# Differentials and distances in probabilistic coherence spaces

## TL;DR

This paper explores derivatives in probabilistic coherence spaces, enabling computation of execution time expectations and establishing Lipschitz properties that relate observational and model distances, thus enhancing probabilistic programming analysis.

## Contribution

It introduces the use of derivatives in probabilistic coherence spaces to compute execution expectations and prove Lipschitz properties, linking observational and model distances.

## Key findings

- Derivatives enable expectation computation of execution time in probabilistic PCF.
- A local differential notion proves Lipschitz continuity of morphisms.
- Lipschitz properties relate observational and model distances in probabilistic languages.

## Abstract

In probabilistic coherence spaces, a denotational model of probabilistic functional languages, mor-phisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful.