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.2021.30
URN: urn:nbn:de:0030-drops-142686
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14268/
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Arkor, Nathanael ; McDermott, Dylan

Abstract Clones for Abstract Syntax

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LIPIcs-FSCD-2021-30.pdf (0.9 MB)

Abstract

We give a formal treatment of simple type theories, such as the simply-typed λ-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed λ-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as β- and η-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.

BibTeX - Entry

@InProceedings{arkor_et_al:LIPIcs.FSCD.2021.30,
  author =	{Arkor, Nathanael and McDermott, Dylan},
  title =	{{Abstract Clones for Abstract Syntax}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14268},
  URN =		{urn:nbn:de:0030-drops-142686},
  doi =		{10.4230/LIPIcs.FSCD.2021.30},
  annote =	{Keywords: simple type theories, abstract clones, second-order abstract syntax, substitution, variable binding, presentations, free algebras, induction, logical relations}
}

Keywords: simple type theories, abstract clones, second-order abstract syntax, substitution, variable binding, presentations, free algebras, induction, logical relations
Collection: 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)
Issue Date: 2021
Date of publication: 06.07.2021

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