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.<<circum.abs[f83,jmc]	Circumscription abstract for Parikh at Brooklyn College>>

.require "memo.pub[let,jmc]" source;
.cb APPLICATIONS OF CIRCUMSCRIPTION TO ARTIFICIAL INTELLIGENCE

Abstract: My "Circumscription - A Method of Non-monotonic
Reasoning" %2Artificial Intelligence%1, April 1980, introduces
a "rule of conjecture" based on minimizing the extension of
a predicate while holding an axiom true.  The present lecture
extends the idea to minimizing a wff while holding an axiom
true, which gives a more symmetric theory.  We will discuss
some of the mathematical properties of circumscription
and some applications to artificial intelligence, particularly
to default reasoning and to the frame problem.

	If  ⊗P  is a vector %2(P1, . . . ,P%4n%*)%1 of predicate
symbols, not necessarily having the same numbers of arguments,
⊗x is a vector %2(x%41%*,_._._._,x%4m%*)%1 of individual variables,
⊗A(P) is the axiom satisfied by the %2P%1s, and  %2E(P,x)%1 is
a wff, then the circumscription of ⊗E(P,x) relative to ⊗A(P) is
the sentence

	%2A(P) ∧ ∀P'.A(P') ∧ (∀x.E(P',x) ⊃ E(P,x)) ⊃ (∀x.E(P,x) ≡ E(P',x))%1.

It essentially says that  ⊗P  is chosen to satisfy ⊗A(P) and
to minimize the set of values of ⊗x for which  ⊗E(P,x) is true.

	In many cases of AI interest this second order formula is
equivalent to a first order formula.  Of special interest also are
the cases where ⊗P is uniquely determined by the circumscription.

	Some, but not all, of the applications to AI
use axiom sets in which objects that are not "abnormal" in certain
aspects are asserted to have certain properties.  A key requirement
is that an object can be abnormal in some aspects while remaining
normal in others.

.begin verbatim
John McCarthy
Computer Science Department
Stanford University
Stanford, California 94305
.end