Scott information system
Encyclopedia
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 domain
s.
satisfying
Here
means 
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
.
gives us a very simple Scott information system as follows:
. Then we may define an information system as follows
Let
be the mapping that takes us from a Scott domain, D, to the information system defined above.
, 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.
Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, a Scott information system is a primitive kind of logical deductive system
Deductive system
A deductive system consists of the axioms and rules of inference that can be used to derive the theorems of the system....
often used as an alternative way of presenting Scott domain
Scott domain
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete cpo. It has been named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory...
s.
Definition
A Scott information system, A, is an ordered triple
satisfying
Here
means 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 calculusPropositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
gives us a very simple Scott information system as follows:
Scott domains
Let D be a Scott domainScott domain
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete cpo. It has been named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory...
. Then we may define an information system as follows
-
the set of compact elementCompact elementIn the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
s 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
Given an information system,, we can build a Scott domainScott domain
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded complete cpo. It has been named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory...
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.