On the Lattice of Program Metrics

TL;DR

This paper explores and compares various program metrics for higher-order calculi, introducing a new interactive metric and analyzing their relationships to better understand their discriminative power.

Contribution

It introduces the interactive metric via the Int-Construction, and analyzes how semantic, observational, and equational metrics relate and differ in discriminating programs.

Findings

Semantic metrics lie between observational and equational metrics.

In some cases, these metric inclusions are strict.

Denotational metric is less discriminating than the interactive metric.

Abstract

In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie \emph{in between} the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.