Generalised Information Systems Capture L-Domains

Abstract

A generalisation of Scott's information systems [14] is presented that captures exactly all L-domains. The global consistency predicate in Scott's definition is relativised in such a way that there is a consistency predicate for each atomic proposition (token) saying which finite sets of such statements express information that is consistent with the given statement.It is shown that the states of such generalised information systems form an L-domain, and that each L-domain can be generated in this way, up to isomorphism. Moreover, the equivalence of the category of generalised information systems with the category of L-domains is derived. In addition, it will be seen that from every generalised information system capturing an algebraic bounded-complete domain a corresponding Scott information system can be obtained in an easy and natural way; similarly for Hoofman's continuous information systems [9] and the continuous bounded-complete domains captured by them.

Information Systems with Witnesses: The Function Space Construction

“…Note that we sometimes write X ∈ CON(i) instead of (i, X) ∈ CON. Proofs of the results can be found in [13].…”

“…In [13] it was shown how approximable mappings between two information systems with witnesses A and A ′ as well as Scott continuous functions from |A| to |A ′ | correspond to each other. As we will show next, this correspondence establishes an isomorphism between the domains…”

“…In a recent paper [13], the author introduced information systems with witnesses as a logicstyle representation of L-domains. L-domains have independently been introduced by Coquand [5] and Jung [10,11] and are known to form one of the two maximal Cartesian closed full subcategories of the continuous domains.…”

“…This allows to consider the more general situation in which a finite set of tokens may have different consistency witnesses, and the result of entailment may depend on them. As was shown in [13], the theories, or states, of such a more general information system form an L-domain, and, up to isomorphism, each L-domain can be obtained in this way. Moreover, there is an equivalence between the categories of information systems with witnesses and L-domains.The category of information systems has approximable mappings as morphisms.…”

“…Moreover, as is known from Hoofman's work [9], in the continuous case entailment is required to allow for interpolation.The paper is organized as follows: Section 2 contains basic definitions and results from domain theory. In Section 3 relevant definitions and facts about information systems with witnesses are recalled from [13]. Approximable mappings between such information systems are considered in Section 4.…”