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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.2023.24
URN: urn:nbn:de:0030-drops-180086
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18008/
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Altenkirch, Thorsten ; Kaposi, Ambrus ; Šinkarovs, Artjoms ; Végh, Tamás

Combinatory Logic and Lambda Calculus Are Equal, Algebraically

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LIPIcs-FSCD-2023-24.pdf (0.7 MB)

Abstract

It is well-known that extensional lambda calculus is equivalent to extensional combinatory logic. In this paper we describe a formalisation of this fact in Cubical Agda. The distinguishing features of our formalisation are the following: (i) Both languages are defined as generalised algebraic theories, the syntaxes are intrinsically typed and quotiented by conversion; we never mention preterms or break the quotients in our construction. (ii) Typing is a parameter, thus the un(i)typed and simply typed variants are special cases of the same proof. (iii) We define syntaxes as quotient inductive-inductive types (QIITs) in Cubical Agda; we prove the equivalence and (via univalence) the equality of these QIITs; we do not rely on any axioms, the conversion functions all compute and can be experimented with.

BibTeX - Entry

@InProceedings{altenkirch_et_al:LIPIcs.FSCD.2023.24,
  author =	{Altenkirch, Thorsten and Kaposi, Ambrus and \v{S}inkarovs, Artjoms and V\'{e}gh, Tam\'{a}s},
  title =	{{Combinatory Logic and Lambda Calculus Are Equal, Algebraically}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{24:1--24:19},
  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/2023/18008},
  URN =		{urn:nbn:de:0030-drops-180086},
  doi =		{10.4230/LIPIcs.FSCD.2023.24},
  annote =	{Keywords: Combinatory logic, lambda calculus, quotient inductive types, Cubical Agda}
}

Keywords: Combinatory logic, lambda calculus, quotient inductive types, Cubical Agda
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 (Formalisation): https://bitbucket.org/akaposi/combinator archived at: https://archive.softwareheritage.org/swh:1:dir:5275bf1e26eb4aa1e2391d0ace5953c3f4898ef4

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