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 ${\displaystyle (T,Con,\vdash )}$

• ${\displaystyle T{\mbox{ is a set of tokens (the basic units of information)}}}$
• ${\displaystyle Con\subseteq {\mathcal {P}}_{f}(T){\mbox{ the finite subsets of }}T}$
• ${\displaystyle {\vdash }\subseteq (Con\setminus \lbrace \emptyset \rbrace )\times T}$

satisfying

1. ${\displaystyle {\mbox{If }}a\in X\in Con{\mbox{ then }}X\vdash a}$
2. ${\displaystyle {\mbox{If }}X\vdash Y{\mbox{ and }}Y\vdash a{\mbox{, then }}X\vdash a}$
3. ${\displaystyle {\mbox{If }}X\vdash a{\mbox{ then }}X\cup \{a\}\in Con}$
4. ${\displaystyle \forall a\in T:\{a\}\in Con}$
5. ${\displaystyle {\mbox{If }}X\in Con{\mbox{ and }}X^{\prime }\,\subseteq X{\mbox{ then }}X^{\prime }\in Con.}$

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

Examples

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:

• ${\displaystyle T:=\mathbb {N} }$
• ${\displaystyle Con:=\{\emptyset \}\cup \{\{n\}\mid n\in \mathbb {N} \}}$
• ${\displaystyle X\vdash a\iff a\in X.}$

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

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

Propositional calculus

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

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

Scott domains

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

• ${\displaystyle T:=D^{0}}$ the set of compact elements of ${\displaystyle D}$
• ${\displaystyle Con:=\{X\in {\mathcal {P}}_{f}(T)\mid X{\mbox{ has an upper bound}}\}}$
• ${\displaystyle X\vdash d\iff d\sqsubseteq \bigsqcup X.}$

Let ${\displaystyle {\mathcal {I}}}$ 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, ${\displaystyle A=(T,Con,\vdash )}$, we can build a Scott domain as follows.

• Definition: ${\displaystyle x\subseteq T}$ is a point if and only if
  • ${\displaystyle {\mbox{If }}X\subseteq _{f}x{\mbox{ then }}X\in Con}$
  • ${\displaystyle {\mbox{If }}X\vdash a{\mbox{ and }}X\subseteq _{f}x{\mbox{ then }}a\in x.}$

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

• ${\displaystyle {\mathcal {D}}({\mathcal {I}}(D))\cong D}$
• ${\displaystyle {\mathcal {I}}({\mathcal {D}}(A))\cong A}$

where the second congruence is given by approximable mappings.

See also

• Scott domain
• Domain theory

References

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