Generalised Information Systems Capture L-Domains

Abstract

A generalisation of Scott's information systems \cite{sco82} 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, and vice versa; similarly for Hoofman's continuous information systems \cite{ho93} and the continuous bounded-complete domains captured by them; for Chen and Jung's disjunctive propositional logic \cite{cj06} and algebraic L-domains (as well as for Wang and Li's \cite{wll20} finitary version and Lawson-compact algebraic L-domains); and for Wang and Li's conjunctive sequent calculi \cite{wlbc20} and proper continuous bounded-complete domains. The proofs always contain syntactic translations between the logical calculi involved.