On Domain Theory over Girard Quantales

摘要: This paper is about a generalization of Scott's domain theory in such way that its definitions and theorems become meaningful quasimetric spaces. The achieved by change logic: the fundamental concepts original (order, way-below relation, Scott-open sets, continuous maps, etc.) are interpreted as predicates valued an arbitrary completely distributive Girard quantale (a CDG quantale). quantales known to provide sound complete semantics for commutative linear logic, distributivity adds notion approximation our setup. Consequently, this we speak based on logic with some additional reasoning principles following from between truth values. Concretely, we: (1) show how define Q-domains, i.e. domains over Q; (2) study their (3) rounded ideal completion Q-abstract bases. As case study, (4) demonstrate domain-theoretic construction Hoare, Smyth Plotkin powerdomains dcpo can be straightforwardly adapted yield corresponding constructions Q-domains.