Scott information system

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 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, or "infinite recursion," represented by.
Of course, the same construction can be carried out with any other set instead of.

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

the set of compact elements of

Let 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.

Definition: is a point if and only if
*
*

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.