An Infinitary Affine Lambda-Calculus Isomorphic to the Full Lambda-Calculus

Abstract

It is well known that the real numbers arise from the metric completion of the rational numbers, with the metric induced by the usual absolute value. We seek a computational version of this phenomenon, with the idea that the role of the rationals should be played by the affine lambda-calculus, whose dynamics is finitary; the full lambda-calculus should then appear as a suitable metric completion of the affine lambda-calculus. This paper proposes a technical realization of this idea: an affine lambda-calculus is introduced, based on a fragment of intuitionistic multiplicative linear logic; the calculus is endowed with a notion of distance making the set of terms an incomplete metric space; the completion of this space is shown to yield an infinitary affine lambda-calculus, whose quotient under a suitable partial equivalence relation is exactly the full (non-affine) lambda-calculus. We also show how this construction brings interesting insights on some standard rewriting properties of the lambda-calculus (finite developments, confluence, standardization, head normalization and solvability).

Infinitary Lambda Calculi from a Linear Perspective

“…Another closely related work is a recent one by Mazza [24], that shows how the ordinary, finitary, λ-calculus can be seen as the metric completion of a much weaker system, namely the affine λ-calculus. Again, the main commonalities with this paper are on the one hand the presence of infinite terms, and on the other a common technical background, namely that of linear logic.…”

A Functorial Bridge Between the Infinitary Affine Lambda-Calculus and Linear Logic

“…The reduction is confluent, as the rules never duplicate any subterm. The results of [10] are formulated in an infinitary calculus with exponential variables only, whose terms and reduction are defined as follows:…”

“…In [10], it was shown that usual λ-terms may be recovered as uniform infinitary affine terms. The categorical version of this result is that, in certain conditions, a model of the infinitary affine λ-calculus is also a model of linear logic.…”

“…Although not valid in general, this condition holds in several Lafont categories of very different flavor, such as Conway games and coherence spaces. The second approach, due to the first author [10], is topological, and is based directly on the syntax. One considers an affine λ-calculus in which variables are treated as bounded exponentials: in a term of this calculus, a variable x may appear any number of times, each occurrence appears indexed by an integer (each instance, noted x i , is labelled with a distinct i ∈ N).…”

“…The contribution of this paper is to draw a bridge between the two approaches presented above. Indeed, we develop a categorical version of the topological construction of [10], which turns out to:…”