An adequate and efficient left-associated binary numeral system in the λ-calculus

Abstract

This paper introduces a sequence of λ-expressions modeling 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, i.e. one in which all recursive functions are lambda-definable. These functions can be computed efficiently; To this end, we introduce a notion of complexity that is independent of the order of evaluation.