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Catalogue of Artificial Intelligence Techniques

Catalogue of Artificial Intelligence Techniques

   

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Circumscription

Keywords: Domain circumscription, minimal inference, minimal models, predicate circumscription

Categories: Inference and Reasoning


Author(s): Helen Lowe

Circumscription captures the idea of `jumping to conclusions', namely that the objects that can be shown to have a certain property are all the objects that satisfy that property. For example, in the missionaries and cannibals problem, we are told that three missionaries and three cannibals want to cross the river in a two-person boat such that the cannibals never outnumber the missionaries in the boat or on either bank. We assume that there are no more cannibals around; the three mentioned in the problem are all there are. More generally, we conjecture that the tuples x,y,...,z that can be shown to satisfy a relationP(x,y,...,z)

are all the tuples satisfying this relation. Thus we circumscribe the set of all relevant tuples. Circumscription is a formalised rule of conjecture. Domain circumscription (also known as minimal inference) conjectures that the known entities are all there are. Predicate circumscription assumes that entities satisfy a given predicate only if they have to on the basis of a collection of known facts; since this collection can be added to subsequently. Circumscription together with first order logic allows a form of Non-monotonic Reasoning. Suppose A(P) is a sentence of first order logic containing a predicate symbol P(x¯) , where x¯=x1,x2,,xn , and that A(Φ) is the result of replacing all occurrences of P in A by the predicate expression Φ . Then the predicate circumscription of P in A(P) is the schema:

A(Φ)x¯(Φ(x¯)P(x¯))x¯.(P(x¯)Φ(x¯))

The sentences that follow from the predicate circumscription of a theory are those which are true in all the minimal models of that theory.


References:


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