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.
A Scott information system, A, is an ordered triple
(
T
,
C
o
n
,
⊢
)
T
is a set of tokens (the basic units of information)
C
o
n
⊆
P
f
(
T
)
the finite subsets of T
⊢⊆
(
C
o
n
∖
{
∅
}
)
×
T
satisfying

If
a
∈
X
∈
C
o
n
then
X
⊢
a

If
X
⊢
Y
and
Y
⊢
a
, then
X
⊢
a

If
X
⊢
a
then
X
∪
{
a
}
∈
C
o
n

∀
a
∈
T
:
{
a
}
∈
C
o
n

If
X
∈
C
o
n
and
X
′
⊆
X
then
X
′
∈
C
o
n
.
Here
X
⊢
Y
means
∀
a
∈
Y
,
X
⊢
a
.
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:
T
:=
N
C
o
n
:=
{
∅
}
∪
{
{
n
}
∣
n
∈
N
}
X
⊢
a
iff
a
∈
X
.
That is, the result can either be a natural number, represented by the singleton set
{
n
}
, or "infinite recursion," represented by
∅
.
Of course, the same construction can be carried out with any other set instead of
N
.
The propositional calculus gives us a very simple Scott information system as follows:
T
:=
{
ϕ
∣
ϕ
is satisfiable
}
C
o
n
:=
{
X
∈
P
f
(
T
)
∣
X
is consistent
}
X
⊢
a
iff
X
⊢
a
in the propositional calculus
.
Let D be a Scott domain. Then we may define an information system as follows
T
:=
D
0
the set of compact elements of D
C
o
n
:=
{
X
∈
P
f
(
T
)
∣
X
has an upper bound
}
X
⊢
d
iff
d
⊑
⨆
X
.
Let
I
be the mapping that takes us from a Scott domain, D, to the information system defined above.
Given an information system,
A
=
(
T
,
C
o
n
,
⊢
)
, we can build a Scott domain as follows.
Definition:
x
⊆
T
is a point iff
If
X
⊆
f
x
then
X
∈
C
o
n
If
X
⊢
a
and
X
⊆
f
x
then
a
∈
x
.
Let
D
(
A
)
denote the set of points of A with the subset ordering.
D
(
A
)
will be a countably based Scott domain when T is countable. In general, for any Scott domain D and information system A
D
(
I
(
D
)
)
≅
D
I
(
D
(
A
)
)
≅
A
where the second congruence is given by approximable mappings.