probably infer that they both saw the same bear. If we let Yogi and Booboo be constants denoting the bears seen by the first child and second child, respectively,.
Circumscription Equality
PETER Stanford
with
Homomorphisms:
and Counterexample
Solving
the
Problems
K. RATHMANN Unitwsity,
MARIANNE
Stanford,
Calfomiu
WINSLETT
UniL’ers@’ of Illmols,
Urbana,
Illinols
AND MARK Digital
MANASSE Equipment
Cotpotution,
Palo Alto,
Callfomi(l
Abstract. One important facet of common-sense reasoning is the abdity to draw default conclusions about the state of the world, so that one can, for example, assume that a given bird fhcs in the absence of information to the contrary. A deficiency in the circumscriptive approach to common-sense reasoning has been its difficulties in producing default conclusions about equallty; for example, one cannot, in general, conclude by default that Tweety # Blutto using ordinary circumscription. or conclude by default that a particular bird flies, if some birds are known not to fly. In this paper, we introduce a new form of circumscription, based on homomorphisms between models, that remedies these two problems and still retains the major desirable properties of traditional forms of circumscription. Intelligence]: Deduction and Theorem ProvCategories and Subject Descriptors: 1.2.3 [Artificial 1.2.4 [Artificial Intelligence]: Knowledge Repreing—}~or~rrto~zofo~z~creasorztng and belief reuwon; logic; 1.2.0 [Artificial Intelligence]: General —plulosentation Formalisms and Methods—predicate sophical
foundation
General Additional
Terms: Theory Key Words
and Phrases: Circumscription,
common
sense reasoning
This work was supported by DARPA under grant N39-84-C-211 (KBMS Project, Gio Wiederhold, principal investigator) and by a Presidential Young Investigator award from the National Science Foundation (NSF IRI 89-58582, Marianne Winslett, principal investigator). A preliminary version of portions of this paper appeared as “Circumscribing Equality,” by Peter Joint Conference on Artificial Rathmann and Marianne Winslett, in ProceeditLgs of the International Zntelhgence (Detroit, 111.,Aug.). Morgan-Kaufmann, San Mateo, Calif., 1989, pp. 468–472. Authors’ addresses: P. K. Rathmann, Computer Science Department, Stanford University, Stanford, CA 94305; M. Winslett, Computer Science Department, University of Illinois, Urbana, IL 61801; M. Manasse, Systems Research Center, Digital Equipment Corporation, Palo Aho, CA 94301. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of tbe Association
for
Computing
Machinery.
specific permission. 01994 ACM 0004-541 1/94/0900-0819
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of the As>owitmn
for Computmg
Md’hmcry.
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or
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$03.50 Vd
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requires
a fee
and/or
820
P. K. Rf\TIIilfANN
ET AL.
1. Introduction Circumscription models that
is a means
of are
a logical true
introduce
in
all
a new
of reaching
theory
to
the
form
preferred of
default
others,
and
conclusions
accepting
models
of
circumscription
the
in
by preferring
as true theory.
which
the
those In
some
statements
this
paper,
preference
we
relation
models is based on homomorphisms instead of the subset inclusion tests among that characterize ordinary circumscription. This new form, called sfructum/ cim[nwcription, has several advantages over ordinary circumscription. These properties will most important
be explored here.
in the main
body
of the paper:
(1) Structural circumscription, unlike ordinary both equalities and inequalities as default tural circumscription Blutto. ” (2)
produce
circumscription, known
that
the
default
one can conclude
Structural
conclusion
“Tweety
exist,).
(i.e., For
the
gives
circumscription For example.
by default
that
naturally
to
Tweety
#
deduc-
are robust in with structural flies,
even if it is
do not fly.
circtunscription
assumptions do
produced by structural kinds of counterexamples.
some birds
the
circumscription, easily produces conclusions. For example, struc-
Structural circumscription, unlike ordina~ circumscription, tions about the equality of unnamed universe elements.
(3) Default conclusions the face of certain
(4)
can
we summarize
extends conclusion
that
if desired,
example,
only
those
include
domain
individuals
we can conclude
that
who
Tweety
closure must
exist
is the only
bird. (5)
circumscription sometimes prefers one model over another when Ordinary common sense dictates that the models are equally desirable. For example. isomorphic models are indistinguishable from the viewpoint of first-order logic, but ordinary circumscription may prefer one isomorphic model to
another. predicate
Structural extensions
sion by subset undesirable; isomorphic, Structural Both models
circumscription are infinite,
inclusion,
does gauging
as is done
not do this. In addition, the “size” of a predicate
in ordinary
it is all too easy for the “smaller” in some meaningful sense, to the circumscription
circumscription,
when exten-
is probably
predicate extension to be original “larger” extension.
does not use subsetting.
ordinary and structural circumscription have the that are “as small as possible”; the common-sense
goal of preferring motivation for this
preference is that we can write down expressions that describe properties that typically do /zot hold in the world, and prefer models in which the extension of those expressions is as small as possible. In this paper, we will examine the case where the expressions to be “minimized” correspond to a subset of the predicate symbols of the language, called minimized predicates. In other words, we prefer models whose extensions of the minimized predicates are as small as possible. Intuitively, preferred models are the simplest models, echoing Ockham’s injunction that “ . . . plurality is never to be posited without necessity.” 1 1John
McCarthy
[198S] has suggested that circumscription E, a modern attempt to codify As a methodological doctrine, the law of parsimony dates back at least to
Ockham’s
razor.
Aristotle’s
statement
They
should,
[Aristotle
in fact
(Stocks
“obviously be
1922):
as few Ariew
then
it would
as possible, 1977].
be better consistently
to assume with
a fimte
proving
number
what
has
of principles. to be
proved”
Circumscription More reflexive
with Homomorphisms
formally, under the circumscriptive and transitive binary relation) on
preferences
between
those that
are minimal
is true those
models.
under
Given
under
in all the preferred sat@ying
than
821
ordinary
a theory
the preorder.2
models.
T, more
Since
sentences
entailment;
paradigm, a preorder models is introduced T, the preferred A sentence
intuitively,
from these
models
of T are
is considered
the preferred
can follow
“ < “ (a to record true
iff it
models
are a subset
T under
circumscription
new
sentences
of
are default
conclusions. The preorders
traditionally
used in circumscription
sirable properties. Section 2 introduces scription, and shows how it overcomes ordinary structural
circumscription. circumscription.
semantics,
including
of circumscription theory result.
to assume
that
only those
individuals
5 gives an axiom for structural given in Section 3. One annoying
is that
they do not always
preserve
that
of the must
circumscription that property of all forms
consistency:
a consistent
may have no preferred models, with wild default conclusions as the Section 6 discusses the cases in which we can guarantee that structural
circumscription
will
our conclusions
appear
2. Introducing
based
preserve
consistency.
in Section
Structural
The preference
order,
and hence
and model,
sentences
in a logical
remaining arities
and then
always
symbols
be included
related
in
Set F contains that
work,
and
reasons
circumscription
define
the
between
is
concepts
models.
of
A signa-
of the terms
we can use to create
the available
function
of the language, for
first
are to be minimized,
and predicate P,
of structural We
of homomorphisms
symbols
of all the function
models.
is a description
theory.
the predicate predicate
7 discusses
the semantics,
between
Q = [F’, P, V, Arifies]
P contains
Section
8.
Circumscription
on homomorphisms
signature ture
of unde-
Section 3 proves some basic technical properties Section 4 gives several useful extensions of
the ability
exist do exist. Section matches the semantics
have a number
the preorder used in structural circumseveral of the difficulties encountered by
and the relation
symbols.
The
explained
equality
later.
symbols,
set V contains
We
Arities
lists the
predicate do
set the
not
must
consider
constants separately, choosing instead to treat them as functions that take no contains at arguments. For technical reasons,3 we assume that every signature least one constant symbol. We use the word theo~ to refer to a finite set of first-order only
sentences,
finite
not
closed
axiomatizations
of
under
theories).
logical
implication
Except
where
consider only theories over finite signatures. A model over a signature consists of a nonempty (often called symbols. The
interpretations) for all extension of a function
arity) on U, while arity) on U. Let M’ and M of
M’
and
M,
the extension be models respectively.
universe
of a theory
symbol
we consider noted,
we
U and extensions
the function and nonequality symbol is an actual function
of a predicate
Then,
(i.e., otherwise
is a relation
predicate (of proper (of proper
T, and let U’ and U be the universes
a homomorphism
from
M’
to
M
is a
2 See Bossu and Siegel [1985], Etherington [1988], Etherington et al. [1985], Lifschitz [1986], McCarthy [1980; 1986], Perlis [1987], and Shoham [1987]. 3 We invoke this restriction to simplify the statements of our results for universal theories; its removal has no other impact except for the statement of Theorem 5.
g~~
P. K. RATHMANN
function
h: U’ +
the following
U from
‘dal
h preserves with
(2) The truth predicate
that
=f~~(lz(al no
),. ... h(a,, )).
arguments,
this
condition
predicates is preserved: If P is a minimized n-ary T’s signature with extensions PAf, and P,~l in M’
. . . a,, G U’, no
will
are
PL1(al,..
condition
at all
said
sometimes we
be
convenient
can
write
is a homomorphism
we
wish
to
~ M to refer
refer
on
the
effect
The
),. ... h(a,, )).
of the
homomorphism
predicates
denoted
by
on these
to L!ary. to
/z(ii)
for
from to
a~) + PA~(h(al
symbols.
allowed
to be
.,
is put
predicate
Z, so that
If there M’
are functions that take constants are preserved.
nonminimized
symbols
Where
a,,))
.,
M, then
Note
It
of M, satisfying
functions: If ~ is an n-ary function symbol from f~r and f~ in M’ and M, respectively, we
h(f~(a,,..
of minimized symbol from
Val
vectors
to the universe
extensions
. . . a,, G U’,
Since constants guarantees that
the
of M’
conditions:
(1) The mapping T’s signature, require that
and
the universe
ET AL.
treat n-tuples such as al, . . . . a,, /z(al ), . . . . h( a,,), and so on.
model
a specific
to homomorphism
M’
to model
M, we write
homomorphism
g from
M’
by
name.
as
M’
+ M.
we
write
to M.
h maps from a source model A to a target model When a homomorphism B, predicate extensions may “grow”, in the sense that P(h(.V)) may be true in
Z3, even
though
P(i)
is false in xl. In addition,
since
h
might
not be one-to-one
or onto, distinct universe elements in A can be mapped to the same element in B. There may also be elements in B that are not mapped onto by any element in A. Usually, any of these effects—extending predicates, adding equalities, or adding new elements—means that B is more complicated than ,4, and hence less desirable
from
the viewpoint
we shall
consider
the
source
preferable Under
as its target model. structural circumscription,
of common-sense
model
of
reasoning.
a homomorphism
a model
M < M‘
(read
Consequently,
to be M
at least
as
is as pi-efewed
[is M’) iff M + M’. We say that M < M’ holds (read M is preferred to M’) M + M’ and M’ + M. The preferred models of theory T are those models
iff M
of T such that no model of T is preferred to M, that is, those that are minimal under s and B
k
A
flies
Ab
h
h k
[\
h “
h
c
D ~
of t a FIG, 1,
What
Four models of’ the Twecty
are the notable
of the four three
‘o
Ab
Ab
in Figure
theory,
characteristics 1? First,
and homomorphisms
of model
Tweety
is not
between
A, the most
abnormal
those
models,
preferred
in A, unlike
model
the other
models: +tb(
tweeh ).
Second, Tweetv and Blutto are two separate the unique name axiom A satisfies tweep
+ blLlttO
birds
in model
A. In other
words,
.
Third, in model A, Tweety is a flying bird, even though there is another bird in A that does not fly. To appreciate these characteristics of structural circumscription’s preferred model in Figure 1, let us examine the preference order of ordinary circumscription with the same four ordinary circumscription, interpretations, so model
models. For two they must have D is not
comparable
models to be comparable under the same universe and function to A,
B, or
C
under
ordinary
Circumscription circumscription. ble model
Under
ordinary
circumscription
&Z iff the extensions
those
extensions
proper
subset
models
B
of its extension C
preferred overall. Tweety and Blutto
under
M‘
of minimized
in J4, and some
and
and cannot
825
with Honlomorphisrns
minimized
in &f. This
ordinary
is preferred
predicates predicate’s
means
that
are subsets
extension
model
circumscription,
to a compara-
in M‘
in Jl’
of’ is a
A is preferred
so that
A
and
D
to are
With A and D both preferred, we cannot conclude that are distinct birds, cannot conclude that Twccty is normal,
conclude
that
Tweety
flies.
Under structural circumscription, model A, in which Tweety is a normal flying bird, is preferred among the models in Figure 1. However, we have yet to show that model A is preferred among all possible models of the theory, or that jZies(twee@ ) holds in all preferred models of the theory. To show that the general conclusions suggested by the above preferences do in fact hold, we must study the technical properties of the preference relation of structural circumscription
in more
3. Basic Properties
detail.
The
of Structural
next section
we explore
The Tweety only functions
theory example suggested and the truth of predicates,
1 formalizes
such an exploration.
Circurnscriptiotz
In this section,
tion
begins
the basic properties
of structural
that but
circumscription.
homomorphisms preserve not also simple formulas. Proposi-
this observation.
PROFIosrrloiN 1. Homomorphisms preserle ground terms and tllc troth of ally ground atomic formula constmcteci using equality or a nzinimi~edpreciic atesynlbol. PROOF.
We prove
the proposition
base case, for any constant preserve t. Now suppose that itself
a ground
term;
the
for ground
t and
terms
homomorphism
f( al,...,
depth
a,,, ) is a ground of nesting of function
least one level shallower than for f( al,..., can assume that h preserves each a,, that
using
a,. ). By the induction is, that
b[
, . . . . b,), ))
=
As the
}Z maps
f~a,~,~(h(bl)
h must
term, m > 0. Each a, is invocations in a, is at hypothesis,
the
of each a, in the source model for h to the interpretation model. Then, by the definition of a homomorphism,
~z(~,ouux(
induction.
h, by definition
. . . ..h(b..l
interpretation
of a, in the target
)).
The right-hand side of this equation is the interpretation of f( al,..., the target model. We conclude that homomorphisms preserve all ground Next, source
consider model
a ground
and where
atomic
formula
P is a minimized
P( al,. predicate
... a,,,), which symbol.
we b,
k true
a,,, ) in terms. in the
By the definition
of homomorphism (second condition) if P,OU,CC( b ~, ..., bn ) is true in the source model, then P ,~,~e,(h(bl), . . . . h(b,,, )) must also be true in the target model. As we have just seen, h must map the interpretation b, of each ground term a, in the source model to the interpretation of ai in the target model. Thus, P(al,.
... u,,, ) is true
in the
target
model,
because
Pt.,~Ct(h(b,
), ...,
h( b,,,)) is
true and for each i, h(bi) is the interpretation of at in the target model. Finally, consider the special case of a ground atomic formula over the equality predicate, namely a, = a., and assume this formula is true in the source model. Let b be the interpretation of a, in the source model. This means that b is also the interpretation of az in the source model, since a ~ = az is true in the source. Because homomorphisms preserve ground terms, the extension of al in the target model is Mb). target model is also Mb). Homomorphism
Similarly, the extension of ac in the h is a function, so formula a, = az
826
P. K. RATHM.ANN
is true
in
the
target
model.
Thus,
we
ground terms and truth of ground minimized predicate. ❑
conclude
atomic
that
over
equality
The proof of Proposition 1 shows that homomorphisms equalities because homomorphisms are functions on model even if equality were not listed in the signature as a predicate equalities would still be preserved: Structural circumscription minimizes the equality predicate. This is why we require that one of the predicates We showed that
O}
O=xrl
O=xo
S(z,) = Xz+l,vz >0
S(zt) = X,+l,vz >0 S(.zz)= Z,+l, V2 >0
S(zt) = ,?*+l,V2 >0 Predicate
Predicate
interpretations:
Blue
: zw,
ZIOO,
interpretations.
Blue
ZIOI,
Homomorphism
h from
h(zt)
= z,,
h(z,)
= ,z,+~,
Homomorphism g(z,
A to
: Zloo, ZIOI, ZIOZ,
B:
Vz >0 V7, >0
g from ) = z,,
II
to A:
Vt 20
g(z, ) = z,, QZ >0
—
->
– –>
blue FIG. 8.
blue Two
models
of the blue
integers
Example 5. A L’ai-iant of the blue integers problem. integers example by adding two additional constants, own successors.
Let
T contain
Vx.Bhe(x)
lBlue
Let us change a and b, which
the blue are their
the formulas
+ Bhe(S(x))
v.x3y.x
S(a)
problem.
= s(y)
=a
S(b)=b
( a)
Tlllue(
b)
3w.Blue(w) a =b
V ~y.(y
=a
Vy
=b
VBhle(y)).
Figure 9 shows some models of T. Model M contains a partly blue Z-chain, 11 and has a = b. M’ has a # b, and a completely blue Z-chain. Even though M does not satisfy the unique name axiom a # b, M and 11’ are both preferred. We explain why and then show how Theorem 2 applies to Al. First, T requires that its models contain nonblue interpretations of a and b that are their own successors. It also requires the existence of an additional element, the first
11A Z-chain [Enderton
w, such that blue element
is a chain 1972].
the successors on the Z-chain.
of elements
related
of w are all blue, For M. we let w be z~;~,
by the
successor
function,
infinite
in both
directions
P. K. RATHMANN
856 Model
Model
M:
Universe
= {yI,
z,, z,
Function
interpretations.
I 2 E -Z}
ET AL.
A[’:
Universe
= {y I,y2,
z,, ~,
Function
interpretations.
I Z E Z)
a=yl
a=yl
b=y,
b=yl S(Y1)
= yl
S(zt) = A+l, Predicate
vz
=
S(Y2)
= !/2
S(zt)= Zt+],vz Predicate
interpretations.
Blue
y,
IS(Y1)
Blue
: ,z,, VZ >99
lnterpretatlons: : z,, V%
/-
0“
blue
L.b
a,b
M
M’
D
a,b
blue
N FIG. 9.
Models
of a variant
of the blue
Let us see why w must lie on a Z-chain: Let a = b and w does not lie on a Z-chain. Then, after S’(z)
Integers
N be a model of T in which finitely many elements come
(“succeed”) w under the S function. Therefore, = z, for some z that follows w and some number
There
is a homomorphism
from
M
pmblcm
to N, mapping
S must i.
“loop”
in
N:
Zqq to w and mapping
SJ(zgg) to S](w). The jth predecessor of zg~ in M we can map to the Jth predecessor of w in a selected infinite chain of predecessors of WM.Therefore, S’(z) = z is true in N M ~ N. We cannot have N ~ M, because the equality but false for every choice of z on M’s Z-chain. We cannot map z to yl, either, because z follows w and hence is blue, but y, is not blue. Therefore, M is preferred to N. We conclude that in any preferred model, w must lie on a Z-chain. Since there is a homomorphism from M to every model in which w lies on s Z-chsin, we conclude that no model where D = b is o = b md preferred to M. Therefore, if M is not preferred, it must be because a model where o # b is preferred to M. Consider now those models where a + b is true. The arguments used above for M can be recycled to show that M’ (shown in Figure 9) is as preferred as any model where a # b. The question of whether M is a preferred model, then, reduces to the question of whether M’ is preferred to M. But M’ * M, because all of M”s Z-chain is blue, but only half of M’s, so there is no way to BILIE?. We conclude that M is a map M”s Z-chain to M while preserving preferred model. Because a # b is true in M’ and false in M, we have model of T. M * M’, so M’ is also a preferred
Circumscription Theorem that
A
with Homomo@isms 2 tells
us that
has a satisfying
there
857
is a set A of minimized
assignment
in
M
but
not
atomic
in M’.
Let
formulas A
such
contain
the
formulas Blue( w,)
Blue(
w, = S(W2) saying
that
Blue(w3)
W2 )
W3 = S(W4)
W2 = S(W3)
w, and all its predecessors
“.”
are blue.
Then,
““”,
A is not true
in M for
any choice of w,, because only part of the Z-chain in M is blue; but every finite subset of A is true in M, because one can choose WI to be a point on M‘s Z-chain that is arbitrarily far from the non-blue elements. 6.1
CONSISTENCY
preserves
AND UNIVERSAL
consistency
THEOREM
11.
Let
T is well founded
for universal
THEORIES.
T be a satisfiable
under
Structural
circumscription
theories.
structural
uniuer-sal
theoy
circumscription,
and
oler
signature
hence
Q. Then
T has prefewed
models. PROOF. By Theorem 10, it suffices to show well-foundedness for the case where Q has but a single priority level. Let M be a model of T. We show the existence
of a preferred
model
Let % be the set of models
M’
such that
satisfying
c2’={BIBk The
set X’ is partially
nonempty.
By
Zorn’s
ordered lemma,
M’
+ M.
T and possibly TAB+
has
to M, that
is,
M}.
(by the < relation)
12 3’
preferred
and, since
a minimal
member
preferred model) if every totally ordered subset has a minimal Let Y = Ml > M, > M3 .-. be such a totally ordered subset construct a model-which is minimal for 5’. Let W be the set of all negative ground literals
it contains (i.e.,
M,
contains
a
member in .7. of X. We now
1, over the equality
predicate
and minimized predicates, such that 1 holds in some model M, of Y. (Note that 1 also holds in all M, < M,; for if 1 were false in MJ, then because M, + M,, 1 would also be false in Ml.) holds, and for any finite subset W’
c
Let wit(l) be a model of Y where 1 W, let wit(W’) be the member of
{wit(l)ll G W’} that is smallest under
