2/2)
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Algorithmic Typing Judgment
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Algorithmic Typing Rules
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Correctness

• Soundness
• Completeness

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Outline

• Purely synthetic typechecking V
• Bidirectional typechecking
• both declarative interpretation and algorithmic
  interpretation

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Do we need bool?
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Bidirectional Typechecking

• No type annotation in ?-abstraction
• Some expression explicitly annotated

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Bidirectional Typechecking

• Analysis phase
• Synthesis phase

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Which rule?
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Analyze ? x. e
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Which rule?
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Synthesize for e e'
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Variable
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(No Transcript)
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Can we infer the type of (e A) ?

• Yes, it's A,but only if e indeed has type A.

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Bidirectional Typechecking Rules
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Intro and Elim Expressions

• Intro expression I
• always check I against a type A.
• never infer its type.
• uses an introduction rule.
• Elim expression E
• always infer its type.
• never check E against a type A.
• uses an elimination rule.

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Typing Rules for Bidirectional Typechecking