Scott information system

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
A Scott information system, A, is an ordered triple $$(T, Con, \vdash) $$ satisfying
 * $$T \mbox{ is a set of tokens (the basic units of information)} $$
 * $$Con \subseteq \mathcal{P}_f(T) \mbox{ the finite subsets of } T$$
 * $${\vdash} \subseteq (Con \setminus \lbrace \emptyset \rbrace)\times T$$
 * 1) $$\mbox{If } a \in X \in Con\mbox{ then }X \vdash a$$
 * 2) $$\mbox{If } X \vdash Y \mbox{ and }Y \vdash a \mbox{, then }X \vdash a$$
 * 3) $$\mbox{If }X \vdash a \mbox{ then } X \cup \{ a \} \in Con$$
 * 4) $$\forall a \in T : \{ a\} \in Con$$
 * 5) $$\mbox{If }X \in Con \mbox{ and } X^\prime\, \subseteq X \mbox{ then }X^\prime \in Con.$$

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

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:
 * $$T := \mathbb{N}$$
 * $$Con := \{ \empty \} \cup \{ \{ n \} \mid n \in \mathbb{N} \}$$
 * $$X \vdash a\iff a \in X.$$

That is, the result can either be a natural number, represented by the singleton set $$\{n\}$$, or "infinite recursion," represented by $$\empty$$.

Of course, the same construction can be carried out with any other set instead of $$\mathbb{N}$$.

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


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

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

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

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


 * Definition: $$x \subseteq T$$ is a point if and only if
 * $$\mbox{If }X \subseteq_f x \mbox{ then } X \in Con$$
 * $$\mbox{If }X \vdash a \mbox{ and } X \subseteq_f x \mbox{ then } a \in x.$$

Let $$\mathcal{D}(A)$$ denote the set of points of A with the subset ordering. $$\mathcal{D}(A)$$ will be a 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.
 * $$\mathcal{D}(\mathcal{I}(D)) \cong D$$
 * $$\mathcal{I}(\mathcal{D}(A)) \cong A$$