A journey through the semantics of higher-order probabilistic languages,domain theory, and topology

This is a talk I gave at the yearly seminar of LMF in Étiolles, France, on June 13th, 2023.


I have been asked to give an introduction to topology in computer science “for dummies”. I understand this as an introduction to my own view of topology, as seen through computer science, and I have decided to concentrate on the semantics of programming languages; more precisely, of pure, higher-order, probabilistic programming languages, with a focus on denotational semantics based on directed-complete partial orders (dcpos).

This will give me the opportunity to introduce some of the notions of domain theory, with a glimpse of topology, or of that bit of topology we need for the task. In the end, there will not really be much of topology in this talk, but I hope you can still get something out of it in terms of semantics of programming languages.


The full slides, with all animation steps (229 pages); the shorter presentation, without animation steps (72 pages).


In six parts:

  1. Introduction, topology, domain theory, Plotkin’s classic study of the language PCF (17:32)
  2. Probabilistic PCF, and continuous valuations (23:17)
  3. Continuous dcpos, a.k.a. domains (7:02)
  4. Statistical programming languages, and ISPCF (20:57)
  5. Commutativity, and Fubini-Tonelli theorems (15:07)
  6. Concluding remarks (7:06)

Jean Goubault-Larrecq