Note: [] Address for correspondence: Theoretical Computer Science,
Jagiellonian University, ul. Gronostajowa 3, 30-387 Kraków,
Poland
Abstract: This paper is about a generalization of Scott's domain
theory in such a way that its definitions and theorems become meaningful in
quasimetric spaces. The generalization is achieved by a change of logic: the
fundamental concepts of original domain theory (order, way-below relation,
Scott-open sets, continuous maps, etc.) are interpreted as predicates that are
valued in an arbitrary completely distributive Girard quantale (a CDG
quantale). Girard quantales are known to provide a sound and complete semantics
for commutative linear logic, and complete distributivity adds a notion of
approximation to our setup. Consequently, in this paper we speak about domain
theory based on commutative linear logic with some additional reasoning
principles following from approximation between truth values. Concretely, we: (1) show how to define continuous Q-domains, i.e.
continuous domains over a CDG quantale Q; (2) study their way-below relation,
and (3) study the rounded ideal completion of Q-abstract bases. As a case
study, we (4) demonstrate that the domain-theoretic construction of the Hoare,
Smyth and Plotkin powerdomains of a continuous dcpo can be straightforwardly
adapted to yield corresponding constructions for continuous Q-domains.
