The Bayesian Ontology Language mathcal BELBEL We introduce the new probabilistic description logic (DL) mathcal BEL BEL, which extends the lightweight DL mathcal ELEL with the possibility of expressing uncertainty about the validity of some knowledge. Contrary to other probabilistic DLs, mathcal BELBEL is designed to represent classical knowledge that depends on an uncertain context; that is, some of the knowledge may hold or not depending on the current situation. The probability distribution of these contexts is expressed by a Bayesian network (BN). We study different reasoning problems in mathcal BELBEL, providing tight complexity bounds for all of them. One particularly interesting property of our framework is that reasoning can be decoupled between the logical (i.e., mathcal ELEL), and the probabilistic (i.e., the BN) components. We later generalize all the notions presented to introduce Bayesian extensions of arbitrary ontology languages. Using the decoupling property, we are able to provide tight complexity bounds for reasoning in the Bayesian extensions of many other DLs. We provide a detailed analysis of our formalism w.r.t. the assumptions made and compare it with the existing approaches.

A Domain Theory for Statistical Probabilistic Programming We give an adequate denotational semantics for languages with recursive higherorder types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixedvariance datatypes. Our semantics justifies important program equivalences including commutativity.  :61,new semantic model is based on quasiBorel predomainsu0027. These are a mixture of chaincomplete partial orders (cpos) and quasiBorel spaces. QuasiBorel spaces are a recent model of probability theory that focuses on sets of admissible random elements. Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher order functions. By contrast, quasiBorel predomains do support both a commutative probabilistic powerdomain and higherorder functions. As we show, quasiBorel predomains form both a model of Fioreu0027s axiomatic domain theory and a model of Kocku0027s synthetic measure theory.