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Hereditary Substitution for the -Calculus Harley Eades and Aaron Stump Computer Science Saturday, June 22, 13 1 The Big Picture Goal: Prove weak normalization of the -calculus. Tool of choice: hereditary substitution. Novelty:

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Hereditary Substitution for the λΔ-Calculus

Harley Eades and Aaron Stump Computer Science

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The Big Picture

Goal: Prove weak normalization of the λΔ-calculus. Tool of choice: hereditary substitution. Novelty: normalization by hereditary substitution has never been applied to any classical type theories.

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Why Hereditary Substitution

It provides a directly defined substitution which preserves normal forms.

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Overview

Hereditary substitution and STLC. The λΔ-calculus. A naive extension of hereditary substitution to the λΔ- calculus. The correct extension. Normalization of the λΔ-calculus.

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The Simply Typed λ-Calculus

Syntax:

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The Simply Typed λ-Calculus

Typing: Reduction:

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Hereditary Substitution

Syntax: Usual termination order: Like ordinary capture-avoiding substitution. Except, if the substitution introduces a redex, then that redex is recursively reduced. Example:

[t/x]At

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Hereditary Substitution

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Hereditary Substitution

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Properties of Hereditary Substitution

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Properties of Hereditary Substitution

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Properties of Hereditary Substitution

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The λΔ-Calculus

Syntax Negation: Rehof:1994

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The λΔ-Calculus

Typing: Reduction: Rehof:1994

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Problems with a Naive Extension

The naive extension is a simple extension to the hereditary substitution function for STLC:

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Problems with a Naive Extension

The naive extension is a simple extension to the hereditary substitution function for STLC: ctype tells us:A ≥ A00 → A0 < ¬(A00 → A0)

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Problems with a Naive Extension

The naive extension is a simple extension to the hereditary substitution function for STLC: ctype tells us:A ≥ A00 → A0 < ¬(A00 → A0)

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How do we fix this?

Consider the following example:

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How do we fix this?

Consider the following example:

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How do we fix this?

Consider the following example:

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A Correct Extension of Hereditary Substitution

Hereditary structural substitution: Is a multi-substitution defined by induction mutually with the hereditary substitution function. Syntax: , where New termination metric:

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A Correct Extension of Hereditary Substitution

New termination metric:

[t/x]At

(<, −, −) (=, <, −) (=, =, <) (<, −, −) (=, =, <)

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A Correct Extension of Hereditary Substitution

Variables:

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A Correct Extension of Hereditary Substitution

Abstractions:

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A Correct Extension of Hereditary Substitution

Applications:

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A Correct Extension of Hereditary Substitution

Applications:

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A Correct Extension of Hereditary Substitution

The hereditary substitution function:

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A Correct Extension of Hereditary Substitution

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A Correct Extension of Hereditary Substitution

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A Correct Extension of Hereditary Substitution

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Concluding Normalization

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Concluding Normalization

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Concluding Normalization

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Related Work

The key notion of using a lexicographic ordering on an

rdering on types and the strict subexpression ordering on

proofs dates all the way to Prawitz 1965. STLC: Lévy:1967, Girard:1989, and Amadio:1998. Hereditary substitution was first made explicit by Watkins: 2004 and Adams:2004.

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Related Work

Abel:2006 implemented a normalizer using sized heterogeneous types. Abel:2008 uses hereditary substitution as a normalization function at the kind level in the metatheory of higher order subtyping. Keller:2010 formalized the hereditary substitution for STLC in Agda.

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Related Work

David:2003 show strong normalization of the simply typed λΔ-calculus using a lexicographic ordering.

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Conclusion

Hereditary substitution is a proof method which shows promise as an effective tool to prove normalization of typed λ-calculi. We showed how to adapt this proof method to a type theory with control. The key notion was to eliminate auxiliary redexes during reduction.

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Thank you!

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