Combinatory Logic and Lambda Calculus Are Equal, Algebraically

Authors Thorsten Altenkirch , Ambrus Kaposi , Artjoms Šinkarovs , Tamás Végh

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Thorsten Altenkirch
  • School of Computer Science, University of Nottingham, UK
Ambrus Kaposi
  • Eötvös Loránd University, Budapest, Hungary
Artjoms Šinkarovs
  • Heriot-Watt University, Edinburgh, UK
Tamás Végh
  • Eötvös Loránd University, Budapest, Hungary

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Thorsten Altenkirch, Ambrus Kaposi, Artjoms Šinkarovs, and Tamás Végh. Combinatory Logic and Lambda Calculus Are Equal, Algebraically. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Combinatory logic
  • lambda calculus
  • quotient inductive types
  • Cubical Agda


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