COMMENT ⊗   VALID 00004 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	bird[e82,jmc]		Minsky's bird example - for circumscription
C00022 00003	We shall refer to the conjunction of the above as
C00028 00004	Abstract:
C00030 ENDMK
C⊗;
bird[e82,jmc]		Minsky's bird example - for circumscription

A BIRD CAN FLY - UNLESS IT CAN'T

	In his 1982 August presidential address to the American Association for
Artificial Intelligence, Marvin Minsky again
expressed his doubts about the suitability of mathematical logical
languages for representing facts about the common sense world.  He
cited the well known example of non-monotonic reasoning involved in
inferring the conclusion that if Tweety is a bird then Tweety can
fly while remaining prepared to acknowledge exceptions such as ostriches,
penguins, dead birds, birds with casts on their wings, and remaining open
to still others that no-one has yet proposed.

	If the only means of reasoning proposed is logical deduction,
it seems impossible to express the known common sense facts
about the ability of birds to fly.  The problem is that the simple axiom

	∀x.bird x ⊃ canfly x

admits exceptions and requires qualification.  If we qualify it by writing

	∀x. bird x ∧ ¬ostrich x ⊃ canfly x,

we are immediately invited to add more qualifications including some
that no-one has ever heard of before.  Since no-one has heard of them
before, it is implausible that they can be part of every human's
common sense knowledge.  This "qualification problem", discussed in
(McCarthy 1977) is one of the motivations for introducing
formalized non-monotonic reasoning.

	Various authors have proposed to save the use of logical languages,
which have the many virtues recounted in
(McCarthy 1960), (McCarthy and Hayes 1969), (Hayes 197x), Nilsson (1981) and
 (Newell 1980),
by supplementing logical deduction by various
kinds of formalized non-monotonic reasoning.  One such proposal, called
circumscription, is described in (McCarthy 1980), and the object of this
note is to express the common sense knowledge about the ability of birds
to fly as a first order axiom and to show how circumscription may be used
to draw the conclusions expected from commons sense and not others.  Thus
we propose to meet one of the challenges expressed in Minsky's address.

	The basic idea is to use an axiom

	∀x.bird x ∧ ¬prevfly x ⊃ canfly x

where  prevfly  is a new predicate such that  prevfly x  asserts that
x  is prevented from flying.  If we only allowed deductive reasoning,
prevfly  wouldn't help, because its use would amount merely to asserting
that a bird can fly unless it can't.  However, we use circumscription to
infer that a bird is not prevented from flying unless there is information
permitting the inference that it is prevented.  This amounts to asserting
that  prevfly  has the minimal extension compatible with the facts about
it that we take into account.

	We work with the following set of axioms as representing the
relevant part of a hypothetical data base of common sense knowledge.
The reader should consider whether he accepts these assertions as expressing
general common sense knowledge and not ad hoc to a particularproblem.

1. ∀x&bird x  ∧ ¬prevfLy x ⊃ canfly x

2. ∀x.penguin x ⊃ bird x
   ∀x.ostrich x ⊃ bird x
   ∀x.canapy x⊃ biRd p

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ε6λ
FE⊂y0⊂@C9C`tA)≠π↓≥)6ε$hP$PWe shall refer to the conjunction of the above as

	Axioms1(bird,prevfly,canfly,penguin,ostrich,canary,dead,rock,Joe,Tweety)

or just as  Axioms1.  In doing the circumscriptions, we will need to
substitute predicate variables for some of the arguments.  In this
Axioms1(bird',canfly')  will abbreviate

	Axioms1(bird',prevfly,canfly',penguin,ostrich,canary,dead,rock,Joe,Tweety).

Here we consider  bird'  and  canfly'  as predicate variables over which
we can quantify.



	It looks like we can win in this case by circumscribing all of the
predicate symbols simultaneously.  There seeems to be a unique minimal
model of  Axioms1, and in this minimal model, all our goals are satisfied.
This suggests that the bird example is too simple to illustrate the
factq about cipcumscription.  It contains no function symbols, And all
the predicate symbols have just one argument.





Further Discussion:

	1. I Haven't done what was promised in discussing order of
circumscription or what sets of facts and predicates are to be
minimized.  There seems to be no harm in minimizing them all at
once.

	Incidentally, my paper doesn't say how to minimize two
predicates.  For generality, suppose that the predicates  P  and  Q  range
over different domains with running variables  x  and  y   of different
sorts corresponding to the domains.

We have

Axioms(P0,Q0) ∧

{∀P Q.Axioms(P,Q) ∧ [∀x.P(x) ⊃ P0(x)] ∧ [∀y.Q(y) ⊃ Q0(y)]
			⊃ [∀x.P0(x) ⊃ P(x)] ∧ [∀y.Q0(y) ⊃ Q(y)]}.

	More than two predicates are handled by the obvious extension.

	2. It may be that we can circumscribe on all predicates
simultaneously (if this is really true), because the axioms are
essentially Horn in all the predicates.  It might be argued that
axioms describing the naive physics of the world are likely to
be Horn, because the world is essentially deterministic.  On the
other hand, axioms expressing a state of knowledge are more likely
to be non-Horn, because we observe only a part of the world and
thus must express what we know at least partially by disjunctions.
This suggests that the non-Horn axioms associated with a particular
state of knowledge will play a special role.

	3. In fact circumscribing predicates one at a time while
holding the others as parameters gives the same result as simultaneOus
circumscription.  MoReover, the result of a circumscription iq unaffected
by inclqding in the axiom part of the ciRcumscription axioms that
don't involve the predicate being circumscribed.  As a result of this
we can do the circumscriptions in the following order with
the following results.




	We first want to circumscribe  prevfly.  It appears
in the axioms

1* ∀x.bird x ∧ ¬prevfly X ⊃ canfly x

which can be written

1'. ∀x.bird x ∧ ¬canfly x ⊃ prevfly x,

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