An Adequate Left-Associated Binary Numeral System in the lambda-Calculus

An Adequate Left-Associated Binary Numeral System in the -Calculus

This paper introduces a sequence of λ-expressions modelling the binary expansion of integers. We derive expressions computing the test for zero, the successor function, and the predecessor function, thereby showing the sequence to be an adequate numeral system. These functions can be computed efficiently. Their complexity is independent of the order of evaluation.

Counting Terms in the Binary Lambda Calculus

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent lambda terms and derive results from their generating functions, especially that the number of terms of size n grows roughly like 1.963447954n.

An example of a non adequate numeral system

Résumé Un système numérique est défini par la donnée de trois λ-termes clos: un λ-terme normal d0 pour Zéro, un λ-terme Sd pour le Successeur, et un λ-terme pour le Test à Zéro, tels que les λ-termes (Sd i d0) sont normalisables et possèdent des formes normales differentes. Un système numérique est dit adéquat ssi il possède un λ-terme clos pour le Prédécesseur. Dans cette Note nous présentons ...

Binary Lambda Calculus and Combinatory Logic

In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parser-interpreters for these binary languages. Along the way we also present new results on list representations, bracket abstraction, and fixpoint combinators. In the second part we review Algorithmic Information Theory, for whi...

Encoding Left Reduction in the Lambda-Calculus with Interaction Nets