License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2023.11
URN: urn:nbn:de:0030-drops-179956
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17995/
Go to the corresponding LIPIcs Volume Portal

Barenbaum, Pablo ; Sottile, Cristian

Two Decreasing Measures for Simply Typed λ-Terms

pdf-format:
LIPIcs-FSCD-2023-11.pdf (0.8 MB)

Abstract

This paper defines two decreasing measures for terms of the simply typed λ-calculus, called the ?-measure and the ?^{?}-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^{?}-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 ?-measure maps each λ-term to a natural number, and it is obtained by evaluating the term in the λ^{?}-calculus and counting the number of remaining wrappers. The ?^{?}-measure maps each λ-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.

BibTeX - Entry

@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/opus/volltexte/2023/17995},
  URN =		{urn:nbn:de:0030-drops-179956},
  doi =		{10.4230/LIPIcs.FSCD.2023.11},
  annote =	{Keywords: Lambda Calculus, Rewriting, Termination, Strong Normalization, Simple Types}
}

Keywords: Lambda Calculus, Rewriting, Termination, Strong Normalization, Simple Types
Collection: 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)
Issue Date: 2023
Date of publication: 28.06.2023

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI