Section: New Results

Continuous Random Variables

Participant : Jean Goubault-Larrecq.

Continuing work on probabilistic and non-deterministic choice in a domain-theoretic setting, Jean Goubault-Larrecq and Daniele Varacca (PPS, University Paris 7) proposed a new monad for probabilistic choice, that of continuous random variables [38] . The usual Jones-Plotkin monad of continuous valuations, although simple enough, suffers from the defect that no category of continuous domains is known that would be both Cartesian-closed (i.e., would allow one to interpret functions) and stable under the Jones-Plotkin monad.

Jean Goubault-Larrecq and Daniele Varacca managed to show that a related monad, that of continuous random variables, inspired from the notion of a random variable in probability theory, did not suffer from this defect: the category of bc-domains is indeed both Cartesian-closed and stable under this monad. Moreover, the authors showed that using one or the other monad gave semantics to higher-order probabilistic programs that were indistinguishable at ground types. Finally, they used this to solve an open problem by Escardò, namely that observational equivalence of probabilistic higher-order programs is recursively enumerable.