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.2023.22
URN: urn:nbn:de:0030-drops-180068
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Frontull, Samuel ; Moser, Georg ; van Oostrom, Vincent


LIPIcs-FSCD-2023-22.pdf (1 MB)


When substitutions and bindings interact, there is a risk of undesired side effects if the substitution is applied naïvely. The λ-calculus captures this phenomenon concretely, as β-reduction may require the renaming of bound variables to avoid variable capture. In this paper we introduce α-paths as an estimation for α-avoidance, roughly expressing that α-conversions are not required to prevent variable capture. These paths provide a novel method to analyse and predict the potential need for α in different calculi. In particular, we show how α-path characterises α-avoidance for several sub-calculi of the λ-calculus like (i) developments, (ii) affine/linear λ-calculi, (iii) the weak λ-calculus, (iv) μ-unfolding and (iv) finally the safe λ-calculus. Furthermore, we study the unavoidability of α-conversions in untyped and simply-typed λ-calculi and prove undecidability of the need of α-conversions for (leftmost-outermost reductions) in the untyped λ-calculus. To ease the work with α-paths, we have implemented the method and the tool is publicly available.

BibTeX - Entry

  author =	{Frontull, Samuel and Moser, Georg and van Oostrom, Vincent},
  title =	{{\alpha-Avoidance}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{22:1--22:22},
  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 =		{},
  URN =		{urn:nbn:de:0030-drops-180068},
  doi =		{10.4230/LIPIcs.FSCD.2023.22},
  annote =	{Keywords: \lambda-calculus, variable capture, \alpha-conversion, developments, safe \lambda-calculus, undecidability}

Keywords: λ-calculus, variable capture, α-conversion, developments, safe λ-calculus, undecidability
Collection: 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)
Issue Date: 2023
Date of publication: 28.06.2023
Supplementary Material: Software (Alpha Avoidance Tool): archived at:

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