Domains, information frames, and their logic

In [10], continuous information frames were introduced that capture exactly all continuous domains. They are obtained from the information frames considered in [9] by omitting the conservativity requirement. Information frames generalise Scott’s information systems [8]: Instead of the global consistency predicate, there is now a local consistency predicate for each token. Strong information frames are obtained by strengthening the conditions for these predicates. Let $\mathrm{CIF}$ and $\mathrm{SIF}$ be the corresponding categories.

In [11] another generalisation of Scott’s information systems was introduced which also exactly captures all continuous domains. As shown in [6], the definition can be simplified while maintaining the representation result. Let $\mathrm{CIS}$ and $\mathrm{SCIS}$ be the corresponding categories. It is shown that all these categories are equivalent. Moreover, the equivalence extends to the subcategories of (strong) continuous information frames with truth elements. Such information frames capture exactly all pointed continuous domains.

Continuous information frames are families of rudimentary logics, associated with each token is a local consistency predicate and an entailment relation. However, they lack the expressive power of propositional logic. In an attempt to make each of this logics more expressible, continuous stratified conjunctive logics are introduced. These are families of conjunctive logics. The category $\mathrm{CSL}$ of such logics is shown to be isomorphic to ${\mathrm{SIF}}_t$, the category of strong continuous information frames with a truth element.