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

Abstract

We explore a quantitative interpretation of 2-dimensional intuitionistic type theory (ITT) in which the identity type is interpreted as a "type of differences". We show that a fragment of ITT, that we call difference type theory (dTT), yields a general logical framework to talk about quantitative properties of programs like approximate equivalence and metric preservation. To demonstrate this fact, we show that dTT can be used to capture compositional reasoning in presence of errors, since any program can be associated with a "derivative" relating errors in input with errors in output. Moreover, after relating the semantics of dTT to the standard weak factorization systems semantics of ITT, we describe the interpretation of dTT in some quantitative models developed for approximate program transformations, incremental computing, program differentiation and differential privacy.

An automatic differentiation system for the age of differential privacy

“…As shown above however, this approach does not attempt to derive a tight bound on sensitivity in the setting of unbounded queries, declaring sensitivity as infinite and relying exclusively on clipping. More general approaches, including category-theoretical views on the intersection of differentiable programming and differential privacy such as [27] have also recently been proposed.…”