A duality theory for quantitative semantics

Abstract

A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain well-understood hypotheses an isomorphism between continuous functions on points and supremum preserving functions on open sets, both with values in a fixed lattice. The functions on open sets provide a topological foundation for possibility theories in Artificial Intelligence, revealing formal analogies of quantitative predicates with continuous valuations. Three applications of this duality demonstrate its usefulness: we prove a universal property for the space of quantitative predicates, we characterize its inf-irreducible elements, and we show that bicontinuous lattices and Scott-continuous maps form a cartesian closed category.