Abstract Clones for Abstract Syntax

Authors Nathanael Arkor , Dylan McDermott

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Author Details

Nathanael Arkor
  • University of Cambridge, UK
Dylan McDermott
  • Reykjavik University, Iceland

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Nathanael Arkor and Dylan McDermott. Abstract Clones for Abstract Syntax. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Equational logic and rewriting
  • Theory of computation → Higher order logic
  • Theory of computation → Proof theory
  • simple type theories
  • abstract clones
  • second-order abstract syntax
  • substitution
  • variable binding
  • presentations
  • free algebras
  • induction
  • logical relations


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