Metric Reasoning about $\lambda$-Terms: the Affine Case (Long Version)

TL;DR

This paper investigates the concept of metric reasoning for affine lambda-terms, proposing trace distances to quantify behavioral similarities in probabilistic lambda calculus, and establishing their theoretical properties and practical applicability.

Contribution

It introduces and characterizes trace distances for affine lambda-terms, extending the notion of context equivalence to a probabilistic setting with fully-abstract metrics.

Findings

Trace distance characterizes behavioral similarity of affine lambda-terms.

Bounded by a coinductive Kantorovich-based distance.

Tuple-based trace distance handles nontrivial examples.

Abstract

Terms of Church's $\lambda$-calculus can be considered equivalent along many different definitions, but context equivalence is certainly the most direct and universally accepted one. If the underlying calculus becomes probabilistic, however, equivalence is too discriminating: terms which have totally unrelated behaviours are treated the same as terms which behave very similarly. We study the problem of evaluating the distance between affine $\lambda$-terms. The most natural definition for it, namely a natural generalisation of context equivalence, is shown to be characterised by a notion of trace distance, and to be bounded from above by a coinductively defined distance based on the Kantorovich metric on distributions. A different, again fully-abstract, tuple-based notion of trace distance is shown to be able to handle nontrivial examples.