Strict Bidirectional Type Checking

Adam Chlipala, Leaf Petersen, and Robert Harper

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Type Systems

Less Type Annotation More Type Annotation

ML

•Concise syntax.•Compact representation.•Complex (global) type reconstruction problem.•Undecidable past a certain point.

•Verbose syntax.•Representation issues.•Simple or no type reconstruction.•Powerful type systems are still decidable.

Haskell JavaTILT IL

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Type Systems

Less Annotation More Annotation

ML

•Concise (enough) syntax.•Compact representation.•Simple syntax directed typechecking.•Powerful and decidable type systems.

Haskell JavaTILT IL

Pierce, Turner (POPL

‘98)

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Simply Typed Calculus

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Terms

Types

1 0

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Terms

Types

1 0

2 3

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Terms

Types

1 0

2 3

3 8

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Terms

Types

1 0

2 3

3 8

4 15

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Terms

Types

1 0

2 3

3 8

4 15

… …

i i2-1

Quadratic!

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Algorithmic view

Inputs are black.Outputs are red.

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Algorithmic view

Inputs are black.Outputs are red.

Redundant

Redundant

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Algorithmic view

Inputs are black.Outputs are red.

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Algorithmic view

Inputs are black.Outputs are red.

Redundant

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Bidirectional Typechecking

Synthesis: Typed terms.

Analysis: Checkable terms.

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Terms

Types

1 1

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Terms

Types

1 1

2 3

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Terms

Types

1 1

2 3

3 5

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Terms

Types

1 1

2 3

3 5

4 7

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Terms

Types

1 1

2 3

3 5

4 7

… …

i 2i-1

Linear!

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Unnecessary!

Much simpler technique works fine.

• Good asymptotic behavior.• Syntax directed.• Extends directly to sum types.

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Recursive types.

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Simple term

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Fold Annotations Only

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With Sum Annotations

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Bidirectional version

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Let Binding

• Surprisingly, bidirectional typechecking interacts badly with let binding.– Quadratic increase in annotation size in

bad cases.

• Bad - programmers use let.• Bad - compilers use let.

– Put code in “named form” (sequentialization)

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Binding forms

• Analysis binding:

• Synthesis binding:

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Sequentializing bidirectional terms may lead to quadratic increase in type size!

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Binding forms (revisited)

• Analysis binding:–

• Synthesis binding:–

• Undecorated analysis binding?– – Might solve our problem.

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Can we typecheck it?

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Key points: • Typecheck body first.• Context of occurrence gives a unique type to z0 because it occurs in an analysis position.

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Algorithmically Speaking

• Divide context into input and output zones.

• Input zone variables must declare types.– Explicitly:– Implicitly:

• Output zone variables get types from occurrences:– – Check A after getting type for u from E.– Must ensure variables actually occur!!

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Formally

• Look to strict logic – Substructural logic in which hypotheses must

be used at least once.

• Normal variables in unrestricted context ( may occur zero or more times.

• Strict variables in restricted context ( must occur at least once.

• Contrast with linear logic:– Linear variables in restricted context () must

occur exactly once.

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Strict Bidirectional Typechecking

• Judgments:– Analysis:– Synthesis:

• Unrestricted context ( is input mapping variables to their types.

• Restricted context ( is output mapping variables to their types.

• Strict variables must occur in analysis positions.

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Cool Fact #1

Theorem: The strict bidirectional typechecking rules constitute an algorithm.– Syntax directed.– Uniquely determined outputs.– Proof is in paper.

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Cool Fact #2

Theorem: Sequentialization into the strict language never increases type size.– Size of sequentialized form is bounded

above by the size of the original.– Proof is in paper.

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Recap

• Bidirectional representation yields asymptotic improvements in type size.

• Let binding destroys these benefits.

• Recognizing strictness allows benefits to be recaptured without unification.