In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains.

Definition

A Scott information system, A, is an ordered triple satisfying Here X \vdash Y means

Examples

Natural numbers

The return value of a partial recursive function, which either returns a natural number or goes into an infinite recursion, can be expressed as a simple Scott information system as follows: That is, the result can either be a natural number, represented by the singleton set {n}, or "infinite recursion," represented by \empty. Of course, the same construction can be carried out with any other set instead of \mathbb{N}.

Propositional calculus

The propositional calculus gives us a very simple Scott information system as follows:

Scott domains

Let D be a Scott domain. Then we may define an information system as follows Let \mathcal{I} be the mapping that takes us from a Scott domain, D, to the information system defined above.

Information systems and Scott domains

Given an information system,, we can build a Scott domain as follows. Let denote the set of points of A with the subset ordering. will be a countably based Scott domain when T is countable. In general, for any Scott domain D and information system A where the second congruence is given by approximable mappings.