A Logic for Lawson Compact Algebraic Ldomains Abstract   In this paper, we build a logic which is named Nsequent calculus. Based on this logic, we provide two kinds of logical representations of Lawson compact algebraic Ldomains: one in terms of logical algebras and the other in terms of logical syntax.  The first representation takes the corresponding logical algebras as research objects. The use of prime filters achieves the connection between our logic and Lawson compact algebraic Ldomains. This approach is inspired by Abramskyu0027s SFP domain logic and the disjunctive propositional logic on algebraic Ldomains introduced by Yixiang Chen and Achim Jung. However, there are essential differences between them at the morphisms part. For the second representation, we directly adopt Nsequent calculi themselves as objects instead of the logical algebras. Then we establish the category of Nsequent calculi with consequence relations equivalent to that of Lawson compact algebraic Ldomains with Scott continuous maps. This demonstrates the capability of the syntax of the logic in representing domains.

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.