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
editA Scott information system, A, is an ordered triple
satisfying
Here means
Examples
editNatural numbers
editThe 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
editThe propositional calculus gives us a very simple Scott information system as follows:
Scott domains
editLet 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
editGiven 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.
See also
editReferences
edit- Glynn Winskel: "The Formal Semantics of Programming Languages: An Introduction", MIT Press, 1993 (chapter 12)