2 Search Results for "Sottile, Cristian"

Document

DOI: 10.4230/LIPIcs.FSCD.2024.8

Mechanized Subject Expansion in Uniform Intersection Types for Perpetual Reductions

Authors: Andrej Dudenhefner and Daniele Pautasso

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We provide a new, purely syntactical proof of strong normalization for the simply typed λ-calculus. The result relies on a novel proof of the equivalence between typability in the simple type system and typability in the uniform intersection type system (a restriction of the non-idempotent intersection type system). For formal verification, the equivalence is mechanized using the Coq proof assistant. In the present work, strong normalization of a given simply typed term M is shown in four steps. First, M is reduced to a normal form N via a suitable reduction strategy with a decreasing measure. Second, a uniform intersection type for the normal form N is inferred. Third, a uniform intersection type for M is constructed iteratively via subject expansion. Fourth, strong normalization of M is shown by induction on the size of the type derivation. A supplementary contribution is a family of perpetual reduction strategies, i.e. strategies which preserve infinite reduction paths. This family allows for subject expansion in the intersection type systems of interest, and contains a reduction strategy with a decreasing measure in the simple type system. A notable member of this family is Barendregt’s F_∞ reduction strategy.

Cite as

Andrej Dudenhefner and Daniele Pautasso. Mechanized Subject Expansion in Uniform Intersection Types for Perpetual Reductions. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dudenhefner_et_al:LIPIcs.FSCD.2024.8,
  author =	{Dudenhefner, Andrej and Pautasso, Daniele},
  title =	{{Mechanized Subject Expansion in Uniform Intersection Types for Perpetual Reductions}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.8},
  URN =		{urn:nbn:de:0030-drops-203371},
  doi =		{10.4230/LIPIcs.FSCD.2024.8},
  annote =	{Keywords: lambda-calculus, simple types, intersection types, strong normalization, mechanization, perpetual reductions}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2023.11

Two Decreasing Measures for Simply Typed λ-Terms

Authors: Pablo Barenbaum and Cristian Sottile

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract

This paper defines two decreasing measures for terms of the simply typed λ-calculus, called the 𝒲-measure and the 𝒯^{𝐦}-measure. A decreasing measure is a function that maps each typable λ-term to an element of a well-founded ordering, in such a way that contracting any β-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the λ-calculus. In this system, dubbed the λ^{𝐦}-calculus, each β-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its λ-abstraction. The 𝒲-measure maps each λ-term to a natural number, and it is obtained by evaluating the term in the λ^{𝐦}-calculus and counting the number of remaining wrappers. The 𝒯^{𝐦}-measure maps each λ-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.

Cite as

Pablo Barenbaum and Cristian Sottile. Two Decreasing Measures for Simply Typed λ-Terms. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{barenbaum_et_al:LIPIcs.FSCD.2023.11,
  author =	{Barenbaum, Pablo and Sottile, Cristian},
  title =	{{Two Decreasing Measures for Simply Typed \lambda-Terms}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.11},
  URN =		{urn:nbn:de:0030-drops-179956},
  doi =		{10.4230/LIPIcs.FSCD.2023.11},
  annote =	{Keywords: Lambda Calculus, Rewriting, Termination, Strong Normalization, Simple Types}
}

Refine by Author
1 Barenbaum, Pablo
1 Dudenhefner, Andrej
1 Pautasso, Daniele
1 Sottile, Cristian

Refine by Classification
1 Theory of computation → Equational logic and rewriting
1 Theory of computation → Lambda calculus
1 Theory of computation → Type theory

Refine by Keyword
1 Lambda Calculus
1 Rewriting
1 Simple Types
1 Strong Normalization
1 Termination
Show More...

Refine by Type
2 document

Refine by Publication Year
1 2023
1 2024