An alternate way of introducing formula circumscription is by means of an ordering on tuples of predicates satisfying an axiom. We define by

That *P*0 is a relative minimum in this ordering is expressed by

where equality is interpreted extensionally, i.e. we have

Assuming that we look for a minimum among predicates *P*
satisfying *A*(*P*), (46) expands to precisely to the
circumscription formula (1). In some earlier expositions of
circumscription this ordering approach was used, and Vladimir Lifschitz in
a recent seminar advocated returning to it as a more fundamental and
understandable concept.

I'm beginning to think he's right about it being more understandable, and there seems to be a more fundamental reason for using it. Namely, certain common sense axiomatizations are easier to formalize if we use a new kind of ordering, and circumscription based on this kind of ordering doesn't seem to reduce to ordinary formula circumscription.

We call it *prioritized circumscription.*

Suppose we write some bird axioms in the form

The intent is clear. The goal is that being a bird and not
abnormal in *aspect*2 prevents the application of (49). However,
circumscribing with the conjunction of (49) and
(50) as *A*(*ab*) doesn't have this effect, because (50)
is equivalent to

and there is no indication that one would prefer to have abnormal rather than to have abnormal. Circumscription then results in a disjunction which is not wanted in this case. The need to avoid this disjunction is why the axioms in section 5 (page ) included cancellation of inheritance axioms.

However, by using a new kind of ordering we can leave (49) and (50) as is, and still get the desired effect.

We define two orderings on *ab* predicates, namely

We then combine these orderings lexicographically giving priority over getting

Choosing *ab*0 so as to minimize this ordering yields the result
that exactly birds can fly. However, if we add

we'll get that ostriches (whether or not ostriches are birds) don't fly without further axioms. If we use

instead of (55), we'll have to revise our
notion of ordering to put minimizing at higher priority
than minimizing and at higher priority than
minimizing *aspect*1.

This suggests providing a partial ordering on aspects giving their
priorities and providing axioms that permit deducing the ordering on *ab*
from the sentences that describe the ordering relations.
Lifschitz (1985) further develops the idea of prioritized circumscription.

I expect that will turn out to be the most natural and powerful variant.

Simple abnormality theories seem to be inadequate also for the blocks world described in section 11. I am indebted to Lifschitz for the following example. Consider

where *S*0 is a situation with exactly blocks *A* and *B* on the table.
Intuitively, the second action is unsuccessful,
because after the first action *A* is on *B*, and so *B* isn't
clear. Suppose we provide by a suitable axiom that when the block
to be moved is not clear or the destination place is not clear,
then the situation is normally unchanged. Then *S*2 should be the same
situation as *S*1 = *result*(*move*(*A*,*B*),*S*0). However, simple circumscription
of *ab* won't give this result, because the first move is only
normally successful, and if the first move is unsuccessful for
some unspecified reason, the second move may succeed after all.
Therefore, circumscription of *ab* only gives a disjunction.

Clearly the priorities need to be arranged to avoid this kind of unintended ``sneak disjunction''. The best way to do it by imposing priorities isn't clear at the time of this writing.

Sat Jun 1 13:54:22 PDT 1996