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 formula_1 satisfying Here formula_10 means formula_11 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: That is, the result can either be a natural number, represented by the singleton set  formula_15, or "infinite recursion," represented by formula_16. Of course, the same construction can be carried out with any other set instead of formula_17. Propositional calculus. The propositional calculus gives us a very simple Scott information system as follows: Scott domains. Let "D" be a Scott domain. Then we may define an information system as follows Let formula_25 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, formula_26, we can build a Scott domain as follows. Let formula_30 denote the set of points of  A with the subset ordering. formula_30 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.