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circum.abs[e84,jmc]	Abstract on Mathematical Problems of Circumscription

The abstract of my paper follows.  If your printing facilities includes
the letter lambda, please replace the written  lambda in the abstract.
Please count me in on the Shaw play expedition.

Mathematics of Circumscription

Abstract: Circumscription (McCarthy 1980, 1984) is a method of non-monotonic
reasoning proposed for use in artificial intelligence.  We take a certain
Let  A(P)  be a sentence expressing the facts "being taken into account",
where  P  stands for a "vector" of predicates regarded as variable.  Let
E(P,x) be a wff depending on a variable  x  and the  Ps.  The circumscription
of  E(P,x)  is a second order formula in  P  expressing the fact that
P  minimizes  lambda x.E(P,x)  subject to the facts  A(P).
The non-monotonicity arises, because augmenting  A(P)  sometimes
reduces the conclusions that can be drawn.  Circumscription raises
mathematical problems similar to those that arise in analysis in
that it involves minimization of a functional subject to constraints.
However, its logical setting doesn't seem to permit direct use of
techniques from analysis.  Here are some open questions that will be
treated in the lecture.

	1. What is the relation between minimal models and the theory
generated by the circumscription formula?

	2. When do minimal models exist?

	3. The circumscription formula is second order.  When is it
equivalent to a first order formula?

	4. There are several variants of circumscription including
successive circumscriptions and prioritized circumscription.  What
are the relations among these variants?

References:

McCarthy, John (1980): 
"Circumscription - A Form of Non-Monotonic Reasoning", Artificial
Intelligence, Volume 13, Numbers 1,2, April.

McCarthy, John (1984):
"Applications of Circumscription to Formalizing Common Sense Knowledge".
This paper is being given at the 1984 AAAI
conference on non-monotonic reasoning
and is being submitted for publication to Artificial Intelligence.