Information systems revisited — the general continuous case

Abstract

a b s t r a c tIn this paper a new notion of continuous information system is introduced. It is shown that the information systems of this kind generate exactly the continuous domains. The new information systems are of the same logic-oriented style as the information systems first introduced by Scott in 1982: they consist of a set of tokens, a consistency predicate and an entailment relation satisfying a set of natural axioms.It is shown that continuous information systems are closely related to abstract bases. Indeed, both categories are equivalent. Since it is known that the categories of abstract bases and/or continuous domains are equivalent, it follows that the category of continuous information systems is also equivalent to that of continuous domains.In applications, mostly subclasses of continuous domains are considered. For example, the domains have to be pointed, algebraic, bounded-complete or FS. Conditions are presented that, when fulfilled by a continuous information system, force the generated domain to belong to the required subclass. In most cases the requirements are not only sufficient but also necessary.

Representing L-Domains as Information Systems

“…As has been shown in [16], continuous domains can be represented in a logicoriented way by information systems.…”

“…In [16] a Scott-style treatment of the general case was presented. Continuous information systems were introduced and the equivalence of the category of these information systems with approximable mappings with the category of abstract bases and approximable relations was derived.…”

“…In Section 3 continuous information systems are defined and important facts from [16] are recalled. Moreover, a terminal object and finite products are constructed.…”

“…L-information systems are introduced in Section 4. It is shown that under the relationship between informations systems and domains considered in [16], L-information systems and L-domains correspond to each other. In Section 5 approximable mappings are studied and the equivalence between the category of L-information systems and approximable mappings and the category of L-domains and Scott continuous functions is derived.…”

Partial information systems and the Smyth powerdomain

“…al [9] or Xu and Mao [11]). In particular, information systems have been used to provide compelling representations of complex objects (see for example Bedregal [2] or Droste and Göbel [3]).…”