Differentials and Distances in Probabilistic Coherence Spaces

Author Thomas Ehrhard

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Thomas Ehrhard
  • CNRS, IRIF, Université de Paris, France


We thank Raphaëlle Crubillé, Paul-André Melliès, Michele Pagani and Christine Tasson for many enlightening discussions on this work. We also thank the referees for their precious comments and suggestions.

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Thomas Ehrhard. Differentials and Distances in Probabilistic Coherence Spaces. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • Theory of computation → Probabilistic computation
  • Theory of computation → Abstract machines
  • Theory of computation → Linear logic
  • Denotational semantics
  • probabilistic coherence spaces
  • differentials of programs


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