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.

1 Definition
2 Examples
- 2.1 Propositional calculus
- 2.2 Scott domains
3 Information systems and Scott domains
4 See also

[edit] Definition

A Scott information system, A, is an ordered triple $(T, Con, \vdash)$

$T is a set of tokens (the basic units of information)$
$Con \subseteq \mathcal{P}_f(T) \mbox{ the finite subsets of T}$
$\vdash \subseteq Con\times T$

satisfying

$\mbox{If } a \in X \in Con\mbox{ then }X \vdash a$
$\mbox{If } X \vdash Y \mbox{ and }Y \vdash a \mbox{, then }X \vdash a$
$\mbox{If }X \vdash a \mbox{ then } X \cup a \in Con$
$\forall a \in T : \{ a\} \in Con$
$\mbox{If }X \in Con \mbox{ and} X^\prime\, \subseteq X \mbox{ then }X^\prime \in Con.$

Here $X \vdash Y$ means $\forall a \in Y, X \vdash a.$

[edit] Examples

[edit] Propositional calculus

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

$T := \{ \phi \mid \phi \mbox{ is satisfiable} \}$
$Con := \{ X \in \mathcal{P}_f(T) \mid X \mbox{ is consistent} \}$
$X \vdash a\mbox{ iff }X \vdash a \mbox{ in the propositional calculus}.$

[edit] Scott domains

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
$Con := \{ X \in \mathcal{P}_f(T) \mid X \mbox{ has an upperbound} \}$
$X \vdash d\mbox{ iff }d \sqsubseteq \bigsqcup X.$

Let $\mathcal{I}$ be the mapping that takes us from a Scott domain, D, to the information system defined above.

[edit] Information systems and Scott domains

Given an information system, $A = (T, Con, \vdash)$ , we can build a Scott domain as follows.

Definition: is a point iff
- $\mbox{If }X \subseteq_f x \mbox{ then } X \in Con$
- $\mbox{If }X \vdash a \mbox{ and } X \subseteq_f x \mbox{ then } a \in x.$

Let $\mathcal{D}(A)$ denote the set of points of A with the subset ordering. $\mathcal{D}(A)$ will be a countable Scott domain when T is countable. In general, for any Scott domain D and information system A