From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved.
POINTWISE
CIRCUMSCRIPTION:
PRELIMINARY
Vladimir
Department
REPORT
Lifschitz
of Computer
Science
St an ford IIniversity
Stanford,
CA 94305
Abstract
Let us accept
valued
Circumscription
subject
We propose
instead
an
is the
to restrictions
a modified
of being
“infinite
tions;
notion
a single
formulas.
of circumscription
condition,
of “local”
conditions
of changing
the value of a predicate
We argucl that
this
it becomes
from
than
the traditional
time,
leads
with
atl&tionaJ.
flexibility
of commonsense
at
circumscrip-
simpler
“global”
to generalizations
needed
in applications
cm
expressed
be dcsc*ribcd
predical,e
other
the intuitions
behind
tliey suggest
dilfercsnt
i(.ittC"
Illigllt
If a
;uitl
predicate
satisfying
a given
derslantling
OII
These
li”.
with
two ap-
dill’erent,,
“minimizing
of
its
a predi-
equivalent;
are somewhat,
views on what
a set tlicn
predicates
it is natural
of a prctlicate
stronger
of making
makes
at other
as minimnlity
is a stronger
condition
without
to this order.
A predicate
if it cannot
tl~c corltlition.
Ieatls to the usual
This
be
un-
they
of them
be required
cm
rc~sc~nrch was partiall<y
406
/ SCIENCE
remain
to remain
definition
new
expresses,
icate
“at every
nite
conjunction”
same;
way;
fixed,
or we
or some
and the others
of circumscription
intuitively,
point”.
minimality
expresses
the impossibility
from true
conjunction”
‘Ynfinitc
by a universal
that
this
is in some
traditiortal
at one point.
will be represented
“pointwise”
approach
wa,ys conceptually
“global”
to
each
of changing
quantilier).
We argue
scription
as an “inf-
conditions;
to fulse
in
of a pred-
It can be interpreted
of “local”
(Formally,
this
proposed
the minimality
the value of a predicate
nccdccl
the
to vary.
The
leads
As far *IS
we can require,
in an arbitrary
approach
gc.licr;~lizat,iOIis
in applications
alid,
with
to circum-
simpler
at
than
t,he iLdtlitiOllid
to the theory
the
the* sa111e time,
Urxibility
of commonsense
yea-
soning.
Proofs
be published
of the
mathematical
facts
stated
below
will
in the full paper.
Basic
Cost
Let us start
one predicate
as parameters.
icate
constant
with the simplest
with all other
Let n(P)
P.
for
Circumscripl,ion
case of circumscribing
non-logical
constants
treated
b e a sentcncc
containing
a pred-
R.ccall that
A(P)
p < P stands
s~~pport~ctl b,y DAIU’A
of Point,wiso
of P in rl( 1’) is, by delinilion,
definii~ion of
-
uodrar (:or~t,~~;\(.t,NOO:l!)-,Y't-C-0'250.
are concerned,
to change
them
Here p is a predicate
This
points
case, that
allow
circurnscriplion.
. - -----
at a poht ( E II” as chang-
from true to false.
point
as
now we can also think
can
by
the mini-
predicate.
is minimal
violating
of nliniltla1it.y
relative
but
of the
“smaller”
a pred-
are orderc’d
to urltlcrstatid
sense;
in the simplest
but
and
of elements
IJnderstanding
smaller
at that
as Boolean-
as families of truth
words,
is a family
true}.
a predicate
ing its value
2. The
])rcYlici~l~cis
A smaller
made
a predicate
lncnn,
set inclusion,
mality
a k-nry
II is the universe
Inathem;~t,ically
them
in a model
is to represent
function
of coItrsc,
sub-
formulas.
syn~bol
identifies
possibility
by R 13oo1c;u1-vnlucd
are,
of predicates
One is I.0 represent
of I/“‘, where
‘I‘his approach
is logical
1986)
by predicate
in two ways.
The
proaches
1980,
of a predicate
by ;I srlbsct
extension.
cate
(McCarthy
iriterprct,ation
the model.
still
this paper
that, ;s, the minimization
to restrictions
The
predicate
{false,
“stronger”
to
reasoning.
or, in other
of these conditions
Circumscription
nhirnimtiion,
Each
set
allowed
1. Introduction
ject
values.
ordered
the values
true to f&e
“pointwise”
and, at the same
the
condi-
the impbssibility
approach
the theory
so that,
minimality
expresses
one point.
tion is conceptually
of predicates
by predicate
minimality
conjun(*tion”
each of these
minimization
expressed
now the view of predicates
functions,
the (glohl)
circumscription
the second-order
A Vpn( [l(P) A p < P).
variable
of the sntne arity
formrlla
(1)
as P, and
(Z is a tuple
Circum(
of object
The
Notice
that
by Cp(A(
cannot
A VxjPx
denote
by
(1)
this
P)).
is a first-order
A model
quantifier
mentioned
minimality
the
as a minimality
that
these
Several
Predicates
case is important,
cumscription
because
(McCarthy
To illustrate
1986)
both
Any
and b, is a model
x.
Cp(A)‘.
of cir-
predicates
in the axioms.
(I)
and (2), take
P identically
with
P) and Cp(A).
false
In addition,
at exactly
of the pointwise
special
applications
between
P is true
are posi-
This
version,
any
two points,
a
but not of the
one.
Even
when
applications
wc usually
gcncral
1ll0rc
to formalizing
common-
riecd forms of circuxiisc.ription
t,liati tliosc
defiiicd
in the previous
section.
Let us start
A(P, Z), where % is a tn-
with a formula
ple of predicate
and/or
function
constants.
circumscription
of P iI1 A( P, 2)
Tllc
(global)
with 2 dlowecf
A v
+x(P;x
minimization
different
is
which
the
predicates
show
and
that
circumscrib-
circumscribing
A, Circum(A;
Pi; Z).
of PI, . . . , P,, in A are positive,
also holds;
to
hence,
in this case,
/\; Cr. (A; Z).
whenever
one
is changed
in
arbitrary
an
counterpart
Let, 11s turn
considrr,
7’11c c.irciiInscript,il)ll
a higher
s1g11s
CircuIti(
by the sarnc
pret,ed
lexicogral,bic;l.lly;
circumscription
p,
of
structure
formula
is the
circumscription;
there
definition.
circumscription,
14; 1’1 >
to the task
formula
that,
PI ,“‘,
(3),
1’2; Z),
which
of minimizing
but
for details,
is equivalent
and
PI, Pz.
the case of two prcdicntcs
priority
defined
This
This
to prioritized
now
is equiv-
asserts
predicates
of A.
a special
for simplicity,
11; P; 2)
and the interpretation
of parallel
need to introduce
no
case when
way, the resulting
be a model
implies
the converse
conjunction
tr?Le to false,
possibly
special
Circum(
Tl lis
valuck of one of the
from
2 is changed
cannot
In the important
each
P; Z)
Circum(A;
all occurences
alent
with
below).
between
parallel
to
is
it from the
of P are minimized
is discussed
relationship
in
Pl , * - * , Cl
It is easy
Pi?
variables
3 p,x).
of several
members
priorities,
What
ing
is
in simple
knowledge
slightly
of joint
pointwise
3. A Generalization
of predicate
with
as
parallel circumscriptior~ (to distinguish
different
The
by (3),
i=l
form
case
the minimized
model
Circum(A;
of A in which
called
occurences
the difrerence
A t.o be Pu E Pb.
This
the
again
n
vx p;x 2 Pi,)
(
A
it asserts
of P in A(P)
for a tuple
i=l
conjunction”
P) implies
time
n
by
the quantifier
are equivalent.
this
circumscription
PI,. . . , P,,.
is given
and p < P understood
Pl,***,Pn,
which
of A(P)
of global
constants
P; 2)
of Circum(,l;
p standing
“infinite
in standard
have no negative
meaning
following
Circum(A;
all occurences
(2)
generalization
P is a tuple of predicate
at one point.
condition:
two formulas
(2)
of A(P)
model
of P at point
of the value
If we assume
tive then
satisfies
another
aud the formula
(3),
We denote
of (2) is a model
into
It is easy to check that
model
formula.
Vx represents
above,
can be viewed
srusc
4. Minimizing
In a further
is
A 2 # y))].
the value of P from true to false
changing
usually
of P in A(P)
A il(Xy(Py
be transformed
global
We
circumscription
pointwise
A(P)
The
variables).
A( P); P).
with
as-
1’1, is
p < P intcr-
see (l,ifschitz
1985).
to
to vary
is
A( I’, Z) A vpz- I( A(p, z) A p < P),
whcrc
t
is
variables
Circum(
cannot
il
of
t4plc
similar
to
%.
tl( P, Z); P; Z),
bc made smaller
intcrprebations
prcdicatc
This
asserts
the
Cuticbioti
dcnotcd
included
by
of P
extetlsion
even at the price of chauging
of the symbols
The corrcspotltling
formula,
that
(9
and/or
the
in 2.
wlic~tic~vc~r/‘, ,
cori.iuuc4ion
cumscription:
5. A Further
Our
variables
Z,
is cquivalcnt
positive.
(4)
is,
110
value
it1 /I. ‘I’his
of prioritize-d
cir-
of PI can be cl~i~~~g~dfrom
true
COUIlt?rpiWt
of changing
P2 arbitrarily.
Generalization
form of pointwise circumscri~~tion
A(P, 2) A VXZl[Pa: A A(Xy(Py
be denoted
0lll.y posil,ivcb occ1ircticcs
tlitV(’
to false even at the price
is
It will
I’2
is the pointwisct
by C,(
gCIlcsrillly,
A 2 # y), z)].
A(P, Z); Z).
Zk
second-order
to (3) iT all owllmiws
Because
f0~tIllll~.
(4)
of the
(4)
of I’ it1 /l( I’, Z) are
forms
next
goal
of poinlwisc
tivating
on the
some
more
We start
general
with a mo-
example.
Consider
which
is to introduce
circuliiscripliorl.
a l)lock
tnblc
n simple
catI
or
OIL
version
of the
blocks
world,
in
bc irl 0111,~ ou(’ of two i)IiL(*cs: cit,hcr
ttlc
floor.
KNOWLEDGE
We want
to tlcscribc
REPRESENTATION
the
/ 407
effect
of one
table.
This
stants,
particular
action,
can be done
using
ONTABLE
block
of bloclcs
before
B
which
and after
What
on the
predicate
and ONTABLEI,
the configurations
There
putting
two unary
con-
above
to the circumscription
new notation?
values of ONTABLE
represent
ues corresponding
the action.
constant
x > (ONTABLE
x f ONTABLE
x)
(5)
list
and then
must
ONTABLE
Here AB is the “abnorma1it.y”
axiom
expresses
what
they are.
This
tion for solving
under
axiom
McCarthy
normally,
formula
the frame
consideration:
The
calls
first
the
“com-
remain
where
(McCarthy
basic
1986).
property
The
of the action
in the new configuration
AB?
to be able
in 2,
the part
fixed,
be varied
The purpose
in the process
of our axiom
new configuration
of blocks;
AU with
a circumscription
set is to characterize
hence
ONT,4BLEl
(global
it is natural
allowed
or pointwise)
predicate
of minimizing
the
to cir-
to vary.
Such
gives
This is exactly
rameters
which
changes
pens
if it was not
can bc shown
sanic
on thr
that
expert;
and
table
prior
~i~clltnscl-iI)tion
to the
dots
or both
ONTABLE
fixed;
in difFerent
Let us change
in this
examplr
SitJJtltiOJl
CakJJhS
dit,ion
to
not Irad
the formal
to tlic
of (McCarthy
for I)locks,
supposed
‘l’lt(~tT
prcdicatc
to have a situation
III t,hc new notation,
1969).
used
111 ad-
ilI‘(’
tW0
Instcnd
a~14 ON’Z’A BLEI,
ON7’11BLE,
term as its second
which
argument.
(5) and (6) bccornc
-lAB z > (ONTABLl+,
St,) G ONTAU,!,~(x,
identically
rather
true
varyiltg
tflli111
get
the
then
P nor
a
Consider
a
which
2.
has no pa-
Intuitively,
of 2 on which
we propose
desired
rffect,
becomes
it
2 may
the following
f- S,,); ONTABLE
other
than
(3).
( I).
V
Making
In the cxnmplc
and WC cn~l
by taking
1s
’ allowed
(7)
2 as a parameter
Z is ON7’,4llLE,
for instance,
Xys(s
The
(7)
t 0 t,reating
it; (7) hcotnc5
scctiotl,
V to be
to vary in situations
S,,.
counterpart
of (7) for pointwisc
circumscription
is
is
We can allow
for x to affect
which
cvcn more
the
ditional
flexibility,
for more
of the
and %. Intuitively,
value
not nccdcd
domain
on
(This
ad-
at x.
is essential
Let T/’bc a X-expression
equals
the sulli of 1,hc arit,ics
tlic function
the set of all values
we will write
form of ~ir~utllscriI)t,ioll
it possible
of the
in this example,
1/ reprcscnts
of 2 into
by making
part
.P is minimized
c’olnplc>x applications.)
XZ?LV( T, 74 w I 10s~ arity
S,))
ilcxibility
choice
2 may vary when
T/(x, u); accordingly,
/ SCIENCE
= gx)).
one symbol,
z, 2) A A(p) z) A p < P].
Calsc is equivalent
fro111 the prrvious
IIC~
408
as 2,
circumscription,
IT I/ is idcnt,ically
every
and
of the same arity
> (fx
sikrta-
two sit8uations
f1 on the table.
ONTdl1Z;E,,
q, lr
.we write
(“p and q are equal
constant.
domain
to be
If p,
then
of only
arity
neither
of the
A(P, 2) A ‘++WV(
of the
wc ittl ro~ltt(‘~ n scc*ottd sort
S,, and S, , which rcprc~sctlt
predicates
language
formalism
and llaycs
for SilflitliOtts.
we have now one binary
and
ways.
by the act ion of placing
of two tmary
It
t,o the
ON?‘Al1LE,,
now slightly
vnriahlcs
tion constatlt,s,
treat
event.
/I I, I<, arc b0l.h
and move closer
Vllriill)lPS
Vilriil,l)lCS,
separated
we must
consists
its val-
values
useful.
arit,y
symbols
or a function
contains
the part
be
same
and
vary.
the oldy
is H, and tlkis only hap-
rcs~~lt. if ONT/1/1 /,I{,, alit1 ON’I’,jl
varied
of
its location
2
other
for Vx(lrx
V of the same
X-expression
specifies
stands
that
We
function
formula:
II).
what we wol~ltl intuitively
block
will
of the
> (pz ZE qz))
constant
For global
AB x z (x = B A lONTABL&
notation
EQ1.( f,g)
first
of minimization.
of its domain on which
7”‘). If f, g are function
as r then
of blocks,
it in
or function
for each
and allow the
symbols
EQ,,(p, 4) for VO(TX
outside
include
predicate
to specify,
remain
following
(4),
it in 2, and then all of its values
predicate
The
of that
in the ‘process
ues must
Assume
should
cumscribe
like
would
B is on the table.
What
fixed
are predicate
the use of circumscrip-
problem
the
will be cir-
(6).
objects
exemplifies
expresses
which
of (5) and
John
law of inertia”:
(6)
predicate
in the conjunction
monsense
second
B.
remain
(3),
or function
we have to either
all values
the val-
Definitions
for any predicate
in the language,
2,
described
like to vary the
s) for s = S1, and have
to s = So fixed.
may vary, or not include
and
cumscribed
We would
do not allow us to do that;
are two axioms:
TAB
corresponds
in this
is
whicli
of P
nlaps
of u satisfying
Vx for AuV(x,
u), The
We will denote
V is identically
this formula
by Cp(A(P,
true then Cp(A;
If V is identically
Z/V)
Cp (A; Z/V)
false then
Z/V).
2);
If
Cp(A;
2).
is equivalent
to
becomes
set along with a circumscriptiorl policy, a metamathematical statement
vary,
scription
CP(A).
tion
6. More
some values
possibility:
mized
predicate
We start
schema
how to perform
of 2 allowed
esting
to vary.
we may
proposed
policies
which
priorities
with
circumscription
There
vary some
is another
values
inter-
of the mini-
As another
(Hanks
here allows
example,
of these
ai
the case
when
2 is empty.
The
#
V is a X-expression
(8)
expresses
the
values
value
of P
A.
than
Cp(A)
the basic
signing
into
at different
arbitrarily
then
change
loosing
its
the prop-
circumscript,ion
V is identically
it when
shows
to the
Applying
the
tasks
the
basic
forms
the
P
of global
(2)
to
Pa V Pb gives
Using
5 x = a) v Vx(Px
the form
of pointwise
in this
section,
we can
higher
priority
to the
circumscription
5 2 = b).
circumscription
express
task
the
idea
kind.
Applying
Vx(Px
and
To this end, introduce
second
a binary
P at b; this
formula
shows
language
of pointwise
at b is given
this
case,
An interesting
matsion 011 priorities
which
is iucludtd
circumsrriptive
P may
reasoning
“minimization
priority.
at point
expresses
that
It is rasy
a
in the
minimizing
P
to see that,
in
(9).
feature
tant
but
is rcprcscntc>tl
example
by the axiom
iu the datnl~asc~ along
Ihc>or,y is
tISlIiLI1~
is that
with
tl1ollgIlt,
inforV(b,n),
l’u V 1’6.
ask
can
of some
(2) to A(P)
(1))
kind
the
instants
temporal
apparently
what
capture
the second
ysis shows that
reasoning
non-
of preferring
of time”,
disjunctive
cannot
of formal
idea
which
term.
would
Their
anal-
of this kind is impor-
be captured
by the existing
formalisms.
problem
Extend
is clearly
the theory
The additional
similar
to the one discussed
by thcsc
“policy”
axioms:
(0 5 i < j 5 3).
axioms
points
“later
“gives
prefcrcnce
satisfies
than
about
A
of as an axiom
method
moving
is
a11
One
expressing
that,
1 leacls
cxccp
tion.
by side on the
also
using
to
Imagine
Vi’C
tlic
11ow
(see
like
two
(Mc-
to use
the effect
moving
case
a diffireasoning
the
of
is “111~ law of mo-
in which
iL1 t.Clll[)l
KNOWLEDGE
resolve
to tlescribc
norn~nlly,
can
true.
to formalize
of the axioms
a2 -f a3
mottcl
circumscription
calculus
us that.
table.
used
WC would
to a situation
tells
with
a “bc11,er”
attempts
12).
Sit~lliLtiOIi
a block.
axiom
world
If a model
tllcll
at the
at x; in this way, it
Pa2 false and Pa3
be
cm1
Section
of
location
past”.
of (IO)
in recent
198G),
formalism
minimized
by making
thcl blocks
Carthy
other
tcrui
uncovered
P may be varied
tell us that
z”when
to tlic
I,llc> lirst
This
culty
tion”
of this
S as the
G (x = a1 v x = a3)].
at earlier
leacl to selecting
V, and
b, a). The
V(
be varied
condition
circumscription
a higher
(8) implies
that
at b. This
when it is minimized
constant
and
of time,
“abnormality”
E (x = al v 2 = a;z)]
v(ai,aj)
result
prcdicatc
P as an
and
a
(9)
of Pa V Pb
be tl le conjunction
TPao.
x) > Py,
McDermott
monotonic
introduced
zi x = a).
= a3 A y = a&
any of circumscriptions
be constructed
let A(P)
in
be
gives
a b ove.
of assigning
of minimizing
will lead to the stronger
posed
Let A(P)
3),
of a0 , . . . , a3 as instances
relation,
The
Vx(Px
7.1.
= ao)V (x = a2 Ay = a)
A s(y,
v Vx[Px
of as-
of minimizing
circumscription
We can think
successor
Hanks
use
effect
the problem
Section
(0 < i < j 5
aj
Vx[Px
false.
how we can
to create
or pointwise
(1)
of
form of pointwise
example
points.
and
term
It is, generally,
priorities
circumscription
to change
without
of circumscription
different
has no pa-
second
by Cp(A;P/V).
(8)
following
form
The
: V(X, y)},
on {y
and turns
The
new
y) which
P.
it is impossible
We denote
stronger
contain
true to false,
at x from
erty
hyV(z,
not
that
circumscrip-
metamathematical
axioms:
G [(x = qAy
~Px
does
than
to
of circum-
new
S(x,y)
and
are allowed
form
us to describe
1985),
V(x
Here
The
consider
and McDermott
the conjunction
is
rameters
predicates
are.
axioms rather
by
P itself.
with
the
expressions.
on Priorities
Now we know
describing
and what
a block
when
blocks
t0
x to a
2 is at 2.
pl;L’.C
Rn-
x is riot clear
A
and
It
011
REPRESENTATION
R side
top
Of
/ +O’-)
B and then
to move
B
second
action
action
B is not clear.
will be “normal”
second
will be
after
the
first
Unfortunately,
the
the first
1.
action
and the
circumscrib-
the possibility
remains
clear,
result.
In this
of the
blocks
and the second
alternative
, and the
first
that
unchanged,
action
model,
is not.
to a minimal
leads
the
Each
value
the first
2. There
more
of the
B
action
is
two models
AB; circumscription
of
than
prcdicat
The
erence
“bad”
will be given
ideas
in the full paper.
of the previous
definition
section,
above.
This
wise circumscription
The
solution
which
most
symbol
be a 0-ary
stant).
We want
and Sz,...,
this
is a
all the forms
definition
of point-
section.
mathematical
function
say, Si.
S,, correspond
The pointwise
where
symbol,
to minimize
list,
We hope
relatively
should
each
S; is
(in particular,
i.e.,
an object
con-
one of the predicate
(Thus
sym-
5’1 corresponds
to 2 in the notation
circumscription
to P
used be-
of 5’1 in A with S;
A
The
present
cases
determining
Here S stauds
for ,!?I,. . . , S,,
icate
and function
icate
atld
predicate
frrllctiott
without
Sl , * * * , s,, and
arity Of Si.
We denote
variables
~~~lIlStiLlltS
parameters
whose
(11)
arity
A x
Y),
Z
32,.
s is a list 31,.
V;
(;
which
is the
to the
=
1,.
does
arity
not
of SI
pred-
. . ,72)
is a
contain
plus
the
by
Cs,(A; Sl/Vl,. . . ,S,/V,,).
If K
is identically
true then we will drop /Vi in this notation.
If V, is idrutically
false Illen we can drop the term Si/Vi
altogether.
8.
point wise
following
atlvautnges
410
/ SCIENCE
to circumscription
over the traditiounl
globill
hits
the
approach.
proposed
for formalizing
of commonsense
of pointwise
and error,
to extend
principles
regard-
policies
(such
a higher
priority
when
Fu-
circumscription
of general
of time
many
reasoning.
as, for into min-
in temporal
the
policy
is clearly
reason-
is selected
unsatisfactory.
the existing
of circumscription
methods
for
to more general
in applications.
to Michael
Gelfond,
Nils Nilssotr,
for useful
Benjamin
Raymond
Grosof,
Reitcr
and Yoav
discussions.
References
S.
and
nrrn Ucthult,
#/430,
Yale
I,ifwltil.x,
1985,
McDermott,
Log&,
011 Arlificial
I’rcw.
$111 [II-
11~telligezlce
1,
121-127.
J., Circumscription
lY’iLSOJlillg,
Artificial
M&artlly,
- a form of non-monotonic
hlk~fipJtce
*J., A pplicntiotts
(1980),
13
27-39.
to formal-
of ~irc.tttnscription
lmowlrtlgc~,
~‘OlIllJJOltS~‘J1S~’
J3. iLtt(l
brlrgll
Reasoning
YALEU/CSD/RR
c,irc.rlttts~riotiott,
C:ohfcrellcc
McCarthy,
ixirtg
Temporal
Report
(I 985).
(:otttl)lll,ittg
Joirlt
D.,
Tccbuical
1Tuivcrsity
V.,
t~~~llili.iOll~ll
ilrtiliichl Iu.kl!igcnce
28
89-118.
McCartlty,
i1pprOdl
to
of circumscription
situation,
J. and Hayes,
from 1,ltv st,;m(lpoittt
The
point
and actions.
of assigning
the res.llt
from
useful in appli-
Acknowledgements
(198(i),
Conclusion
of meta-
time
instants
by trial
needed
Hanks,
, . , s,, of pred-
corrcspouding
S;
. .>I.
about
of circumscription
at earlier
Sltoharu
(11)
@Y(QY
forms
I aui grateful
i=l
the selection
may vary
powerful
on applications
.Joh u McCarthy,
A(S) A V’ZS~[S~ZA A EQV,,(Si, Si)
the form
It is also important
to vary on I< is, by definition,
&owed
that
the principle
forms
minimized
instead
flexibility
reasoning
lead to the discovery
in many
including
policies
is sufficiently
complex
work
stance,
symbol
more “modular”:
of the
by axioms,
additional
to formalizing
in this paper
ing).
become
definitions.
Circumscription
cations
ture
of
for each
policies,
which provides
imization
or a function
policies
is defined
can be described
ing the choice
Case
it can
from
we need
covers
general
pref-
Details
uses also the
what
A(S1, . . . , S,) b e a sentence,
Let
fore).
The
is given in the next
General
a predicate
bols
by “giving
example.
so that
of circumscription
introduced
7.
can be eliminated
as in the previous
policy
Circumscription
5.
model
to the past”,
the circumscription
es.
4.
point,
is
formula.
is no need to define
of priorities,
only gives a disjunction.
circumscription
of pointwise
one predicate.
a separate
to the normal
second
case
by a first-order
3. Circumscription
action
so that
basic
The
expressed
does not allow us to prove this assertion.
the positions
corresponds
Intuitively,
to the law of motion,
“abnormal”.
It does not eliminate
“normal”
else.
because
In ot(her words,
relative
ing abnormality
leaves
somewhere
will be unsuccessful,
Midtie,
1 Iliiversity
I’., Some
of :~rt,ilic*i;d
I>. (bhls.),
l’ress,
M
il(’
philosophical
it~ttlligc~ttcc,
I Jill<’
I~tliub~~rgli,
IJJt(‘l/igX’JlCt’
I%!)),
problems
itt:
Meltzer,
4
463502.
(l~tiill-