Ellipses and lambda definability

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Ellipses are a meta-linguistic notation for denoting terms the size of which are specified by a meta-variable that ranges over the natural numbers. In this work, we present a systematic approach for encoding such meta-expressions in the λ-calculus, without ellipses: Terms that are parameterized by meta-variables are replaced with corresponding λ-abstractions over actual variables. We call such λ-terms arity-generic. Concrete terms, for particular choices of the parameterizing variable are obtained by applying an arity- generic λ-term to the corresponding numeral, obviating the need to use ellipses. For example, to find the multiple fixed points of n equations, n different λ-terms are needed, every one of which is indexed by two meta-variables, and defined using three levels of ellipses. A single arity-generic λ-abstraction that takes two Church numerals, one for the number of fixed-point equations, and one for their arity, replaces all these multiple fixed-point combinators. We show how to define arity-generic generalizations of two historical fixed-point combinators, the first by Curry, and the second by Turing, for defining multiple fixed points. These historical fixed-point combinators are related by a construction due to Böhm: We show that likewise, their arity-generic generalizations are related by an arity-generic generalization of Böhm’s construction. We further demonstrate this approach to arity-generic λ-definability with additional λ-terms that create, project, extend, reverse, and map over ordered n-tuples, as well as an arity-generic generator for one-point bases.

Original languageEnglish
Article number25
JournalLogical Methods in Computer Science
Issue number3
StatePublished - 30 Sep 2015


  • Arity-generic expressions
  • Bases
  • Definability
  • Fixed-point combinators
  • LISP/Scheme
  • Lambda calculus
  • Variadic functions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


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