Suppose we have a data base of facts axiomatized by a formalism involving the predicate ab . In connection with a particular problem, a program takes a subcollection of these facts together with the specific facts of the problem and then circumscribes . However, many common sense domains are axiomatizable in such a way that the circumscription is equivalent to a first order formula. In this case we call the circumscription collapsible. This can presumably be extended to other cases in which the ranges and domains of the functions are disjoint, so that there is no way of generating an infinity of elements.
Circumscription is also collapsible when the predicates are all monadic and there are no functions.
We can then regard the process of deciding what facts to take into account and then circumscribing as a process of compiling from a slightly higher level nonmonotonic language into mathematical logic, especially first order logic. We can also regard natural language as higher level than logic. However, as I shall discuss elsewhere, natural language doesn't have an independent reasoning process, because most natural language inferences involve suppressed premisses which are not represented in natural language in the minds of the people doing the reasoning. In simple cases they give the same results. However, a computer program using default logic would have to establish the existence of models, perhaps by constructing them, in order to determine that the sentences mentioned in default rules were consistent.