A representation of Lambda terms suitable for operations on their intensions

Abstract

A representation for lambda terms is described based on the scheme of de Bruijn for eliminating variable names. The new notation provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this “delaying” of substitutions. The precise mechanism that is used is, however, more complex than the usual so as to support the ability to examine subterms embedded under abstractions. A virtue of our representation is that it permits substitution to be realized as an atomic operation and thereby provides for efficient implementations of β-reduction. Operations on lgr;-terms are described based on our representation and the relationship of these to the conventional definitions are exhibited.