Scott information system

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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[edit]

A Scott information system, A, is an ordered triple

satisfying

Here means

Examples[edit]

Natural numbers[edit]

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[edit]

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

Scott domains[edit]

Let D be a Scott domain. Then we may define an information system as follows

  • the set of compact elements of D

Let be the mapping that takes us from a Scott domain, D, to the information system defined above.

Information systems and Scott domains[edit]

Given an information system, , we can build a Scott domain as follows.

  • Definition: is a point iff

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[edit]

References[edit]

  • Glynn Winskell: "The Formal Semantics of Programming Languages: An Introduction", MIT Press, 1993 (chapter 12)