# Catalogue of Artificial Intelligence Techniques

## 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 $\u27e8x,y,\mathrm{...},z\u27e9$ that can be shown to satisfy a relation$P(x,y,\mathrm{...},z)$

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 $P(\underset{\xaf}{\overset{}{x}})$
, where $\underset{\xaf}{\overset{}{x}}={x}_{1}^{},{x}_{2}^{},\dots ,{x}_{n}^{}$
, and that $A(\Phi )$
is the result of replacing all
occurrences of $P$
in $A$
by the predicate expression $\Phi $
. Then the
predicate *circumscription* of $P$
in $A(P)$
is the schema:

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:

- McCarthy, J.,
*Circumscription---a form of non-monotonic reasoning*Artificial Intelligence**13**(1980), 27--39.

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