On Generalized Metric Spaces for the Simply Typed Lambda-Calculus

Abstract

Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way. However, the application of generalized metrics to higher-order languages like the simply typed lambda calculus has so far proved unsatisfactory. In this paper we investigate a new approach to the construction of cartesian closed categories of generalized metric spaces. Our starting point is a quantitative semantics based on a generalization of usual logical relations. Within this setting, we show that several families of generalized metrics provide ways to extend the Euclidean metric to all higher-order types.

Effectful program distancing

“…Semantical approaches to program distancing and program approximation for higher-order languages have been proposed in a number of recent papers. Closest to our work are the work by Dal Lago and Gavazzo [2021a];Dal Lago et al [2019]; Pistone [2021] on differential logical relations, and the unpublished work by Westbrook and Chaudhuri [2013] that inspired the former. There are several dissimilarities between the present paper and the aforementioned work on differential logical relations, the main one being that the latter can deal with pure languages only.…”

From Identity to Difference: A Quantitative Interpretation of the Identity Type

“…In Sec. 5 we show that dTT yields a natural language for differential logical relations [13,23,28], an approach to approximate program transformations in which program differences are themselves higher-order entities. In Sec.…”

“…Indeed, this kind of issues is due to the fact that Met is not a cartesian closed category, that is, a model of full STλC, but only of its sub-exponential variant Fuzz. The theory of differential logical relations [13,23,28] (in short, DLR), has been developed to overcome this kind of problems when investigating approximate transformations in STλC. A DLR is a ternary relation ρ Ď X ˆL ˆX relating the elements of some set X with the values of some complete lattice L of "errors over X"; intuitively, ρpx, ǫ, yq is to be read as the fact that the error of replacing x by y is bounded by ǫ.…”

“…However, }x, y} ρ needs not be a metric in general: first of all, the self-distances }x, x} ρ (that we note simply as }x} ρ ) need not be zero (i.e. the bottom element of L); moreover, } , } ρ needs not satisfy the usual triangular law of metric spaces (for a detailed comparison between DLR and -generalized [33] -metric spaces, see [28]). Here we will restrict our attention to separated DLR, i.e.…”

“…A map of DLR pX, L, ρq and pY, M, µq is given by a function f : X Ñ Y (hence no Lipschitz or other continuity conditions are asked) together with an auxiliary map ϕ : X ˆX ˆL Ñ Y which, intuitively, tracks errors in input into errors in output; more formally, ϕ must satisfy, for all x, y P X and ǫ P L, that if ρpx, ǫ, yq holds, then both µpf pxq, ϕpx, y, ǫq, f pyqq and µpf pxq, ϕpx, y, ǫq, f pyqq also hold. (Separated) DLR and their maps form a cartesian closed category DLR (see the Appendix, and [13,28] for further details).…”