Circumscription with homomorphisms: solving the equality and counterexample problems

Circumscription with Homomorphisms: Solving the

Equality and Counterexample Problems

PETER K. RATHMANN

Stanford Unitwsity, Stanford, Calfomiu

MARIANNE WINSLETT

UniL’ers@’ of Illmols, Urbana, Illinols

AND

MARK MANASSE

Digital Equipment Cotpotution, Palo Alto, Callfomi(l

Abstract. One important facet of common-sense reasoning is the abdity to draw default conclu-

sions about the state of the world, so that one can, for example, assume that a given bird fhcs inthe 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 ordinarycircumscription. or conclude by default that a particular bird flies, if some birds are known not tofly. In this paper, we introduce a new form of circumscription, based on homomorphisms betweenmodels, that remedies these two problems and still retains the major desirable properties of

traditional forms of circumscription.

Categories and Subject Descriptors: 1.2.3 [Artificial Intelligence]: Deduction and Theorem Prov-

ing—}~or~rrto~zofo~z~creasorztng and belief reuwon; 1.2.4 [Artificial Intelligence]: Knowledge Repre-

sentation Formalisms and Methods—predicate logic; 1.2.0 [Artificial Intelligence]: General —plulo-sophical foundation

General Terms: Theory

Additional 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 ScienceFoundation (NSF IRI 89-58582, Marianne Winslett, principal investigator).

A preliminary version of portions of this paper appeared as “Circumscribing Equality,” by PeterRathmann and Marianne Winslett, in ProceeditLgs of the International Joint Conference on Artificial

Zntelhgence (Detroit, 111.,Aug.). Morgan-Kaufmann, San Mateo, Calif., 1989, pp. 468–472.

Authors’ addresses: P. K. Rathmann, Computer Science Department, Stanford University, Stan-

ford, CA 94305; M. Winslett, Computer Science Department, University of Illinois, Urbana, IL61801; M. Manasse, Systems Research Center, Digital Equipment Corporation, Palo Aho, CA94301.

Permission to copy without fee all or part of this material is granted provided that the copies arenot made or distributed for direct commercial advantage, the ACM copyright notice and the titleof the publication and its date appear, and notice is given that copying is by permission of tbeAssociation for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or

specific permission.01994 ACM 0004-541 1/94/0900-0819 $03.50

JOLIIIKd of the As>owitmn for Computmg Md’hmcry. Vd 41, No 5, September 1[)94. pp 81 Y-X73

820 P. K. Rf\TIIilfANN ET AL.

1. Introduction

Circumscription is a means of reaching default conclusions by preferring some

models of a logical theory to others, and accepting as true those statements

that are true in all the preferred models of the theory. In this paper, we

introduce a new form of circumscription in which the preference relation

among models is based on homomorphisms instead of the subset inclusion tests

that characterize ordinary circumscription. This new form, called sfructum/

cim[nwcription, has several advantages over ordinary circumscription. These

properties will be explored in the main body of the paper: we summarize the

most important here.

(1) Structural circumscription, unlike ordinary circumscription, easily produces

both equalities and inequalities as default conclusions. For example, struc-

tural circumscription can produce the default conclusion “Tweety #

Blutto. ”

(2) Structural circumscription, unlike ordina~ circumscription, gives deduc-

tions about the equality of unnamed universe elements.

(3) Default conclusions produced by structural circumscription are robust inthe face of certain kinds of counterexamples. For example. with structural

circumscription, one can conclude by default that Tweety flies, even if it is

known that some birds do not fly.

(4) Structural circtunscription extends naturally to include domain closure

assumptions (i.e., the conclusion that only those individuals who must exist

do exist,). For example, if desired, we can conclude that Tweety is the only

bird.

(5) Ordinary circumscription sometimes prefers one model over another whencommon 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. Structural circumscription does not do this. In addition, when

predicate extensions are infinite, gauging the “size” of a predicate exten-

sion by subset inclusion, as is done in ordinary circumscription, is probably

undesirable; it is all too easy for the “smaller” predicate extension to be

isomorphic, in some meaningful sense, to the original “larger” extension.

Structural circumscription does not use subsetting.

Both ordinary and structural circumscription have the goal of preferring

models that are “as small as possible”; the common-sense 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 casewhere 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 Ock-

ham’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

Ockham’s razor. As a methodological doctrine, the law of parsimony dates back at least to

Aristotle’s statement “obviously then it would be better to assume a fimte number of principles.

They should, in fact be as few as possible, consistently with proving what has to be proved”

[Aristotle (Stocks 1922): Ariew 1977].

Circumscription with Homomorphisms 821

More formally, under the circumscriptive paradigm, a preorder “ < “ (a

reflexive and transitive binary relation) on models is introduced to record

preferences between models. Given a theory T, the preferred models of T are

those that are minimal under the preorder.2 A sentence is considered true iff it

is true in all the preferred models. Since the preferred models are a subset of

those sat@ying T, more sentences can follow from T under circumscription

than under ordinary entailment; intuitively, these new sentences are default

conclusions.

The preorders traditionally used in circumscription have a number of unde-

sirable properties. Section 2 introduces the preorder used in structural circum-

scription, and shows how it overcomes several of the difficulties encountered by

ordinary circumscription. Section 3 proves some basic technical properties of

structural circumscription. Section 4 gives several useful extensions of the

semantics, including the ability to assume that only those individuals that must

exist do exist. Section 5 gives an axiom for structural circumscription that

matches the semantics given in Section 3. One annoying property of all forms

of circumscription is that they do not always preserve consistency: a consistent

theory may have no preferred models, with wild default conclusions as the

result. Section 6 discusses the cases in which we can guarantee that structural

circumscription will preserve consistency. Section 7 discusses related work, and

our conclusions appear in Section 8.

2. Introducing Structural Circumscription

The preference order, and hence the semantics, of structural circumscription is

based on homomorphisms between models. We first define the concepts of

signature and model, and then of homomorphisms between models. A signa-

ture Q = [F’, P, V, Arifies] is a description of the terms we can use to create

sentences in a logical theory. Set F contains the available function symbols, set

P contains the predicate symbols that are to be minimized, set V contains the

remaining predicate symbols of the language, and the relation Arities lists the

arities of all the function and predicate symbols. The equality predicate must

always be included in P, for reasons explained later. We do not consider

constants separately, choosing instead to treat them as functions that take no

arguments. For technical reasons,3 we assume that every signature contains at

least one constant symbol. We use the word theo~ to refer to a finite set of

first-order sentences, not closed under logical implication (i.e., we consider

only finite axiomatizations of theories). Except where otherwise noted, we

consider only theories over finite signatures.

A model over a signature consists of a nonempty universe U and extensions

(often called interpretations) for all the function and nonequality predicatesymbols. The extension of a function symbol is an actual function (of proper

arity) on U, while the extension of a predicate symbol is a relation (of proper

arity) on U.

Let M’ and M be models of a theory T, and let U’ and U be the universes

of M’ and M, respectively. Then, 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; itsremoval has no other impact except for the statement of Theorem 5.

g~~ P. K. RATHMANN ET AL.

function h: U’ + U from the universe of M’ to the universe of M, satisfying

the following conditions:

(1) The mapping h preserves functions: If ~ is an n-ary function symbol fromT’s signature, with extensions f~r and f~ in M’ and M, respectively, we

require that

‘dal . . . a,, G U’, h(f~(a,,.. ., a,,)) =f~~(lz(al ),. ... h(a,, )).

Since constants are functions that take no arguments, this condition

guarantees that constants are preserved.

(2) The truth of minimized predicates is preserved: If P is a minimized n-arypredicate symbol from T’s signature with extensions PAf, and P,~l in M’

and M, then

Val . . . a,, G U’, PL1(al,.. ., a~) + PA~(h(al ),. ... h(a,, )).

Note that no condition at all is put on the effect of the homomorphism on

the nonminimized predicate symbols. The predicates denoted by these

symbols are said to be allowed to L!ary.

It will sometimes be convenient to treat n-tuples such as al, . . . . a,, as

vectors Z, so that we can write /z(ii) for /z(al ), . . . . h( a,,), and so on.

If there is a homomorphism from model M’ to model M, we write M’ + M.

Where we wish to refer to a specific homomorphism by name. we write

M’ ~ M to refer to homomorphism g from M’ to M.

When a homomorphism h maps from a source model A to a target model

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 of common-sense reasoning. Consequently,

we shall consider the source model of a homomorphism to be at least as

preferable as its target model.

Under structural circumscription, a model M < M‘ (read M is as pi-efewed

[is M’) iff M + M’. We say that M < M’ holds (read M is preferred to M’) iffM + M’ and M’ + M. The preferred models of theory T are those models M

of T such that no model of T is preferred to M, that is, those that are minimal

under s and <. If M is preferred, then all models isomorphic to M are also

preferred, because there are homomorphisms in both directions between them;

so structural circumscription does not share ordinary circumscription’s distress-ing tendency to prefer one isomorphic model over another. The preference

relation s is a preorder, because M s M (reflexivity) and because the

composition of two homomorphisms is itself a homomorphism (transitivity).

Note that A s B A B s A does not imply that A and B are equal. They could

differ, for example, in the extensions of predicates which are allowed to vary.

The above definitions of homomorphism and the preference order among

models are rather dry, and use what may be for many readers an unfamiliar

notation. Let us now see how the definitions work, by examining how structural

circumscription answers the most familiar question in nonmonotonic reason-

ing: Can Tweety fly?

Circumscription with Homomo~hisms 823

Example 1. The Tweety Theoy. Let us build a theory encoding information

about birds. First, we need some constants (0-ary functions): tweety, of course,

and, say, blutto. We need unary predicates bird and flies, saying whether their

respective arguments are birds and can fly. These predicates give our theory its

first sentences,

bird( tweety )

and

bird( blutto ),

saying that both Blutto and Tweety are birds; and

Y1 ies( blutto ),

saying that Blutto is not a flyer. We define one additional unary predicate, Ab,

the truth of which indicates that its argument is abnormal in some way. Since

we expect abnormality not to happen, the Ab predicate is minimized, while

bird and jZies are allowed to vary. Finally, we connect abnormality, flight, and

birdhood together with the sentence

Vx.(biz-d(x) A +lb(x)) +flies(x).

Figure 1 lists four models of the Tweety theory, A through D. In A, l?, and

C, the interpretation of Tweety is t and of Blutto is b, and there are no other

universe elements. In model D, both Tweety and Blutto are interpreted as the

element tb, and there are no other universe elements. In other words, tweety =

blutto holds in D but not in the other models. As for predicate interpretations,

Tweety is a bird and Blutto an abnormal nonflying bird in all four models. In

model A, Tweety is a normal flyer; in B, an abnormal flyer; in C and D, an

abnormal nonflyer.

Under the preference order imposed on the models of Figure 1 by structural

circumscription, models A, B, and C are all preferred to model D, because

each has a homomorphism to D (listed in Figure 1), with no homomorphism in

the reverse direction. In addition, model A, where Tweety is a normal flyer, is

preferred to models B and C.

Consider the homomorphism h from model C to D. /z maps both t and b of

C to element tb of D. Thus defined, /z preserves all functions, since it maps

the interpretations of &vee@ and blutto in C to the interpretations of those

same terms in D. Also, h preserves the truth of Ab, the only minimized

predicate. There cannot be any homomorphism g from model D to C, because

in order to preserve all functions, g would have to map the single universe

element tb that is the interpretation of tweety and blutto in model D to the

two separate universe elements t and b that are the interpretations of tweety

and biutto in model C.Also, consider the homomorphism k from A to C. It maps the interpreta-

tion of each term in A to the interpretation of the same term in C, thuspreserving functions. The truth of Ab is also preserved, since b is abnormal in

both models, and t is only abnormal in model C. There cannot be a homomor-

phism in the reverse direction, since in order to preserve tweety it would have

to map t in C to t in A, and then Ab(t ) in model C would not be preserved.

824 P. K. RATHMANN ET AL.

Models of the Tweety Theory

r

Model A

Univers-e {t, b} ;,b} j,b} :b}

Interpretation of functions: tweety {t} {t} {t} {t/l}

blutto

I

{b} {/)} {b} {tb}

Interpretation of predicates: bird {t, b} {t, b} {t, b} {tb}

ffies {t} {t} {} {}

Ab {b} {t, b} {t)b} {tb}

Homomorphism h from A, B, or C to ~

~(t) = tb

h(b) = tb

Homomorphism k from .4 to B or C:

k(t) = t

k(b) = b

00O?

> c!!e?ttflies

flies Ab A k B Ab

[\

h h

k

h h “c D

a

of‘o

~

t

AbAb

FIG, 1, Four models of’ the Twecty theory, and homomorphisms between those models,

What are the notable characteristics of model A, the most preferred model

of the four in Figure 1? First, Tweety is not abnormal in A, unlike the other

three models:

+tb( tweeh ).

Second, Tweetv and Blutto are two separate birds in model A. In other words,A satisfies the unique name axiom

tweep + blLlttO .

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 circumscrip-

tion with the same four models. For two models to be comparable under

ordinary circumscription, they must have the same universe and function

interpretations, so model D is not comparable to A, B, or C under ordinary

Circumscription with Honlomorphisrns 825

circumscription. Under ordinary circumscription M‘ is preferred to a compara-

ble model &Z iff the extensions of minimized predicates in M‘ are subsets of’

those extensions in J4, and some minimized predicate’s extension in Jl’ is a

proper subset of its extension in &f. This means that model A is preferred to

models B and C under ordinary circumscription, so that A and D are

preferred overall. With A and D both preferred, we cannot conclude that

Tweety and Blutto are distinct birds, cannot conclude that Twccty is normal,

and cannot 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 detail. The next section begins such an exploration.

3. Basic Properties of Structural Circurnscriptiotz

In this section, we explore the basic properties of structural circumscription.

The Tweety theory example suggested that homomorphisms preserve not

only functions and the truth of predicates, but also simple formulas. Proposi-

tion 1 formalizes 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 for ground terms using induction. As the

base case, for any constant t and homomorphism h, by definition h must

preserve t. Now suppose that f( al,..., a,,, ) is a ground term, m > 0. Each a, is

itself a ground term; the depth of nesting of function invocations in a, is at

least one level shallower than for f( al,..., a,. ). By the induction hypothesis, we

can assume that h preserves each a,, that is, that }Z maps the interpretation b,

of each a, in the source model for h to the interpretation of a, in the target

model. Then, by the definition of a homomorphism,

~z(~,ouux( b[ , . . . . b,), )) = f~a,~,~(h(bl) . . . ..h(b..l )).

The right-hand side of this equation is the interpretation of f( al,..., a,,, ) in

the target model. We conclude that homomorphisms preserve all ground terms.

Next, consider a ground atomic formula P( al,. ... a,,,), which k true in the

source model and where P is a minimized predicate symbol. By the definition

of homomorphism (second condition) if P,OU,CC(b ~, ..., bn ) is true in the source

,~,~e,(h(bl), . . . .model, then P h(b,,, )) must also be true in the target model. Aswe 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 thesource 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). Similarly, the extension of ac in the

target model is also Mb). Homomorphism h is a function, so formula a, = az

826 P. K. RATHM.ANN ET Al..

is true in the target model. Thus, we conclude that homomorphisms preserve

ground terms and truth of ground atomic formulas over equality and over any

minimized predicate. ❑

The proof of Proposition 1 shows that homomorphisms preserve ground

equalities because homomorphisms are functions on model universes. Thus,

even if equality were not listed in the signature as a predicate to be minimized,

equalities would still be preserved: Structural circumscription “automatically”

minimizes the equality predicate. This is why we require that “ = “ be listed as

one of the predicates to be minimized.

We showed earlier that < is a transitive relation. It is instructive to note

that < also is transitive: assume A, B, and C are models, A < B, and B < C.

This means that A ~ B and B S C, while B + A and C + B. Certainly

A ‘:~ C, by composition of homomorphisms, but is it possible that there is an h11~ ,?

such that C ~ A? If such an h exists, than B + A, contradicting the assump-

tion that A < B. We conclude that no such h can exist, and the transitivity

property is verified.

Example 1 suggests that structural circumscription can produce unique name

axioms; that is, default conclusions stating that distinct terms denote distinct

universe elements. This corresponds to the common-sense reasoning principle

that different names usually denote different objects. Example 1 also suggested

that the extensions of minimized predicates in preferred models are minimal,

in a certain sense, when restricted to ground terms. In the remainder of this

section, we first formalize these two intuitions, by stating guarantees about the

minimization of predicate extensions under structural circumscription. Theo-

rem 1 considers universal theories, and Theorem 2 covers the general case.

After the presentation of the two theorems, we discuss how the guarantees of

minimality given by structural circumscription differ from those given by

ordinary circumscription, by introducing the concept of a ‘-ghost” universe

element and discussing how structural circumscription handles unnamed uni-

verse elements.

Theorem 1 states that if a ground atomic formula over a minimized predicate

is true in a preferred model, then making it false must force the extension of

some minimized predicate to grow. For brevity in what follows, we say rni}zi-

mized atomic formula to refer to an atomic formula over a minimized predicate

symbol.

THEOREM 1. Let T be a uniLersal theo~, and let A be a preferred model of T

where the minimized ground atomic formula a Ilolds. Then jor a~ly model N of Twhere a does not ilold, some minimized ground atotnic formula that is f[dse in A is

true in N.

PROOF. Let &l be a structure formed by restricting the universe of N tocontain only the extensions of ground terms. Since the restriction of a model of

4 A unnerwl theory is a finite set of sentences, each of which can lx put into prenex form with

a possibly empty sequence of universally quantified variables, followed by a quantifier-free for-

mula. Existential theories are defined similarly, except that the quantifiers must be emstential.

Unwersal-existential sentences in prenex form have a possibly empty sequence of unwersally

quantified variables. followed by a possibly empty sequence of existentially quantified variables,

followed by a quantifier-free formula; and so on.


a universal theory to ground term extensions also satisfies the theory (when the

signature contains at least one constant), iii? K T.

Since a is false in N, a is also false in M. By Proposition 1, A -+ M,

because a is true in A but not in M. If there were a homomorphism M + xl,

itwould mean that M < A, by the definition of <. However, by hypothesis, A

is a preferred model and there can be no model that is preferred to A. Hence,

by contradiction, M + A.

Now we shall consider the consequences of M + A. For convenience of

notation, let UJ~ be the universe of M and let 111~be the universe of A. Also,

let us define a relation h, which relates the extensions of ground terms in M to

the corresponding ground-term extensions in model A. This h is not a

homomorphism, and therefore must fail one of the criteria for a mapping to be

a homomorphism:

(1) A function from Ul[ to U~, which(~) presemes functions, and

(3) Preserves minimized predicates.

We shall look at each condition in turn, and see that a failure in any one of

these conditions implies that a ground atomic formula true in M is not true

in A.

First, what if h fails to be a function? It is a relation, and since M’s universe

consists only of ground-term extensions, h covers its domain. The only way

remaining for h to fail to be a function is for h to map an element of U}, onto

more than one element of Ud. This can only happen if two ground terms t, and

t2 designate the same element of U*l but designate two different elements of

CJq. In this case, the ground atomic formula t, = tz is true in M, but not in A.

When h is a function, the second condition cannot fail, because of the way h

is defined. Because h maps ground-term extensions to the corresponding

ground-term extensions, it must preserve functions on those elements. Since

M’s universe consists only of ground-term extensions, functions are preserved

on all model elements.

Now, when h is a function, consider the third condition, that h preserve

minimized predicates. If this is violated, then there is a tuple of universe

elements (xl, . . ., .x.) and a minimized predicate P such that P(.Y1, ..., x,, ) is

true in M and P(h(xl ),..., h(.x,, )) is false in A.

Since M’s universe contains only the extensions (interpretations) of ground

terms, there are ground terms t,,,...,t, such that the extension of t,,,in M is,,xl, and so on. However, h maps ground-term extensions onto the correspond-

ing ground-term extensions, so h( x, ) must be the extension of t,,in A. This

implies that the ground atomic formula P(t,,,, . . . . t,,,,) is true in M but not

in A.

Thus, of the three ways for h to fail to be a homomorphism, the second is

not possible unless the first also fails, and both the first and third require a

minimized ground atomic formula to be true in M but false in A. If this

ground atomic formula is true in M, it must also be true in N, and the theoremis satisfied, ❑

An analog of Theorem 1 holds for the case where T can contain any

sentence. Theorem 2 shows that making an additional unique name axiom true

in a preferred model, or falsifying some other minimized ground atomic


formula, will expand the extension of some minimized predicate as a side

effect. First, a definition: Given a model IV and a set of formulas S, a sutisfjing

assignment for S and IV is a function A mapping each free variable of S to a

universe element of N, such that all the members of S[ A ] are true in N.

Tmzo~F~ 2. Let M be a prefen-ed model of T where the minimized ground

atomic formula a holds, Let M‘ be a model of T where a is false. Then there

exists a set S of minimized atomic formulas that has a satisfying assignment for M‘

but none for M.

PROOF. Let &f and M’ be defined as above: Let us create a set of variables,

one variable L, for each universe element e of AZ. Further, define an assign-

ment .4’ that maps each variable to its universe element, that is, A’(zI, ) = e.

Note that this mapping covers the universe of &f’, so that each universe

element has a unique preimage. It thus makes sense to define a function .4’-1

that maps each universe element e to the variable that is assigned to e under

assignment A’.

Let S contain every minimized atomic formula ~ such that A‘ is a satisfying

assignment for { ~} and M’, that is, such that ~[A’] holds in M’. Note that

every variable 1’, in A‘ occurs in S. in, for example, the formula t’, = LI,. It

remains to show that there is no satisfying assignment A for A4 and S.

Suppose that A is such an assignment. Then let h be a mapping from the

universe of M’ to that of i’kf, defined as follows:

h(e) =A(A’-l (e)).

We now show that h is a homomorphism.

First, 11 preserves the truth of minimized predicates: Suppose that H .7) is

true in M’, for some choice of .L Then, the atomic formula P(A’ - l(1)) is in S.

(Recall that A‘ -1 (2) is the tuple resulting from applying A’- 1 to each elementof 2.) Therefore, P( A‘ -1 (I))[A], which can be rewritten as P( A( A’ - 1(1))) or

P( Iz( ,i)), is true in M. Therefore, the truth of minimized predicates is pre-

served.

Second, h preserves functions: Suppose that the formula f(.~) = ) holds in

M’, for some choice of ~ and y. Then, the atomic formula f(A’ -1 (i)) =A’-l(}) if in S. Therefore, the formula f(A’-l(.t)) = A’- l(Y))[A], which can

be rewritten as f(A(A’-l(.i))) = A(A’-l(jI))) or f(h(z)) = /z(y), is true in M.

Since IZ(JJ) = A( f( 2)), we conclude that /Z preserves functions and is a homo-

morphism.

We have shown that M‘ + M. By assumption, some minimized ground

atomic formula a is true in Lf but false in M’; by Proposition 1, this means

that M + M’. We conclude that M’ is preferred to M. But this contradictsour original assumption that M is a preferred model of T. ❑

In Theorem 2, the set S that differentiates A4 and M’ might be infinite. In

Section 6, we will show why S cannot always be finite.

Ordinary circumscription guarantees that extensions of all minimized

nonequality predicates are minimal in preferred models, where minimality is

measured by set inclusion. Theorem 1 also gives this guarantee for structural

circumscription, but only for ground terms and universal theories. For universe

elements not denoted by ground terms, the situation is more complicated, as

shown by Theorem 2; in the following paragraphs, we explore the source of the

complications.

Circumscription with Homomo@tisms 829

Given a particular universe element in a preferred model under structural

circumscription, we can usually add a “ghost” of that element to the model and

still have a preferred model. For example, if model A contains a universe

element x, we can often create a new model A‘ which is exactly the same as

A, except it contains a new ghost universe element y in addition to the

universe elements of A. We have y satisfy some subset of the predicates

satisfied by x, and have all functions treat y exactly as they do x. Then there is

a homomorphism A + A‘, given by the inclusion, and there is also a homo-

morphism A’ -+,4 that maps both x and y in xl’ onto x in A. So, A’ is

preferred if A is. Thus, unless the theory explicitly rules out ghosts, many

preferred models will include them.

For a formal definition of ghosts, we need the concept of a uniL1erse

restn”ction of a model M of a theory T. We say that M’ is a universe restriction

of Jl if ill”s universe is a subset of J1’s, and M”s functions and relations are

the restrictions of those of M to i14”s universe, and A4’ is also a model of T.

If, in addition, &l + h4’ under a homomorphism that is the identity on the

universe elements that occur in both Nl and M’, then the universe elements

removed from ikl in forming M’ are all ghosts.

For the Tweety theory, Figure 1 showed four models without ghosts. Figure 2

shows a model E that is identical to model A of Figure 1, except that E

contains elements t‘and b‘ that are ghosts of elements t and b, respectively,

in the form of unnamed birds. There is a homomorphism A + E, given by the

inclusion; there is also a homomorphism E + A, in which t‘ and b‘ are

mapped to elements t and b, respectively, in A. Since A is preferred, so is E.

Generalizing, we see that the Tweety theory has preferred models with arbi-

trarily many unnamed birds. Under ordina~ circumscription, preferred models

do not have any birds other than Tweety and Blutto.

Let us look at another example of ghosts. Let T contain just penguin( opus),

with the predicates penguin, bachelor, cat, and = minimized. Then, structural

circumscription has preferred models containing any number of penguins, as

long as they are unnamed. In contrast, ordinary circumscription would con-

clude that Opus was the only penguin extant. If in addition we know that Opus

is a bachelor ( bachelor(opus)), structural circumscription permits any number

of unnamed bachelor and/or penguin elements in preferred models. If we also

know that Bill is a cat (cat(bill)), there may also be unnamed cats in preferred

models under structural circumscription; however, bachelor cats will not be

permitted. Under structural circumscription, any ghost element must enjoy

(minimized) properties that are a proper subset of those of some other

element. Thus, given only the knowledge that there are bachelor penguins and

that there are cats, structural circumscription will not admit the possibility of

unnamed individuals enjoying properties from both these groups (e.g., un-

named bachelor cats); no such individuals will exist in any preferred models.

This is because a bachelor cat can only be mapped homomorphically to an

element that is a bachelor and a cat, since both of these predicates are being

minimized: thus, a model containing bachelor cats will not be preferred to anymodel containing no such individuals.

The Opus example shows that some sentences that follow under ordinary

circumscription, such as Vx.penguin(x) = (x = opus), are not produced by

structural circumscription. In some applications of circumscription, this will be

welcome behavior. In Section 4.3, we show how to capture more of the

830 P. K. RATHMANN ET AL,

FIG. 2, Unnamed birds are allowed.

AfodeJ E

Universe = {t, b, t’, b’}

Function ]nterpretat]ons

tweety = t

blutto = b

Predicate interpretations

bird t, b, t’, b’

flies : t, t’

Ab b, b’

E

fhes Ab

quantified conclusions of ordinary circumscription, by minimizing the predicate

‘< # “.

Structural circumscription gives default conclusions about the equality of

unnamed universe elements, an extremely important property not shared by

some of the other proposals for handling equality in circumscription, as

explained in Section 7. For example, if the description of Blutto as a nonflying

bird is replaced in the Tweety theory by ~x.binf(x) ~ +les(x ). then under

structural circumscription we still obtain the conclusion that Tweety is a flying

bird. Without the ability to conclude by default that an unnamed universe

element is different from a named one, we could not produce this conclusion.~

4. Finer Control of Semantics

Early work on circumscription quickly established the utility of several exten-

sions to the basic circumscriptive paradigm. In this section, we show how to

include these extensions in structural circumscription.

The first extension is to restrict the class of homomorphisms to include only

those which hold certain predicates fixed. In this case, a homomorphism must

also preserve the complement of the extension of a fixed predicate. As a

second extension, we will show how to give different priorities to the minimiza-

tion of different predicates. In Sections 4.3 and 4.4, we consider the effect ofholding equality fixed and allowing it to vary at different priority levels. Finally,

Section 4.5 discusses the relationship between obtaining default conclusions

about equality and obtaining default universally quantified sentences.

4.1 HOLDING PREDICATES FIXED. In some applications of circumscription,we want to disallow homomorphisms between models with significantly differ-

ent extensions of certain “fixed” predicates. We reflect this change in the

5 This problem is discussed as the “counterexample problem” in Etherington et al, [ 1991]. Section

7 discusses Etherington et al.’s method of obtaining such conclusions by applying circumscription

only to selected universe elements.


definition of a signature [F, P, Q, V, Arities], by adding the new set Q

containing those predicate symbols that are to be held fixed. As before, P, Q,

and V must be disjoint. The needed change to the definition of homomorphism

is to add that if h is a homomorphism from M’ to M, and P is a predicate to

be held fixed, then PM,(X) must be true iff P~(h(,i)) is, for all tuples .i of

appropriate arity from the universe of M’. Section 4.2 includes the exact

extended definition of a homomorphism.

Predicates can also be held fixed via a simple first order modification to the

theory. To hold a predicate P fixed, we add to the signature an additional

predicate symbol notP, and add a sentence to the theory,

v~.(p(~) V notp(~)) A -(P(2) A notp(i)),

ensuring that notP and P are complements. We then minimize both P and

notP as usual. How does this hold P fixed? A homomorphism h cannot map a

tuple ,i where P is false to a tuple h(~) where P is true, because notP(l)

would hold but notP(h( 2)) would not, and notP must be preserved under

homomorphism. As usual, if P(i) holds, then P(h(.i)) must also hold. Hence,

by adding notP, we constrain homomorphisms to hold P fixed. For additional

discussion of this technique, and a proof that it gives the correct semantics in

ordinary circumscription, see de Kleer and Konolige [1989].

Under structural circumscription, fixing a predicate can produce new univer-

sal statements about that predicate. For example, consider the theory T

containing just penguin( opus). Circumscribing T with penguin minimized will

produce preferred models containing ghost elements that are not penguins.

However, circumscribing T with penguin fixed will give us Vx. penguin(x). To

see this, let A be a model where every universe element is a penguin, and let B

be a model in which some universe element x is not a penguin. There is a

homomorphism from A to B, that maps every element of A to the interpreta-

tion of Opus in B, holding penguin fixed. There cannot be a homomorphism

from B to A, because such a homomorphism would have to map x to an

element that is a penguin, and then penguin would not be held fixed. There-

fore A is preferred to B when penguin is held fixed. Thus, fixing a predicate P

has the effect of forcing a ghost element to behave exactly like one of its

non-ghost counterparts, with respect to P. Section 4.3 explores in more detail

the effect of holding a predicate fixed.

4.2 PRIORITIZING PREDICATES. We now have a three-level classification of

predicates: those held fixed, those minimized, and those allowed to vary. One

can also consider a proliferation of such classes, minimized with different

priorities [McCarthy 1986]. In this section, we present a simple formalization of

prioritization that follows that used in ordinary circumscription.fi We can

include the priority layering specification in a prioritized signature, by rewriting

the set P of predicates to be minimized as a tuple of disjoint sets of predicate

symbols [PI, ..., P,, ], where each P, is a set of predicates to be minimized at

priority level i. The equality predicate must be included in Pi. Where the

meaning will be clear, we will also sometimes write PI < “”” < P,, to describethe prioritization levels. In the remainder of the paper, we consider only

prioritized signatures.

6 Benjamin Grosof [1991] has suggested the use of a more general approach to pnoritization in

ordina~ circumscription, which could also be adopted for use with structural circumscription.


Prioritization is used, for example, in AI applications with a hierarchy of

types of abnormality, encoded using different Ab, predicates. It is more

important to minimize certain types of abnormality than others; for example,

we would rather conclude that Tweety is an abnormal animal (because animals

normally don’t fly) but a normal bird, than that Tweety is an abnormal bird

(e.g., an ostrich) but a normal animal. As other researchers have shown[Etherington 1988: Lifschitz 1985], this behavior can be obtained by minimizing

the different types of abnormality at different priority levels.

We now extend the definition of homomorphism to allow a prioritized

signature O = [F, [l’[, ..., P.], Q, V, Arities]. Given models M and A4’, we say

that M’ + J4 holds at priority level i if there is a mapping h from the

universe of ilf ’ to that of M, such that for any tuple 1 of elements from the

universe of M’,

(1) The mapping h preserves functions: If ~ is an n-ary function symbol in F,

with extensions &~ and ~~f in M!’ and M, respectively, we require that

Val -.. a,, e u’, h(”fM(al,. . ..cz.l )) =fAf(lL(ul ),.. .7(a,*)))).

(2) The truth of minimized predicates is preserved: If P is an n-ary predicatesymbol from PI U “”” u P, with extensions Pb[i and P,,f in M’ and M, then

Val .“” a,, G U’, Pj~l(al,. ... a,, ) -+ Plf(h(al ),. ... h(a,l )).

(3) F~ed predicates are held constant: If P is an n-ary predicate symbol fromQ with extensions P~ and PA~ in M’ and M, then

Val . . . an E u’, PA~)(al,..., a,l) =P~(h(a, ),... z(a, Z)))).

We say that M’ < M iff there exists a level 1 < i < n such that at level i,

M’ + M and M + M’. The < relation does not seem to extend cleanly to

the prioritized case, so we define the preferred models of theory T as those

models of T that are minimal with respect to <.

PROPOSITION 2. Under a prioritized sigrlature, the relation “ < “ defines a

partial order.

PROOF. Irreflexivity is immediate from the definition. For antisymmetry,

suppose that M’ < M and M < M’. Then, M _ M’ and M’ + M at some

level i, and M’ 3 M and M + M’ at some level j. Clearly, i #j; suppose that

i < j. If there is a homomorphism M’ + M at level j, then there is also a

homomorphism M’ + M at level k, for every k < i. But then M’ ~ M at level

i, a contradiction.

To show transitivity, suppose A < B and B < C. Since A < 1?, there is apriority level i where A + B and B * A. There are two possible situations

regarding homomorphisms between B and C at level i:

— B + C. Then A + C, by composition of homomorphisms, but C + A,

because if there were such a homomorphism, B + C + A would give a

homomorphism B + A.

— B -H C. Since B < C, there exists a level k < i such that C + B and

B + C at level k. There is also a homomorphism A + B at level k, since

~ < i. Then A + C by composition, and C + A, because such a homo-

morphism would mean that C + A + B at level k. contradicting B < C.

Since in both cases A < C, the proposition follows. ❑


In some cases, an alternate definition of prioritization will prove to be more

convenient. Suppose we have a class C of models. For a particular priority level

i in the signature, we can consider all the homomorphisms at level i between

models in C. These homomorphisms define a preorder on the models in C’,

such that M < M’ iff at level i, M + M’ and M‘ + M. In the alternate

definition, a model is preferred according to prioritized structural circumscrip-

tion iff it is minimal in each of the preorders defined by the different priority

levels of the signature. Phrased slightly differently, M is preferred overall if it

is preferred under each of the single-level circumscriptions given by signature

[F, [P, u ... UP,], Q, Vi7P(+1 U “.. U P,l, Amities], foralll <i <n.

This alternate definition gives exactly the same set of preferred models as

the first definition above. To see this, let us assume M is a model not preferred

according to the alternate definition. Then, at some priority level i, M is not

preferred, which means that there is another model M’, such that at level i,

M’ + M and M + M‘. But this means that M’ < M in the partial order of

the first definition, and so M is not preferred according to the first definition

either. Similarly, if M is preferred according to the alternate definition of

prioritization, at no level is another model preferred to M, so M is minimal in

the partial order of the original definition of prioritization as well.

4.3 HOLDING EQUALJm FIXED. In many applications of circumscription,

we expect a slightly stronger form of structural circumscription to be desirable:

structural circumscription with domain closure. Domain closure corresponds to

the common-sense reasoning assumption that only those things that must exist

do exist. Without domain closure, structural circumscription admits the possi-

bility that there are unnamed elements in the universe that are “like” named

elements (ghosts). The existence and nature of these unnamed individuals

follow from the homomorphisms used to pick out preferred models; the model

having a minimal universe is not preferred to a model containing “extra”

elements because the extra elements “fold onto” the minimal universe: One

can construct a homomorphism from the larger to the smaller universe. If one

knows a priori that all the relevant individuals are known and named, that is,

are designated by ground terms, then a domain closure assumption is indicated.

More formally, by the domain closure assumption, we mean that the universe of

a preferred model is an interpretation of the ground terms of the signature.

One might be tempted to say that M has a minimal universe iff M has no

proper universe restrictions.’ This would be a poor definition, however: con-

sider a signature containing just the unary functions left and right, and a theory

that states that the elements in the universe are arranged into complete binary

trees via left and right. One might think that the models with minimal universes

in this theory would be those containing one rooted binary tree. If, however,

one removes the root element from such a tree, two binary trees remain; so in

some sense two rooted trees have as small a universe as one rooted tree. A

better definition of “minimal universe,” which handles this case correctly, is tosay that M has a minimal universe iff every proper universe restriction of M

~ This is often used as the definition for the domuin closure us.wnpfion. The flaws discussed herehave led us tu introduce our own definitions for “domain closure” and “minimal universe.”


has a universe restriction that is isomorphic to A4 ( M is prime, in the parlance

of mathematical logic).

A class of models satisfying the domain closure assumption will satisfy the

minimal universe assumption as well, since one cannot remove the interpreta-

tion of a term from a model and still have a well-formed model. It is a

well-known theorem of logic that those models of a universal theory that satisfy

the minimal universe assumption also satisfy the domain closure assumption,

assuming that the signature contains at least one constant, The minimal

universe assumption is a generalization of the domain closure assumption that

is suitable for use with nonuniversal theories, where it might not be possible

for terms to denote all universe elements.

A minimal universe assumption can be added to structural circumscription,

by holding equality fixed in the manner described below. Since the original

motivation for structural circumscription was to remedy ordinary circumscrip-

tion’s inability to reach default conclusions about equality, it may seem

perverse to disable such an important feature of structural circumscription.

However, this variation can be used without sacrificing the usual default

conclusions about equality, as explained below. Moreover, the variation will

cast additional light cm the relationship between ordinary and structural

circumscription. In this section, we will hold equality fixed by including in the

theory a formula defining the new predicate # :

vay.(.Y #y) = 1(.X =-v).

If # is allowed to vary, then equality is minimized as usual, as though # had

not been defined. However, when # is minimized at some priority level, the

set of preferred models is profoundly affected. To understand the source of the

effect, let us consider preference relations among the three models in Figure 3.

Assume these are models over a signature with no functions, so that only

predicates need to be considered in determining whether there can be homo-

morphisms between these models; and assume for now that the signature has a

single priority level. When equality is not held fixed, model ,4 is at a

disadvantage with respect to models B and C, because A contains an element

where both P and Q hold. Because B and C do not contain such an element,

there can be no homomorphism A + B or A + C when P and Q are both

minimized, so A cannot be preferred to B or C.

When equality is held fixed by minimizing # , the effect is to disallcw ezery

homon~orpilism fronl a larger (finite) unilerse to a smaller unilerse. Such a

homomorphism would not preserve the # predicate, since it would have to

fold two separate elements in the source model onto a single element in the

target model. Because B and C both have larger universes than A, neither of

them will be preferred over Y4 when equality is held fixed, because B + xl and

C-A.

Table I shows the effect of minimizing various combinations of predicates at

a single priority level, with the remaining predicates allowed to vary. For

example, when only # is minimized, model A is preferred over B and C: We

already know B + A and C * A, and since P and Q are allowed to vary, there

are homomorphisms mapping element a of A to any desired element of B or

C. In contrast, when P, Q, and # are all minimized, A * B and A * C

hold, because a satisfies both P and Q, but no element in B or C does; in this

Circumscription with Homonlo@isms 835

Model A Model B

Universe: {a} Un~verse: {b, c}

P(a) P(b)

Q(a) P(c)

Model C

Universe: {d, e}

P(d)

Q(e)

FIG. 3. Models A, B, and C.

TABLE I. THE EFFiZCT OF DIFFIZRENT SIGNATURES ON THE PRLSENCE OFHOMOMORPHISMS BFTWEEN MODELS A, B, AND C.

Predicates minimized Models most preferred among A, B. C

—— A, B, C

P= A, B, CQ= B+= A

PQ = Bp#= A

Q+= A, B

PQ#= A, 13>C

Minimized: A+B B+.4 B+C C-B A+C C+A

Y Y Y Y Y YP= Y Y Y Y Y YQ. n Y Y n Y Y+= Y n Y Y Y n

PQ = n Y Y n n y

P+= Y n n Y Y n

Q#== n n Y n Y n

PQ#= n n n n n n

case all three models are equally preferred. If # and one of P and Q are

minimized, then there is a homomorphism A + C, and so C will not be

preferred: and so on.

When equality is not held fixed, model C is at a disadvantage with respect to

model B, because C’ has nonempty extensions for both P and Q. Since B has

an empty extension for Q, no homomorphism from C to B could preserve Q,

so C is unlikely to be a preferred model. When equality is fixed, however, B no

longer holds that advantage, because a homomorphism from B to C would

have to fold two different universe elements of B onto a single element of C,

which is now disallowed.Consider now a signature containing multiple priority levels. We say that a

predicate P is held fixed at leuel i when both P and the negation of P are

minimized at a level s i, or P or P‘s negation is listed as a fixed predicate in

the signature. If P is held fixed at level 1, we say that P is completely jixed.

836 P. K. RATHMANN ET AI..

As a new complication, the level at which equality is held fixed also affects

the preferred models. To see this in the current example, recall that the result

of circumscribing at multiple priority levels (Section 4.2) is the intersection of

the classes of models obtained by circumscribing at single priority levels, so

that, for example, circumscribing {P} < {Q} < {#} would give {A. B, C} n {B}

n {A, B, C}, from minimizing P, then PQ, then PQ # , according to Table I.

In this instance, even though minimizing # eliminates the homomorphism

from B to A (so that A might conceivably be a preferred model), that

homomorphism is disallowed too late, since model A is eliminated at level 2,

when P and Q are minimized with + still allowed to vary.

Ordinary circumscription cannot be used directly on a theory to produce a

minimal universe assumption [Etherington 1988], though one can use a closely

related technique, called domain circumscription [Davis 1980: McCarthy 1977,

1980], to do so. Alternatively, one can take the theory T whose universe is to

be minimized, introduce a new unary predicate U meaning “in the universe”.

and create T1: a rewriting of T that requires that the formulas of T be true

only on elements of U. (Details of the process can be found in [Etherington

1988] or [Moinard and Rolland 1991]; we use a very similar process in Section 5

to create our formula also called T~,. ) One can apply ordinary circumscription

with U minimized and other predicates held fixed, and then conjoin the result

with b’.YU(.x); ithas been shown that all models of T that are minimal under

the preorder associated with domain circumscription are (contained in) models

of this circumscription, and that this approach is equivalent to domain circum-

scription [Moinard and Rolland 1991]. On the other hand, no one has proven

that domain circumscription produces only models that have no proper uni-

verse restrictions (the definition of “minimal universe” used in the preorder

associated with domain circumscription) [Etherington 1988]. The equivalent

result for structural circumscription is also elusive: Theorem 3 covers a number

of cases. In addition, the first part of the proof of Theorem 3 shows that when

equality is held fixed at any level, all preferred models with finite universes

satisfy the minimal universe assumption.

THEOREM 3. Ij equaliw is held fired at cm}’ lelcl, then the preferred models of’

T hale ttlirlinuzl uni[wses ;f my of the following hold.

( 1) El%ety model of T has a unilerse restrictiotl with a jinite unileme.

(2) All predicates are conlpletel~ jixed.

(3) T is uniLersal.

PROOF. For the case where every model has a finite universe restriction:

Suppose that A is a model of T that does not satisfy the minimal universeassumption. Then there is some model B of T that is a proper universe

restriction of A to a smaller and finite universe. The identity map from B to A

is a homomorphism, so B + A at all levels. On the other hand, any homomor-

phism )z: A ~ B must map some two elements e and e’ of A’s universe to the

same element of B’s universe, because B’s universe is smaller and finite. Then

h(e) = /l(e’) but e # e‘. Therefore, h does not preserve # at the level i

where + is minimized, and so there is no homomorphism from ,4 to B at

level i. We conclude that B <A, so A is not a preferred model of T.

Therefore, every preferred model of T will satisfy the minimal universe

assumption. when every model of T has a finite universe restriction.


For the case in which all predicates are completely fixed, let A be a model of

T that does not satisfy the minimal universe assumption. Then, by the defini-

tion of the minimal universe assumption, A has a proper universe restriction B

such that no universe restriction of B is isomorphic to A. At the single priority

level of the signature, there is a homomorphism B + A, given by the identity

mapping.

Suppose that there is a homomorphism k: A + B. Let k(A) be the universe

restriction of B to universe elements that are in the image of k. Because k

holds equality fixed, k defines a one-to-one and onto mapping between

universe elements of A and k(A). In addition, for any predicate symbol P and

tuple i of elements from A, P(2) holds in A iff P(k(2)) holds in MA),

because k holds P and all other predicates fixed. As k is guaranteed to

preserve all functions, we conclude that k(A) is isomorphic to A, under

isomorphism k. This means that B has a universe restriction that is isomorphic

to A, contradicting our assumption that A had a nonminimal universe. We

conclude that A * B holds when A has a nonminimal universe. But then B is

preferred to A, since we know B + A. We conclude that A cannot be a

preferred model of T, and that all preferred models of T must have minimal

universes when all predicates are held completely fixed. Note that this merely

says that if a preferred model exists, it must have a minimal universe. See

Section 6 for a discussion of when we can guarantee the existence of such a

preferred model.

For the case in which T is universal, recall that we assume that the signature

contains at least one constant. Let A be a model of T. If A contains only

elements that are the extensions of ground terms, then it has a minimal

universe. Therefore, assume A contains an element x that is not the interpre-

tation of a ground term. Since T is universal, A must have a proper universe

restriction B that is the restriction of A to universe elements that are

interpretations of ground terms. Then, the identity mapping of B into A is a

homomorphism. Does there exist a homomorphism A ~ B? If so, then h(x)

must be the interpretation of some ground term t in B. Since /z must preserve

ground terms, h also maps some other element y, the interpretation of t in A,

to the interpretation of t in A. Since x # y but h(x) = A(y), h does not hold

equality fixed. We conclude that no such homomorphism /z: A + B exists.

Thus, the assumption that A has a nonminimal universe implies that A is not

preferred since its restriction B is preferred to A, and we conclude that every

preferred model of T has a minimal universe when T is universal. ❑

When equality is completely fixed, we do not obtain default conclusionsabout equality, such as the unique names assumption. To see this, consider two

models A and B, where t, = t2 is true in A, but not in B. Any homomorphismA + B would not preserveequality,while any homomorphism B + A would

not preserve the # predicate. Thus, models that differ on the equality of

ground terms are incomparable.

Placing # at priority levels 2 and beyond gives the full set of defaultconclusions about equality, because preferred models must satisfy the level 1

structural circumscription in which inequality is allowed to vary. The level 1

circumscription eliminates all models not satisfying the usual conclusions about

equality that structural circumscription gives.

In the case of universal theories, minimizing # at any level gives us all the


conclusions of ordinary circumscription,x plus domain closure (assuming that

the signature contains at least one constant). To state this formally, we have

the following theorem:

THEOREM 4. Let T be a unil’er-sal theo~ ouer a signature in which # is

trr.inimized. Consider the follo~’ing three statements:

(1) M is a preferred model of T under structural circumscription.

(2) M is a prefened model of T under ordinary circumscription.

(3) M satisfies the minimal uniuerse assumption for T.

Then statement (1) implies statements (2) and (3). If equality is con~pletely ftied,

statement (1) holds iff statenzents (2) and (3) both ~lold.

PROOF. (1) implies (2) and (3). By Theorem 3, the preferred models of T

under structural circumscription have minimal universes, so (1) implies (3). To

prove by contradiction that (1) implies (2), suppose that A is a preferred model

of T under structural circumscription (1 ), but not under ordinary circumscrip-

tion (2). Then some model B of T must be preferred to A under ordinary

circumscription. Since A and B are comparable under ordinary circumscrip-

tion, they must have identical universes and function interpretations. Let h:B + A be the identity mapping on the universe of A and B. Then, h preserves

all functions. Suppose that level i is the lowest level at which. under ordinary

circumscription, B can be shown to be preferred to A. Then, at level i, all

minimized predicates have extensions in B that are subsets of their extensions

in A; all fixed predicates have identical extensions in A and B; and some

minimized predicate P has an extension in B that is a proper subset of its

extension in A. But this means that h is a homomorphism from B to A at

level i, so B + A at level i.

We have pointed out previously that when T is universal, its minimal-

universe models satisfy the domain closure assumption, so that all universe

elements are the interpretations of ground terms. Since (1) implies (3), A has a

minimal universe and satisfies the domain closure assumption. This means that

in order to preserve all functions, any homomorphism .4 + B at level i must

be the identity map on all universe elements. But such a homomorphism could

not preserve predicate P, since P is minimized at level i and has an extension

in B that is a proper subset of its extension in A. We conclude that .4 + B at

level i, and that B is preferred to A under structural circumscription, violating

our assumption that A was preferred under structural circumscription. We

conclude that (1) does imply (2).

(2) and (3) together imply (1) when equality is completely ftied. Let M be apreferred model of 7“ under ordinary circumscription (2) that satisfies the

minimal universe assumption (3). We have pointed out previously that when T

is universal, its minimal-universe models satisfy the domain closure assump-

tion, so all universe elements of M are the interpretations of ground terms.

Suppose that some model M’ of T is preferred to M under structural

circumscription. Then, there must be a homomorphism h: M’ + M at some

s We have not formally defmed the treatment of prioritized and fixed predicates under ordinarycircumscription. In ordinary circumscription, if a predicate k fixed, then one only comparesmodels with identical extensions for that predicate. Prioritized minimization IS defined in thesame manner as in Section 4.2.

Circumscriptiotl with Homomo~hisms 839

level i where M + M’. The function h must be onto, because h preserves all

ground terms, and all universe elements in M are the interpretations of ground

terms. Since h holds the equality predicate fixed, h is a one-to-one function,

which means that every element in the universe of M‘ must be an interpreta-

tion of a ground term, as otherwise h would have to equate two elements from

M’ and would not be one-to-one. Because h is one-to-one and onto, without

loss of generality, we can rename every element of M’ so that M and M’ have

identical universes and h(x) = x for all x. With its new universe, M‘ is still a

model of T and is still preferred to M under structural circumscription. The

homomorphism h: M’ + M is now just the identity map. Since h preserves

functions, M and M’ must now have identical function interpretations. There-

fore, M’ and M are now comparable under ordina~ circumscription. Because

h is a homomorphism, the extension of each minimized predicate at level i in

M’ must be a subset of its extension in M; and each fixed predicate must have

the same extension in M and M’.

The identity map from M to M‘ preserves functions, since M and M‘ have

the same universes and function interpretations. If it also preserves predicates,

then M + M’ at level i, contradicting our assumption that M ++ M’ at level i.

Therefore, the identity map from M to M’ must not preserve predicates at

level i: There must be some tuple i of elements of M such that minimized

predicate P holds for 1 in M but not in M’. This means that the extension of

P in M’ is a proper subset of its extension in M. We conclude that M’ is

preferred to M under ordinary circumscription, contradicting our original

assumption that M is a preferred model under ordina~ circumscription. We

conclude that in fact M must be preferred under both ordinary and structual

circumscription. ❑

We can give a different characterization of the relationship between the two

types of circumscription:

COROLLARY 1. Let T be a universal theory ol’er a signature in which equalip is

held fixed at some priorigy leuel. Then for any sentence a,

T EC,,C a * T *$,,C,,C a.

If a is ground and equalip is held completely ftied, then

T Fci,C a + T i=s,,c,.C a ,

where KC1,C and >~t,c,,C refer to entailment under ordinary and structural

circumscription, respectively.

PROOF. The forward implication in both formulas follows from Theorem 4.

For the reverse implication in the second formula, suppose M is a preferred

model under ordina~ circumscription in which a is false, so that a does not

follow from the ordina~ circumscription of T. The restriction N of M to a

minimal universe is still a preferred model under ordinary circumscription, and

a is false in N. By Theorem 4, N is a preferred model under structuralcircumscription, so a cannot follow from T under structural circumscription,

either. ❑

Minimizing # as the sole predicate at the last level (level n) has the effect

of restricting the universe of the models that would otherwise be preferred. To


see this, recall the conjunctive definition of prioritized circumscription; mini-

mizing + last can only further restrict the models that would be obtained if

# were always allowed to vary. Further, minimizing + may eliminate

additional models that do have minimal universes. For example, consider a

theory with two models (up to isomorphism), both with universe {a, b}. In Ml,

a and b satisfy P, and b satisfies Q. In Ml, a and b both satisfy P and Q. In

minimizing {P, Q} < { #}, there are homomorphisms in both directions be-

tween Ml and Ml at level 1. At level 2, however, Ml -+ Mz and Mz ++ Ml, so

M~ is not preferred. Note that ordinary circumscription would prefer M, over

Mz, because the predicate extensions in Ml are strictly smaller than those in

A4z. Because of element “folding” under homomorphism, however, M. is not

eliminated under structural circumscription unless equality is held ~ixed at

some level.

Example 2. More about Tweety. Continuing with the Tweety theory, if +

is minimized along with Ab, a preferred model must have in addition a

minimal universe. Therefore, T now has only one preferred model, up to

isomorphism. In this model, Tweety and Blutto are distinct birds, and are the

only elements in the universe.

Holding equality fixed eliminates all uncountable models, as the following

useful proposition shows.

PROPOSITION 3. If equaliy is held ftied at any lel’el, all prefewed models under

structural circumscription haue a countable uni[!erse.

PROOF. Chang and Keisler [1990] show in Theorem 3.1.6 (on page 138) that

every model M has a universe restriction M’ of any desired universe cardinal-

ity down to the cardinality of the language. Since we assume a finite signature,

the language generated from the signature is countable, and hence any model

M has a countable universe restriction M’. Consider any priority level i at

which equality is held fixed. The identity map from M’ to M defines a

homomorphism at level i. However, since the universe of M is uncountable,

any function mapping it into the universe of M’ cannot be one-to-one. Thus,

any candidate homomorphism from M to M’ cannot hold equality fixed, since

it must equate in M‘ the images of elements that are distinct in M. This means

that no model of uncountable cardinality is preferred when equality is held

fixed at any level, since it is always less preferred than its countable universe

restriction, 0

Holding equality completely fixed makes structural circumscription behave

more like ordinary circumscription, and so presents good opportunities tocompare the two. The results in the remainder of this section show that when

equality is held completely fixed and the signature is sufficiently large, models

that are preferred under ordinary circumscription will also be preferred under

structural circumscription.

Let us examine the consequences of adding a new unary function symbol S

to the signature of theory T, without adding any sentences in T referring to

this new function symbol. Because S is unconstrained, every model under the

old signature can be extended to many new models under the new signature,

one new model for each possible interpretation of S. Therefore, such a

signature extension is a conservative extension of T. In addition, the preferred

models of T under ordinary circumscription are unchanged except for the


addition of an interpretation of S. The theorem below shows that with this

seemingly minor change to T’s signature, whenever T has a countable pre-

ferred model under ordina~ circumscription, T must also have a preferred

model under structural circumscription. First, we define a bit of terminology:

by “the reduct excluding S“, we mean the model that results when the

interpretation of S is removed from a model. By fl~, we mean the signature ~

with function symbol S removed.

THEOREM 5. Let T be a theov over signature Cl, in whiclz equality is held

completely ftied. Also, let ~ include a unajy function symbol S t)lat does not

occur in T. Then el’e~ countable model of T which is prefemed by ordinary

circumscription ol’er the signature lllY is a reduct (excluding S ) of a prefeved

model of T under stmctural circl{mscnption OLW’ signature (_).

PROOF. Let us consider a countable model M, which is preferred under

ordinary circumscription with signature fl~. For the first part of the proof, it

will be convenient to assume that the universe of M is countably infinite,

rather than finite. Like any countably infinite set, the universe of M can be

placed in isomorphism with the natural numbers. Let us pick such an isomor-

phism in which the extension of some constant symbol c (recall that we assume

that every signature contains at least one constant) corresponds to the smallest

natural number. This isomorphism to the natural numbers defines an extension

for the function symbol S in which S(c) corresponds to the second natural

number, S( S( c )) corresponds to the third, and so on.

Such an arrangement is called an N-chain. We show that, augmented to

include this extension of S, M is also preferred under structural circumscrip-

tion.

Assume, for purposes of contradiction, that some other model M’ is pre-

ferred to M. This implies that there is a homomorphism M’ S M at some level

i. Let us consider the extensions of the terms c, S(c), S(S(c)), . . . in M’. Since

each of these is distinct in M, they must all be distinct in M’. Otherwise, g

could not preserve S. Further, each element of the universe of M‘ must be the

extension of one of these terms, since otherwise g could not preserve + in

mapping any extraneous elements onto M, which consists solely of an N-chain.

Therefore, the universe of M‘, like that of M, consists of a single N-chain.

Further, since g is a one-to-one and onto function, g-1 is also a function andgog-l is the identity. The remainder of the proof repeats arguments used in

proving Theorem 4.

Since g preserves the functions of Q, g-1 must also preserve functions.

However, although g-1 is a function and preserves functions, it cannot be a

homomorphism, since M’ < M under structural circumscription. Therefore,

g – 1 must fail to preserve some predicate.

Models M’ and M have universes that are isomorphic, but this is not

sufficient for them to be comparable under ordinary circumscription. Such

comparability only occurs when universes and function extensions are identical,rather than just isomorphic. We can circumvent this barrier by building a

model M“, with the universe and function extensions of M, but the predicate

extensions of M’. We define the predicates of M“ with the condition that

P(i) is true in kf” if and only if P(g - 1(-i)) is true in M’. Note that since M“

is isomorphic to M‘, itmust also satisfy T.


Now, consider the relationship between M“ and M. By construction, they

both have the same universe and function extensions. Every predicate fixed at

level i has the same extensions in M and NV, and every predicate minimized

at level i must have an extension in M“ that is a subset of its extension in M.

In addition, some minimized predicate has an extension in M“ that is a proper

subset of its extension in M. Therefore, M“ is preferred to M under ordina~

circumscription. This is a contradiction, since by assumption M is a preferred

model of T under ordinary circumscription. Therefore, we must conclude that

M, as augmented, is a preferred model of T under structural circumscription.

When M has a finite universe, we can choose an interpretation of S that

arranges the universe in one big loop, and then the argument given above also

applies. ❑

Theorem 5 guarantees the existence of preferred models under structural

circumscription when ordinary circumscription has countable preferred models.

We do not know if it is possible under ordinary circumscription to have only

uncountable preferred models. However, the principle underlying Theorem 5

can be generalized to apply to this case. First, a bit of terminology: if we add

new constants to the signature of T, enough to give every element of model M

a name, this is called expanding by constants. The expanded M (in which all

elements are named) is still a model of T.

THEOREM 6. Let T be a theory oL’er a signature itl which equaliv is held

complete~ fired, and let M be a model of T. If M is prefewed under ordinary

circlunseription, then after M is expanded by constants, it will be preferred u~lder

both ordinav and structural circunlscviption. After e.rpansio?l by constants, if M is

prefewed under structliral circumscription, then it is also prefetred under ordinary

circunlseription.

Theorem 6 talks about uncountable preferred models under structural

circumscription. This does not contradict Proposition 3, because after expan-

sion by constants, the language may be uncountable, and Proposition 3 as-

sumes a finite signature, which can only generate a countable language.

PROOF. Suppose that M is preferred under ordina~ circumscription. Ex-

pansion by constants will give every element of M a name. Then, the proof

technique from Theorem 5 can be reused to show that no model can be

preferred to M under structural circumscription.

For the other direction, suppose that M has been expanded by constants and

is now preferred under structural circumscription. Suppose that M’ is pre-ferred to (the expanded) M under ordinary circumscription. Then M and M’

must have the same universe and function interpretations, and there must be a

homomorphism from M’ to M, the identity map. There must be a homomor-

phism from M to M’, also, else M’ would be preferred to M under structural

circumscription. But this second homomorphism must preserve all constants, so

it must also be the identity map. Therefore, M and M‘ must be identical on

the nonvarying predicates, and M‘ cannot be preferred to M under ordinary

circumscription. ❑

4.4 ALLOWING EQUALITY TO VARY. So far, tools have been defined sothat ordinary predicates can be held fixed. minimized, or allowed to vary, while


equality can be held fixed or minimized.’) The remaining possibility—allowing

equality to vary-has applications to the problem of object identity. It is only

when equality is allowed to vary that it is possible to nonmonotonically derive

positive sentences about equality, that is, to tentatively conclude that objects

are equal.

Cartesian philosophy holds that objects may always be distinguished by their

attributes. Such properties as length, weight, and even color are characterized

by potentially infinite gradations, so in a finite class, the chance of finding two

objects with, for example, the same length is infinitesimal. If two objects can’t

even agree on one value, needless to say, the chance they can agree on all of

them need not even be considered.

Computer scientists, particularly the proponents of object-oriented models,

tend to reject the applicability of this Cartesian viewpoint to computer systems.

Computers, after all, are finite systems. Such attributes as length can only be

measured and represented to a finite precision. There is a nonnegligible

possibility that two distinct objects may be identical, if not on their real

attributes, then at least on the finite precision measurements of those at-

tributes that a computer stores and manipulates. To deal with this possibility,

the concept of object identity is introduced, so that the equality relation is

represented explicitly, rather expecting it to be deducible from other proper-

ties.

Despite this, people do frequently make Cartesian inferences. A glimpse is

enough to recognize a face, coincidences are at least suspected of arising from

a common cause, fingerprints are legally assumed to have come from the same

person if they share more than eight common features. This kind of reasoning

is nonmonotonic, since the discovery of a distinguishing property can always

split objects previously thought to be identical.

Can we capture this kind of reasoning with structural circumscription? Let us

look at an example, and see what the formulation might look like.

Example 3. One or two bears? Assume we are on a camping trip, and two

children are gathering firewood. One of them runs back to the campsite

screaming in fear at having seen a bear. Moments later, the other also sees a

bear and runs back to the campsite. Both tell their circumscription-based robot

scoutmaster that they saw a bear that was large, brown, and had a white spot

on its side. Among other things, the robot should 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, we might write down the

inference as

[Color( Yogi)= Color( Booboo) A

Size( Yogi)= Size( Booboo) A

Marking( Yogi)= Marking( Booboo)] -+ Yogi= Booboo

The above is a heuristic inference rather than a logical implication. The

usual way to express such an inference using circumscription is to use the

9As explained earlier, if equality were omitted from the set of minimized predicates in thesignature, then due to the definition of homomorphism, equality would not really be allowed tovary; accomplishing this requires a new kind of homomorphism.


predicate XI to indicate unexpected behavior:

[ Color( Yogi ) = Color( Booboo) A

Size( Yogi)= S&( Booboo) A

Marking( Yogi)= Marking( Booboo) A

~b( Yogi) A +b( Booboo)] + Yogi= Booboo.

We want to be able to conclude a positive sentence about equality, which is

only possible if Ab is minimized with higher priority than the equality predi-

cate. Recall that, in Section 4.2, minimizing with priorities was expressed as an

intersection of classes of models: the preferred models of T under a signature

in which {Ah} < { = } are the intersection of the preferred models when Ab is

minimized and = allowed to vary. and when Ab and = are both minimized at

level 1. This formulation requires equality to be allowed to vary, which was not

provided for in the original definitions. In this section, we define and explore

some of the properties of a generalization of structural circumscription which

can allow equality to vary. The formulation differs from that of Section 2 in

that a generalized homomorphism is a relation from the source to the target

universe, rather than a function. Generalized homomorphisms are used in this

section only.

We begin by adding to the old definition of a model the requirement that a

model contain the extension of the equality predicate, that is, the identity

relation on the universe.

We also define a new kind of homomorphism. Given models A‘ and A, a

generalized homomorphism from A’ to A is a binary relation h, with domain

the universe of A‘ and codomain the universe of A, satisfying the following

conditions:

(1) The generalized homomorphism must cover its domain. In other words, for

all elements .x’ of A’, there must be an element x of A such that h(x’, x )

holds. Since functions satisfi this property automatically, this condition did

not appear in the old definition.

(2) The generalized homomorphism preserves functions. For an n-ary function

symbol ~ of the signature, let ~ have interpretation ~.l in A and ~.i in A‘.

Let I be an rz-tuple of elements of .4, and i‘ an rz-tuple of elements of

A’, such that ~~(.t) = y, ~q(.i’) = y’, and h(xj, xl) A ““” A lz(IJ,, I,, ) hold.

Then. h(y ’, y) must also hold. Figure 4 shows this relationship graphically;

the solid arrows represent the conditions, and the dashed arrow represents

the conclusion.

(3) The generalized homomorphism preserves the truth of all predicates notallowed to vary, so that if a minimized or fixed predicate is true on a tuple

of arguments, the predicate must be true on the generalized homomorphic

images of those arguments. In other words, for any nonvarying n-ary

predicate symbol P with extensions PA and P,i ~ in A and A‘, respectively,

and n-tuples 2 and .i’ from the universes of A and A‘, respectively,

[lJ4(x\,... ,x:, ) Ah(x; ,xl) A ““” Ah(x;l, x,,)] +<4(x1, . . ..x.I). (1)

If P is held fixed, we also require

[IF’’r(x ;,..,, x:) Ah(x; ,xl) A ..’ A/z(x~ ,x,,)] + -+4(x 1,. ... x,I). (2)


M

fM,(~/}. ..__. ___. !------ --_-> fM(~)

$ ~@ h

z

FIG. 4. The generalized homomorphism conditions on functions,

As before, the composition of generalized homomorphisms is a generalized

homomorphism; and the “ < “ relation is a preorder. Without prioritization,

M’ < M iff M’ ~ M and M * M’. We say that M is a preferred model of

theory T if no other model of T is preferred to M. Prioritization is defined as

before. These semantics are identical to those of Section 2, if equality is

minimized or held fixed:

THEOREM 7. Let T be u theory oler a prioritized signature in which equali@ is

mitlimized, or ftied, at priority level 1. Then the preferred models of T are the same

whether ordinary or generalized homomoiphisrns are used.

PROOF. The only difference between ordinary and generalized homomor-

phisms is that generalized homomorphisms do not require h to be a function:

Mxj, x~), h(xj, X2), and xl #x2 (3)

might all hold. But if equality is minimized or held fixed at level 1, then by

formula 1, any homomorphism satisfies

[=.4(x;, ~;) A ~l(xj, xl) A ~2(x;, X2)] + ‘A (x,, X2),

so formula 3 cannot come to pass. ❑

Let us now return to the bear example. Let A be a model in which Yogi and

Booboo both denote the same bear, and let B be a model in which they are

different. As shown in Figure 5, in model A, both Yogi and Booboo denote

universe element a, while in model B they denote y and ~, respectively. (For

clarity, the figure omits the functions Color, Size, and Marking.) Note that in

model B, the universe element denoted by Yogi is abnormal; otherwise, B

would not satisfy the theory. If we circumscribe, prioritizing Ab over =, we

find that there is a generalized homomorphism A ~ B at level 1, that maps ato both ~ and y. There is no generalized homomorphism in the reverse

direction at level 1, because there is no abnormal element in xl. Thus, the

model where Yogi and Booboo denote the same bear is preferred, because this

gives less abnormality in the preferred models.


Yog2 Booboo

Model A

FIG. 5, Example of a gencrahzed homomorphism,

45 TRADE-OFFS BETWEEN EQUALITY AND UNIVERSAL SENTENCES. Struc-

tural circumscription allows us to derive new sentences about equality, but at

the same time we lose some of the power of ordinary circumscription to

conclude universal sentences, due to the presence of ghosts. Other researchers

have noted an apparent tension in nonmonotonic reasoning mechanisms be-

tween the ability to draw default conclusions about equality and the ability to

conclude universal sentences (Yoav Shoham, personal communication). For

example, default logic, autoepistemic logic, and unprioritized structural circum-

scription are good for producing conclusions about equality, but are not very

helpful in concluding universal statements, such as “Tweety and Blutto are the

only birds. ” In the case of structural circumscription, this inability is due to the

presence of ghosts. In default and autoepistemic logic, the inability arises from

the method of constructing extensions, which is oriented towards terms. At the

other extreme, ordinary circumscription produces universal statements easily,

but no conclusions about equality.

As an interesting compromise for those applications where strong universal

conclusions are desirable, we suggest prioritized structural circumscription with

equality held fixed at a level greater than 1. This circumscription gives all the

conclusions about equality that are ordinarily obtained under structural cir-

cumscription, guarantees that all finite preferred models have minimal uni-verses, and for many theories (Theorem 3) guarantees a minimal universe

assumption for all preferred models. In moving toward a minimal universe,

fixing the equality predicate tends to eliminate ghosts from preferred models,

and ghosts are the stumbling block for obtaining universal conclusions under

structural circumscription.

For some applications, perhaps a minimal universe assumption will be too

strong. (Its cousin, the domain closure assumption, is certainly too strong for

many AI applications. ) For example, the node-path example in Section 6 doesnot have a minimal-universe model. When a minimal-universe assumption is

too strong, one can obtain both universal and equality conclusions by combin-


ing the preorders of ordina~ and structural circumscription. The models

preferred overall will be those that are preferred under both structural and

ordinary circumscription. Thus, we can get the power of both formalisms, at the

cost of an increased chance of there being no preferred models.

We illustrate the various options possible with an example, summarized in

Table II. In Figure 6, we picture four models for the theory T = {P( -4), P( B)}.

In model 1, there is only one element and A and B are equal. In model 2,

there are two elements, one each for A and B. Both models 3 and 4 have an

extra element; in model 3, P is true for this extra element, and in model 4, P is

not true on this element.

If we circumscribe T with ordinary circumscription, minimizing P and = , we

can rule out model 3, since it has an extra element for which P is true. The

other models—1, 2 and 4—are minimal for ordinary circumscription. Thus, we

obtain

VX.P(X) + (x =/4 VX =B), (4)

but we do not get A + B.

Applying structural circumscription, minimizing P and = as before, rules out

model 1, in which A = B. The extra elements in models 3 and 4 are ghosts,

and models 2, 3, and 4 are all preferred models of the structural circumscrip-

tion of T, so we do not get the universal conclusion of formula 4. If we use

structural circumscription and in addition hold equality completely fixed, giving

a minimal universe assumption, ghost elements are ruled out and only models

1 and 2 are allowed, giving formula 4 but not A # B. Minimizing # at level 2

gives model 2, so we obtain both A + B and formula 4. Conjoining ordinary

and structural circumscription gives preferred models 2 and 4, which gives both

formula 4 and A + B, without a minimal universe assumption.

5. An Axiom for Structural Circumscription

So far, we have been developing the properties of structural circumscription

entirely on the basis of the model-theoretic definitions. One can also pick out

the preferred models of a theory by encoding the preference criteria in an

axiom in second-order logic.2 When this axiom is added to the theory, the

models satisfying the enhanced theory will be exactly the models preferred

according to the original model-theoretic conditions.

For the first part of this section, we will consider only signatures with a single

priority level. Recall that such a signature can be written as Q = [F, P, Q, V,

Arities], including a set F of function symbols, and nonoverlapping sets Q, P,

and V, corresponding respectively to those predicate symbols held fixed, those

minimized, and those allowed to vary. As usual, the equality predicate must be

included in P. We write S to designate a tuple containing all the symbols in F,

P. Q. and V, except the equality predicate.

We change our notation for theories slightly to include the functions and

predicates as a parameter of the theory. Thus, T(S) denotes T with its usual

functions and predicates, and T(S’) denotes the result of replacing in T all thefunction and predicate symbols of S with the corresponding predicate and

function variables of S‘. For those familiar with ordinary circumscription, a

first cut at an axiom for structural circumscription might be:

T(S) /A VS’.1[(S’ < S) ~ T(S’)],


TABLE II. THE EFFECT OF DIFFERENT SEMANTICS ON FIGURE 6.

Type of Circumscription Prioritization Scheme Preferred Models

Ordinary {P, =} 1, ~, 4

Structural {P, =} 2,3,4

Structural {P, =, +} 1, 2Structural {P, =} <{#) ~

IIA.P

B.

2 0A.P

b.

.

3

FIG, 6. Four models for the theory (P( A), H B)}.

where S‘ < S is shorthand for a formula specifying that there

phism from the “model” defined by S’ to that defined by S,

aA.P

B.

4

is a homomor-

but not in the

reverse direction. This simple approach cannot capture the semantics of

structural circumscription, however, because the predicate variables of S‘ and

predicates of S are over the same universe, and homomorphisms need to be

able to compare models with different universes. 1“ The most convenient way to

allow different universes for S and S‘ is to extend the signature and use a

two-sorted logic, where one sort (hereafter called the test SCM) is used for the

universe of S‘, while the other sort (the original sort) is used for the universe of

S. We could get by with using single-sorted logic, using a very similar construc-

tion to the one below; but the two-sorted construction is a bit cleaner, and we

lose nothing by adopting it. It is convenient to assume that the elements of the

test sort are disjoint from those of the original sort, and we do so. When {1 is

changed by adding a test sort, we call the new signature O” = [F, P, Q, V,

Arities, TestSort], where TestSort is a miniature signature for the new sort: a

tuple giving the sets of function and predicate symbols over the test sort (other

than the equality predicate), followed by their arities. Except for the equality

predicate, members of F, P, Q, and V are defined only on the original sort;

and the functions and predicates of TestSort are defined only on the test sort.Proposition 4 shows that the test sort need not be very large: it suffices, when

testing to see if a model of a theory T is preferred, to consider only other

models with countable universes.

PROPOSITION 4. Let M be a model ti~at is not prefemed under structural

circumscription. Then there is a nlodel with a countable uniL1erse that is preferred

to M.

‘(] For example, consider a theory containing just (Vxy.x = y) + VZ. P(Z). Structural circumscrip-

tion prefers the model with two universe elements and an empty extension for P to the model

with a single element and a nonempty cxtens]on for P.


PROOF. Let N be a model that is preferred to M. Since the signature is

countable, by Theorem 3.1.6 on page 138 of Chang and Keisler [1990] N has a

countable universe restriction N’. There is an identity homomorphism from

N’ to N, so by composition of homomorphisms, IV’ ~ M. If there were a

homomorphism from M to N’, then, by composition, we would have M + N,

which is not the case. ❑

To ensure that there are at least a countably infinite number of elements in

the test sort, we can add axioms to T and rename it T{):

Vx.s(x) # oVxy.s(x) = s(y) +x =y,

(5)

where s is a new unary successor function and O a new constant, both over thetest sort. Thus, we have the expanded signature Q” = [F, P, Q, V, Arities,

TestSort], where TestSort = [{s, 0}, { }, TesrAtitiesl. Note that s, O, and = are the

only function and predicate symbols defined over the test sort, and none of

them are included in S. The members of S have arguments and results only in

the original sort; the predicate and function variables of S‘ range over

predicates and functions whose arguments and results are confined to the test

sort. The quantifiers of T(S) range only over the original sort, and the

quantifiers of T( S‘ ) range only over the test sort. From a model M of T[) over

signature flo, one can obtain a model of T over signature Q by restricting M

to 0.

With this notation, the structural circumscription of T is given by

StrCirc(To, Q,,) = TO(S) A VS’U.l [(S’ < S) A TL,(S’)l. (6)

U is a unary predicate variable taking arguments of the test sort. Intuitively,

the predicate variables of U and S‘ pick out a universe and a set of predicate

valuations, respectively, over the test sort. In essence U and S‘ construct a new

“test model” of T over the test sort, to see whether the test model is preferred

to the model given by the original sort and the extension of S (the restriction

to Q). The conjunct Tu( S ‘), defined precisely below, guarantees that the test

model actually satisfies the formulas of T. The biggest difference from ordi-

nary circumscription comes in the construct S‘ < S, which we explain below.

First, let us construct a second-order definition of the homomorphisms used

to test for minimality. Let k and g be unary function variables, such that /z

takes an argument of the test sort and produces a result in the original sort,

and g does the reverse. We will write hom(h ) as shorthand for the conjunction

of the following formulas:

V,i.U(.i) ~ (h(f’(.i)) =f(h(i)))> Vf~F

Vi.U(i) + (R’(i) -+ R(h(.i))), VRG(P –{=}) (7)

where U(i) is shorthand for U(xl) A “” o A U(x~), where n is the cardinality of

f or R, as appropriate. The quantifiers in formula 7 range over the test sort.Formula 7 is the analog, in second-order logic, of the definition of homomor-


phism given in Section 2. Formula 7 is finite because 0 is finite. Similarly,

rew-seHcvn(g) is shorthand for the conjunction of the following formulas:

The quantifiers of formula 8 range over the original sort. Formula 8 is finite

because f) is finite.

We can now define S‘ < S, using the definitions of horn and relerseHom:

~11 hom(h ) A ~ ~g rcLerseHom( g ). (9)

Formula 9 ensures that h is a homomorphism from the test model to the

restriction of the model to 0, and that there is no homomorphism in the

reverse direction. As described in Section 2, this is the condition for the test

model within a model M to be preferable to the restriction of M to Q.

Finally, TU(S’) is formed from T( S’) by (1) ensuring that the test universe

is nonempty, (2) restricting the range of quantifiers to the test universe, and

(3) ensuring that the results of functions are in the test universe, whenevertheir arguments are. This process is the same used in rewriting T before cir-

cumscribing the domain using ordinary circumscription [Etherington 1988;

McCarthy 1980]. Step (1) is accomplished by adding 3x.LXX) to TU(S ‘). Step (2)

is accomplished by first putting the formulas of T( S‘ ) into prenex form, and

then successively replacing all occurrences of subformulas of the form Yx. a,

where x ranges over the original sort, by V.Y. U(.~) + a, and subformulas of the

form 3x. a, where x ranges over the original sort, by 3x. U(X) A a. For

example, if T is

3.xVy .bird( x) A y # motherOf( tweety ),

then Tu ( S‘ ) contains the formula

3x.U(X) A (’v’y.U(y) ~ (bird’(x) Ay + rnotherOf’(tweety ’))).

To accomplish step (3), one must add to T[,(S’) the formula

V.i.u(.i) + u(f’(i)),

for each function f in F. For example, the motherOf function will require the

addition of V.x. iX x ) - U(motherOf’( x)) to Tu (S’). For a O-ary function f such

as tweev, the appropriate formula to add is simply U( f‘ ), as in U(twee& ‘).

Theorem 8 shows that the model-theoretic and axiomatic formulations ofstructural circumscription are equivalent.

THEOREM 8. Under stnlctural circanncription with a signature O hal ing a

single priorip leuel, the preferred models of a theo~ T are the models of StrCirc( T1),

Q{) ), restricted to Q.

PROOF. Let M be a model of StrCirc(To, CiO). Suppose M’ is a model of T

that is preferred to the restriction of M to Q. We show that in fact M fails to

satisfy StrCirc( TO, 00), by constructing a copy of M’ within M’s test sort. In

other words, we find values for U and S‘ in formula 6 such that S‘ < S and

TJS’) both hold.

Circumscription with Homomotphisrns 851

By Proposition 4, we can assume without loss of generality that M“s

universe is countable. Since formula 5 guarantees that the test sort is infinite,

one can rename universe elements of M’ so that each element of M’ is a

member of the test sort of M. We assume that this renaming has been

performed.

To make a copy of M’ inside the test sort of M, we first pick out the

universe of M’: Let U in formula 6 be a predicate that is satisfied by an

element e iff e is in the universe of M’. We must also copy M”s predicate and

function interpretations to the test part of M. To do this, let S‘ in formula 6 be

the tuple of first-order predicates and functions such that

— for each predicate R in S‘, the extension of R‘ in M is the same as that of

R in M’; and

— for each function ~ in F, the extension of ~’ in M is identical to that of ~

in M‘, where their domains coincide; and f ‘(i) = O, elsewhere.

We now show that with this choice of U and S‘, M satisfies Tu( S ‘). Let a be

a quantifier-free formula with free variable x. If VX. a is true in M’, then

VX.U(X) + a is true in M, where the universal quantifier now ranges over the

test sort of M. Similarly, if 3x. a is true in M’, then 3x.U(X) A a is true in M,

where the existential quantifier now ranges over the test sort of M. Through

induction on the number of quantifiers, it follows that this holds for arbitrary a

in prenex form.

Further, if U(1) is true in M, then U( f‘( i)) is also true, by the definition of

U, for any function f from 0.

Finally, because M’ has a nonempty universe, 3x.U( x) is true in M for this

choice of U. We conclude that with this choice of U and S‘, M satisfies 7’U(S ‘).

We now show that with this choice of S‘ and U, M satisfies S‘ < S. Since

M’ is preferred to the restriction of M to Cl, there must be a homomorphism h

from M’ to the restriction of M to Q, and no homomorphism in the reverse

direction. Let h in formula 9 be any first-order function mapping from the test

sort to the original sort in M, such that h agrees with h where their domains

coincide (see Figure 7). On the test sort restricted to U, h preserves predicates

and functions of F, P‘, and Q, and therefore satisfies formula 7; in other

words, hom( h) holds.

If there were a function g satisfying formula 8, then g would also define a

map from M (restricted to Q) to M’, and preserve predicates and functions.

Therefore, g would be a homomorphism from the restriction of M to M’,

which is not possible since M‘ is preferred to the restriction of M to Q. By

formula 9, S‘ < S is true in M for this particular choice of h, U, and S‘; that

is, M is not a model of formula 6, a contradiction; so the restriction of M to Q

must be a preferred model of T.

For the reverse implication, suppose that M“ is a preferred model of T. We

show that M“ is identical to the restriction of a model of SM1-c(TO, Qo) to Q.

Let M be a structure over signature fltl that satisfies formula 5 and that

is identical to M when restricted to 0. Note that M satisfies all the formulasof T(S). Take a particular choice of U and S‘ such that M is a model

of T(S) A 7’U(S ‘). We show that S‘ < S, and the theorem will follow via

formula 6.

Let M’ be a structure over !2, whose universe is the subset of M“s test sort

that satisfies U. Let each function and predicate symbol p of 0 be given the

P. K. RATHMANN ET AL.

M

test sort J

FIG. 7. Diagram for proof.

interpretation in M’ that p’ had in M, restricted to elements in U. Then A4’

must satisfy all the formulas of T. For if there is some formula a of T that is

false in M’, then the “requantified” version of a in TU(S’) would be false in

M, which is not the case.

Further, since M“ is a preferred model, M’ is not preferred to M“. This

means that either there is no homomorphism h from the universe of M’ to the

universe of M“. or there is a homomorphism in the reverse direction. As M’

mirrors the structure of the test part of M, and M“ mirrors the structure of

the original part of M, this means that there cannot be a function h from the

test sort to the original sort of M that satisfies hem(h). unless there is also a

function g satisfying w~erseHom(g). Thus, formula 9 is false in M for this

choice of U and S’. Therefore, S’ < S’ is true in M for this arbitrary choice of

U and ,S’, and the theorem follows. ❑

Let us now consider the case where f) is a prioritized signature. We use

adjacency to denote set union, so that ABC denotes the union of sets A, B,

and C. To include a tuple P = [Pi, ..., P,l ] of disjoint sets of predicate symbols

to be minimized at levels 1 through ~Z in the second-order axiom for circum-

scription, we write a conjunction of single-level structural circumscriptions:

1=?7

= A StrCirc(T(,, [F, P, ~~~P,, Q. P,+l . ~. P,, J’. Arities, TestSor~] ). ( 10)~=1

Theorem 9 shows that the model-theoretic and axiomatic formulations of

prioritized structural circumscription are equivalent:

THEOREM 9. Under structural circumscription With u prioritized signature Q,

the prcfcmed models of a theo~ T are the models of StrCirc( Tc,, 0{] ), restricted

to 0.

PROOF. Suppose that M’ < M, for models A!l and ilj’ of T. Then. for some

priority level i, M’ + M and M + M’. Then, by Theorem 8, M is not a model

of StrCirc( T,), [F, P[ “”” P,, Q, P,+ , . . . P,lV, Arities, TestSort ]), so M is not amodel of StrCim( T,,, Q,,).

For the reverse direction, suppose that there is no model M’ < M. Then, at

each level i, there is no model M’ such that M’ + M and M + M’. There-

fore, by Theorem 8, M is a model of StrCirc(TO, [F, P, . . . P,, Q, P,+, . . ~ P,lV,

Arities, TestSort ]) for 1 < i < n. ❑


6. Consistency

It is most unpleasant to discover that one’s method of common-sense reasoning

allows one to conclude anything and everything. Unfortunately, this situation

arises with many natural preorders on models; for example, problems with

inconsistency in ordinary circumscription are well known [Etherington 1988]. In

this section, we show that for universal theories, structural circumscription

preserves consistency: If a universal theory has models at all, it will have

preferred models under structural circumscription. This holds when all predi-

cates are minimized at once, as well as the variations where certain predicates

are held fixed, allowed to vary, or minimized in a priority order. (Ordinary

circumscription also enjoys these properties [Etherington 1988; Lifschitz 1986]).

We also give examples of other types of theories where inconsistency can arise.

Circumscription produces inconsistency when a theory T has no preferred

models. This only happens when every model of T lies on an infinite descend-

ing chain Ml >Mz >Mj > . . . . where each model is preferred to the one

before. If for every model Ml, either Ml is preferred or there is a preferred

model that is preferred to Ml, then we say that T is well founded for its

preorder [Etherington 1988], and know that circumscribing T with this pre-

order will not produce inconsistency. Universal theories are well founded for

many interesting preorders, as are existential theories, Theories that are not

well founded may still remain consistent under circumscription; we give an

example of this below. Any theory that has a model with a finite universe will

remain consistent under circumscription, since the finite universe prevents

construction of an infinite descending chain.

We have already presented a number of consistency results for the case

where equality is held completely fixed, in Section 4.3. The results in Section

4.3 showed that when every universe element in a model can be given a name

(e.g., if the signature is sufficiently large), then structural circumscription will

preserve consistency if ordinary circumscription does.

Theorem 10 shows that under structural circumscription, when checking for

well foundedness it sufficies to consider the case of a single priority level.

THEOREM 10. Under structural circumscription, if T is a well founded theo~

for all signatures having a single ptioriy leuel, then T is well founded for all

signatures.

PROOF. Let M be a model of T, and let O be the prioritized signature

Q= [F, [PI,..., P,, ], Q, V, Arities]. Let ~, be the single-level signature

01 = [F, PI ““. P,, Q, PL+, .. . F’,lV, Arities]. Recall that prioritized circumscrip-

tion can be expressed as an intersection of single-level structural circumscrip-

tions:

~=~

Preferred models under Q = n Preferred models under !2,. (11)i=l

Finding a model that is preferred under all ~i is an iterative process. If M ispreferred under Cll, we set Ml = M: otherwise, we apply the well foundedness

of T at a single priority level to the first signature in Eq. (11), to find a

preferred model Ml under Q ~ such that Ml + M. We then use Ml as a

starting point for level 2. If Ml is preferred under flz, we set MZ = Ml;

otherwise, we apply the well foundedness of T at a single priority level to Q ~,


to find a preferred model A4z under ilz such that MJ - M(. We continue this

iteration, creating a sequence of models Ml, M?, ..., M,, such that M, is

preferred under 0,, and M, a M,_ ~ at priority level i.

The conditions for a mapping to be a homomorphism become progressively

more strict in successive conjuncts of formula 11, since fewer and fewer

predicates are allowed to vary. In particular, a homomorphism M, ~ Ml_, at

level i is also a homomorphism at all lower levels. By composition of homo-

morphisms, then, kl~ ~ M, at level i + 1, for all levels i. Since M, is preferred

under !2,, M,, is also preferred under f),. Thus, M,, is preferred under Q. It

remains to show well foundedness.

If M,l = M, then M itself is preferred. Otherwise, let k be the lowest

priority level such that M~ # M. By the construction of the sequence, M~ ~ M

and M * kf~ at level k. On the other hand, we know that M,, ~ h4k at levelk + 1, and by composition, M,, ~ M at level k. If M ~ M,, at level k, then by

composition M ~ ML at level k, which is not possible, so M * M,l at level k.

Therefore. M. < M under Q. Thus, for an arbitrary model M, if M is not

itself preferred, we have found a preferred model M,l such that M,, < M. u

Before proceeding further, let us give the canonical example of inconsistency

produced by ordinary circumscription, and discuss how structural circumscrip-

tion handles the same example.

Example 4. The blue integers problem. Consider the theory of the natural

numbers with successor: constant O, unary function S, and the two formulas

Vxy.s(x) = s(y) +x =y (12)

and

Vx.s(x) + o, (13)

plus the two formulas 3x. Blue( .t ) and V.I. Blue( x) - l?h~e(S(.~)). Blue is a

minimized predicate in the signature of T, so the preferred models are those

having as few blue numbers as possible.Figure 8 shows ~o models of T, Both models contain a number line (the

X1’S) and a ghost copy of the number line (the xi’s). Model A in Figure 8 differs

from model B only in that the first blue integer is Zgg in model A and z, ~,. in

model B. B is preferred to A under ordinary circumscription. There are

homomorphisms in both directions between .4 and B, so neither is preferred

over the other under structural circumscription. As one can always start

painting numbers blue at a yet later point on the number line, T has no

preferred models under ordinary circumscription: B is preferred to A, but

another model is preferred to B. and so on. Under structural circumscription,however, both models A and B in Figure 8 are preferred.

Under structural circumscription, it is not even necessa~ to include formulas

12 and 13 in T. Formulas 12 and 13 are true in all preferred models because

structural circumscription produces the equivalent of unique name axioms even

for nonground terms. As mentioned earlier, other forms of circumscription do

not have this property.

Theorem 2 showed the existence of a set of atomic formulas that differenti-

ates a given preferred model M from another promising model M’. In the

remainder of this section, we examine a variant on the blue integers example to

see why the differentiating set cannot always be finite.

Circurnscriptiotl with Homomo~phisms 855

Model A:

Universe = {z,, z, I z > O}

Function interpretations:

O=xrl

S(z,) = Xz+l,vz>0S(zt) = ,?*+l,V2 >0

Predicate interpretations:

Blue : zw, ZIOO, ZIOI,

Model B:

Universe = {z,, z, I z > O}

Function interpretations

O=xo

S(zt) = X,+l,vz>0S(.zz)= Z,+l, V2 >0

Predicate interpretations.

Blue : Zloo, ZIOI, ZIOZ,

Homomorphism h from A to B:

h(zt) = z,, Vz >0

h(z,) = ,z,+~, V7, >0

Homomorphism g from II to A:

g(z, ) = z,, Vt 20

g(z, ) = z,, QZ>0

— -> – –>blue blue

FIG. 8. Two models of the blue integers problem.

Example 5. A L’ai-iant of the blue integers problem. Let us change the blueintegers example by adding two additional constants, a and b, which are their

own successors. Let T contain the formulas

Vx.Bhe(x) + Bhe(S(x))

v.x3y.x = s(y)

S(a) =a S(b)=b

lBlue ( 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, w, such that the successors of w are all blue, For M. we let w be z~;~,the first blue element on the Z-chain.

11A Z-chain is a chain of elements related by the successor function, infinite in both directions

[Enderton 1972].

856

Model M:

Universe = {yI, z,, z, I 2 E -Z}

Function interpretations.

a=yl

b=yl

S(Y1) = yl

S(zt)= A+l, vzPredicate interpretations.

Blue : ,z,, VZ >99

0“ blue

a,b

M

P. K. RATHMANN ET AL.

Model A[’:

Universe = {y I,y2, z,, ~, I Z E Z)

Function interpretations.

a=yl

b=y,

IS(Y1) = y,

S(Y2) = !/2

S(zt)= Zt+],vzPredicate lnterpretatlons:

Blue : z,, V%

/-

M’L.

b

a,b Dblue

N

FIG. 9. Models of a variant of the blue Integers pmblcm

Let us see why w must lie on a Z-chain: Let N be a model of T in which

a = b and w does not lie on a Z-chain. Then, finitely many elements come

after (“succeed”) w under the S function. Therefore, S must “loop” in N:

S’(z) = z, for some z that follows w and some number i.

There is a homomorphism from M to N, mapping 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,M ~ N. We cannot have N ~ M, because the equality S’(z) = z is true in N

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 whicho = b md w lies on s Z-chsin, we conclude that no model where D = b is

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

map M”s Z-chain to M while preserving BILIE?. We conclude that M is a

preferred model. Because a # b is true in M’ and false in M, we haveM * M’, so M’ is also a preferred model of T.

Circumscription with Homomo@isms 857

Theorem 2 tells us that there is a set A of minimized atomic formulas such

that A has a satisfying assignment in M but not in M’. Let A contain the

formulas

Blue( w,) Blue( W2 ) Blue(w3) “.”

w, = S(W2) W2 = S(W3) W3 = S(W4) ““”,

saying that 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 AND UNIVERSAL THEORIES. Structural circumscription

preserves consistency for universal theories.

THEOREM 11. Let T be a satisfiable uniuer-sal theoy oler signature Q. Then

T is well founded under structural circumscription, and hence 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 M’ such that M’ + M.

Let % be the set of models satisfying T and possibly preferred to M, that is,

c2’={BIBk TAB+ M}.

The set X’ is partially ordered (by the < relation) and, since it contains M,

nonempty. By Zorn’s lemma, 12 3’ has a minimal member (i.e., contains a

preferred model) if every totally ordered subset has a minimal member in .7.Let Y = Ml > M, > M3 .-. be such a totally ordered subset of X. We now

construct a model-which is minimal for 5’.

Let W be the set of all negative ground literals 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.) Let wit(l) be a model of Y where 1holds, and for any finite subset W’ c W, let wit(W’) be the member of

{wit(l)ll G W’} that is smallest under <. By the parenthetical remark above,wit( W‘ ) is a model of W‘. As wit( W’ ) also satisfies T, let us add the formulas of

T to W, and every finite subset of W will still have a model.

The compactness theorem for first order logic (e.g., [Enderton 1972]) states

that if every finite subset of a first-order theory has a model, then the entire

theory must have a model. Since W satisfies this condition, it too must have a

model, which we call B‘.

Since T contains only universal sentences, and ground Iiterals are also

universal, W is a universal theory. Universal theories have the property that a

universe restriction of a satisfying model will also satisfy the theory. Therefore,

if we take B to be the universe restriction of B‘ formed by restricting itsuniverse to include only the extensions of ground terms, B will also be a model

of w.

‘2 If X is a partially ordered set such that every chain in X has a lower bound in X, then X

contains a mimmal element. See, for example, Halmos [1970].


Now, let us take a closer look at the model B. W includes a copy of T, so we

know that k~ T. For any M, E Y, consider the mapping h: B + M, formed

by mapping the extension of any ground term in B onto the extension of that

same ground term in M,. We might wonder whether this is a well-defined

function, whether it is possible that the mapping might relate one universe

element of B to two or more universe elements of M,. This could happen if

two ground terms tl and t2 denote the same universe element of B, but

different universe elements in M,. However, if this were the case, then 11 + t2

would be true in M,, but not B. This cannot happen since B satisfies W, a

theory that includes all sentences of the form t,# t,that are true in M,. By a

similar argument, the mapping h: B + M, preserves all minimized predicates

of the signature Q. Thus, h: B + M, is a homomorphism, for all M, in 5’.

Since for any M,, Ml + M, composition of homomorphisms gives B -+ M, and

hence B G 2? and B is a lower bound for L?’. Then, according to Zorn’s lemma,

3’ has a minimal element, M‘, in %, M’ is minimal for models with homomor-

phisms onto M, and hence is preferred.

We know now that M’ is preferred and M’ + M. To prove well-founded-

ness, we must show that either M is preferred, or a preferred model is

preferred to M. If M’ = M, then M is preferred. If M * M’, then a preferred

model is preferred to M. If M is not preferred and M + M’, we can reach a

contradiction as follows: Let N be a model of T that is preferred to M. SinceN+ Mand M+ M’, wehave N+ M’. If M’_N, then wehave M+ Nby

composition of homomorphisms, contradicting the fact that N is preferred to

M. Therefore, N is preferred also to M’, contradicting the assumption that M’

is preferred. We conclude that M * M’, and that structural circumscription

with a single priority level is well founded. ❑

6.2 CONSISTENCY AND EXISTENTIAL THEORIES. Recall that we assume that

our theories are finite. Under this assumption. structural circumscription

preserves consistency for existential theories:

TIHEOREM 12. Existential theories m-e well founded under str[lctural circunl -

scrip tion, so any satisfiable existential theoq has prefen-ed nlodels.

PROOF. By Theorem 10, it suffices to consider the case where there is a

single priority level. Let us handle fixed predicates by defining and minimizing

their negations. Then, Q contains a single priority level, no fixed predicates,

and possibly some varying predicates.

Let T be a finite set of existential sentences over Q. Take a model N of T;

let S be the set of all models of T having a homomorphism to N. This set isordered by the < relation. By Zorn”s Lemma, if every totally ordered sequence

from S has a minimal element in S, then ,S has a minimal element. Take a

totally ordered infinite sequence from S, possibly uncountable; we use O to

refer to an arbitra~ model on this chain.

Let a be a formula of T, by assumption in prenex disjunctive normal form.

Since a is true in all models on the chain, at least one disjunct of a must be

true in every model on the chain. There are only finitely many disjuncts in a,

and there are infinitely many models on the chain; therefore, there is some

disjunct of a such that for any model O on the chain, there is some model

following O on the chain in which that same disjunct is satisfied. That disjunct

must have the form /1 ~ “”” A 1,,, for each 1, a literal; if the prefix of a is


ax, ““” Xm, then let ~ be the formula Elxl “-” .Y,,l.ll A “”” A 1,,. Form a reduced

chain by removing from the chain all models in which /3 does not hold. Repeat

this process, reducing the chain further, until all formulas of T have been

considered; since T is finite, this process will terminate, leaving an infinite

reduced chain. Note that if ~ has been used in a reduction step, ~ is still true

in all models on the final reduced chain.

Let E be the set of all existential sentences a in prenex form, whose bodies

are conjunctions of literals, and such that a is true in all models on the

reduced chain. Loosely speaking, these are the sentences whose truth is

preserved under homomorphism along the chain. Note that all the ~ formulas

used during reduction are members of E.

Let E’ be a skolemized version of E, with corresponding signature Q‘ for

T u E’. Corresponding to each O on the reduced chain there is a model O‘ of

T u E‘, differing from O only in that the skolem constants in Q‘ are inter-

preted in O as witnesses for the existential variables of E from which the

skolem constants sprang. We speak of the skolemized chain, that is, the

sequence of skolemized models corresponding to the unskolemized chain; note,

however, that the process of skolemizing need not preserve the preference

ordering, since the homomorphisms on the unskolemized chain need not

preserve the skolem functions.

Consider the set of herbrand terms of Q‘. We construct a model M whose

universe is a set of equivalence classes over these terms, such that M satisfies

T and is preferred to O. Let the equivalence classes be formed as follows:

Terms tland tz over Q’ are in the same equivalence class, written [t, 1= [t21,iff (T u E’) K (tl = t2). The interpretation of a term t in M is [t]. A ground

atom a is true in M iff (T U E‘) + a. M satisfies all formulas of T, because

M satisfies the skolemized versions in E‘ of the formulas ~ used during

reduction, which in turn entail T.

We next show that M (regarded as a model over Q) is preferred to all

models on the descending chain. Consider the mapping h from M to any

target model O on the reduced chain, constructed as follows: Let h map the

interpretation in M of a term of 0‘ to the element of O that is the

interpretation of that same term in O‘. This is a well-defined mapping, because

if tj = t2 is true in M, then it is also true in O‘ and therefore in O, by the

construction of M. The mapping preserves functions by construction. Any

ground atom a over Q‘ that is true in M is also true in O‘, by construction of

M. Because every tuple of elements in a relation in M is denoted by a ground

atom a over Q‘, and O and O‘ have the same predicate extensions, h

preserves all predicates between M and O. Therefore, h is a homomorphism

from M to O. In addition, there is no homomorphism from O to M, because if

there were, then by composition of homomorphisms there would be a homo-

morphism from O to P, for P any model following O in the reduced chain, so

P could not be preferred to O. We conclude that M is preferred to all models

in the reduced chain. By the construction of the reduced chain, for any model

O on the full chain, there is a model P following O on the full chain. such that

P is also on the reduced chain. Therefore M is also preferred to all models onthe full chain. By composition of homomorphisms, there is a homomorphism

from M to N. By Zorn’s lemma, then, there is a model V in S that is minimal

under <. V must be preferred to N, as otherwise there could be no infinite

descending chain. If V is not a preferred model, then there is some model V’


such that V’ + V and J“ * ~7’. But then by composition of homomorphisms,

V’ + N, so V’ is also a member of S, contradicting our assumption that V is

minimal in S. Therefore, T is well founded, and has a preferred model if it has

any models. ❑

6.3 CONSISTENCY AND ALTERNATIONS OF QUANTIFIERS. As the example

below shows, some universal-existential sets of sentences do not have a

preferred model under structural circumscription. Interestingly, ordinary cir-

cumscription does produce a preferred model for this example.

Example 6. Nodes and paths. Consider the following finite set T of

universal-existential sentences. Let the signature be two-sorted. so that all

universe elements are either paths or nodes, but not both. (If a single-sorted

logic is used, the sentences are still universal-existential. ) The signature con-

sists of the constant node A: the equality predicate: at the same priority level, a

binary predicate prefix, whose first argument must be a node and second

argument a path; and unary functions left and right, which take a node as

argument and produce a node. Intuitively, the nodes are arranged in a

complete bina~ tree, with constant A denoting the root (see Figure 10). A

path through the tree may be traversed by following left and right links down

from the root. The elements of sort patlz are infinite-length paths through the

tree, beginning at the root; prefix( n, p) will be true if path p passes through

node n. We require that there be at least one path through every node. T will

fail to have a preferred model because one can always remove a single path

from a model of T, and still have a model of T.

T contains three formulas about nodes and four formulas about paths, listed

below. We use n to denote variables ranging over nodes, and p for variables

over paths.

(1) A is the root of the standard tree.

Vn.,1 # left(n) A A # rigllt(lz)

(2) Every node has two children.

Vn.left( n) + rigizt(n)

(3) Non-root nodes have exactly one parent.

Vn.n # A -+ 3n’.[it = left(n’) V n = right(n’)]

Vnn’n” .((n = left( n’) V n = rig/2 t(n’)) (14)

A (n = /eft(n”) V ?Z = rig~zt(n” ))) - /z’ = n“

(4) Every node has a path through it.

Vn Elp.prefm( n. p) (15)

(5) Every path includes the root of the standard tree.

Vp.prefti(A, p)

(6) Paths are nonbranching and of infinite length. ( @ is the exclusive-Orlogical connector.)

Vnp.prefti(n, p) - (prefti(left(n), p) @ prefi~(right(n), p)) (16)

(7) A path through a node also goes through the parent of that node.

Vnp.[prefti(n, p) A n # A] + 3n’.prefti(n’, p)

A(rz = left(rz’) V n = right( n’)) (17)

A (L)


left(L) right(L)

/\ Aleft(left( L)) right(left(L)) left(right(L)) right(right(L))

Afi ~ ~

...

FIG. 10. Model of the node-path example,

Let M be an arbitrary model of this set of sentences. We construct a model

M’ that is preferred to M, and therefore show that T has no preferred models.

Let M’ have a universe whose nodes are the herbrand universe for the

signature, that is, in which each term t is interpreted as a distinct universe

element also named t.The paths of M’ are equivalence classes over the paths

of M, formed as follows: Put two paths p and p‘ of M in the same equivalence

class, written [p] = [ p ‘], iff for all terms t,prefix( t, p) = prefix(t, p‘) holds in

M. The paths of M’ consist of these equivalence classes, minus one equiva-

lence class which we will call e. If a node-path tuple (n, p) is in the

interpretation of prefti in M, where [p] # e and n is the interpretation of a

term t of the signature, then let (t,[p])be in the interpretation of prefk in

M’.

We now show that M’ is a model of T. The only point of doubt is whether

M’ satisfies formula 15, the requirement that every node have a path through

it. Let n be a term denoting a node. By formulas 16 and 17, two paths p ~ and

pz must go through n in M, one heading for lefl(n) and one for right(n). These

two paths must be in different equivalence classes, as they have different

prefixes. Therefore, one of pl and p2, say pl, is not in equivalence class e, and

so [p, 1 is an element of M’. We conclude that [p, ] passes through n in M’.We now show that there is a homomorphism from M’ to M. Let h map the

interpretation of a term in M’ to the interpretation of that same term in M.

For a path p in M’, let II(p) be some path p’ of M such that [p’] = p. This

mapping preserves functions and it also preserves the truth of prejix; therefore,

it is a homomorphism.

Suppose there were a homomorphism h from M to M’. Pick a path p of M

that is in equivalence class e. Homomorphism h must map p to some other

equivalence class e‘. Let p‘ be some member of e‘, and let t be the ‘highest’

node in the standard tree for which prefk(t, p) # prefm(t, p‘) in M. Let s be

the term denoting the sibling of t in the standard tree. Path p must go through

one sibling, and p‘ through the other, by formula (16). Since k must preserve

the truth of predicates, both prefix( t, e‘ ) and prefix(s, e‘) must be true in M’.

But this violates formula 16, which we know M’ satisfies. We conclude thatthere is no homomorphism from M LO M’, that M’ is prcfcrrcd to AZ, and that

T has no preferred model under structural circumscription.

It may be instructive to see where the proof of Theorem 12 fails for the

node-path example. We step through (a simplification of) the construction used

in the proof of that theorem, to see where it breaks down. Let N be a model of


the node-path theory, and let S be the set of all models of the theory that

agrees with N except that they omit one or more paths that go through a

particular node (the interpretation of the term n ~) of N. The proof of Theorem

12 shows that for every totally ordered sequence of models from S, there is a

minimal element for the sequence in S. Consider a sequence of models from S,

such that each successive model in the sequence omits a superset of the paths

omitted by the previous model. This sequence of models is totally ordered and

clearly lacks a minimal element in S, because such a minimal element would

fail to satisfy formula 15, which requires every node to have a path through it.

Not surprisingly, formula 15 is a universal-existential formula.

The construction then produces a set E of existential 13 sentences that are

true all along the chain. In our case, one such sentence is =p.pre~i.x(nl, p). The

construction then skolemizes E, producing E’, and including the sentence

pre~id n,, p,), where p ~ is a new skolem constant. The construction thencreates a model M to be the minimal element for the entire chain. The

universe of M, in our case, will be all the nodes of the standard tree (as they

are terms in the original language), plus many interpretations of path skolem

constants. The problem with the construction comes when we reach the point

that says, a ground atom a is true in M iff ( T U E’) + a. What, then, is true

of p,? All models in the chain agree on the prefix of p,, up until p, reaches n.

After this point, the theory requires that p, be of infinite length, so

prefti(left( n, ), p,) or prefix( left( n, ), p,) must hold, and so on to infinity; butthe models in the chain cannot agree on the route taken by the path, Thus, N

does not satisfy the node-path theory, and so is not in S.

If ordinary circumscription rather than structural circumscription is applied

to the node-path example, one finds that the preferred models of T are those

in which there are no nonstandard nodes, that is, in which all nodes are

interpretations of terms. Inconsistency is avoided because ordinary circum-

scription cannot compare models with different universes, and the removal of a

path changes the universe of a model.

Interestingly. T has preferred models but is not well founded under ordinary

circumscription. The nonwell-founded models are those containing a nonstan-

dard “tree” of nodes, in addition to the standard tree. Paths in such models

have a standard part, which travels through the standard tree. and then may

also have a nonstandard part that traces through nonstandard nodes. One can

always remove the nonstandard prefix of a single path, and have the resulting

model still be a model of T. Yet one cannot remove all the nonstandard

prefixes of paths, due to the requirement that every nonstandard node have a

path through it. Thus, under ordinary circumscription there are no preferredtmodcls that are preferred to the nonstandard nlodcla in the node-path cxanr

pie. As all standard models are preferred, T has preferred models but is not

well founded.

Finally, a very slight change in T will make T inconsistent under ordinary

circumscription. The change is to require that there be a blue path through

every node; that is, alter formula 15 slightly:

VrI ~p.prefi~(n, p) A blue(p).

13 Using umvcrsol-existential sentences doesn’t help


There is no preferred model under ordinary circumscription because one can

make any single path of a model nonblue, without changing the universe, and

still have a model of T.

6.4 CONSISTENCY AND INFTNITE THEORIES. A set of sentences with no

alternations of quantifiers at all may fail to have a preferred model under

structural circumscription if the set of sentences is infinite. (In the other

sections of this paper, we explicitly disallow infinite theories.)

Example 7. Infinite encoding of the node-path example of Section 6.3.

Take a finite signature with a single priority level, over two disjoint sorts,

nodeLellels and paths. There is a nodeLellel constant O, and a unary successor

function S mapping nodeLevels to nodeLevels. There are also binary” predi-

cates left and right, whose first argument must be of sort path and second

argument of sort nodeLeLlel. The formulas of T state that:

(1) A path must branch right or left at each level of the tree.

Vpn.left(p, n) = T right(p, n). (18)

(~) For anY finite set of directions of the form “go right(left) at level k of the

tree,” there is a path through the tree that follows those directions. To

implement this, T contains every formula of the form

3p.left(p, cl) A ““” A lefi(p, c,, ) A right(p, dl) A “-. A right(p, dn, ), (19)

where the c,s and d,s are distinct nodeLeL’el terms, and n + m is greater

than O.

Let M be a model of T. We show that there is a model M’ that is preferred to

M, and that therefore T has no preferred model.

Let M’ have a universe whose nodeLevels are the terms generated by O and

S, with the interpretation of term t being t.The paths of M’ are equivalence

classes over the paths of M, formed as follows: Put two paths p and p‘ of M

in the same equivalence class, written [p] = [p ‘], iff for all nodeLet’el terms 11,

left( p, n) = left( p‘, n) holds in M. The paths of M’ consist of these equiva-

lence classes, minus one equivalence class that we will call e. If a

path –nodeLevel tuple (p, n) is in the interpretation of left in M, where

[PI + e and p is the interpretation of a path term t of the signature, then let(t, [n])be in the interpretation of left in M’. Let the interpretation of right in

M‘ be the complement of the interpretation of left there.

We now show that M’ is a model of T. Clearly, formula 18 is still satisfied.

Consider, then, a formula

~x.~eft(x, cl) A ““” A left(x, c,, ) A right(x, dl) A “.. A right(x, d,,, ) (20)

from the remainder of T. There is a path p in M that is a witness for formula

(20). If [PI + e, then [PI is an element in M’, and M’ satisfies formula 20. If

[PI = e, then pick a nodeLeuel term n that does not appear in the formula athand. Either left( p, n) or right( p, n) holds in M: we consider only the case


where le~t( p, n) holds, as the other case is symmetric. T also contains the

formula

3x.le~t(x, cl) /? ““. ~ le~t(x, c,, ) /! right(x, n)

~right(x, d,) A ... A right(x, d~). (21)

Let p‘ be the witness for formula 21 in M. Because paths equivalent to e

turn left rather than right at nodeLevel n, p‘ must be in an equivalence class

other than e, and so [p’] is an element in M‘. Since [p’] is a witness for

formula 21 in M’, it is also a witness for formula 20. We conclude that M’ is a

model of T.

We now show that there is a homomorphism from M’ to M. Let h map the

interpretation of a zzodeLez!el term in M’ to the interpretation of that same

term in M. For a path p in M’, let /z(p) be some path p’ of M such that

[ p‘1 = p. This mapping preserves functions and it also preserves the truth ofleft and riglzt; therefore, it is a homomorphism.

Suppose there were a homomorphism h from M to M’. Pick a path p of M

that is in equivalence class e. Homomorphism h must map p to some other

equivalence class e’. Let p’ be some member of e‘. Because p and p’ are in

different equivalence classes, there is some term t such that (t,p) is in the

interpretation of left and (t, p‘ ) is not (or vice versa; the other case is

symmetric). Since /z must preserve the truth of predicates, this means that (t,

e‘) is in the interpretations of both left and right in M’. But this violates

formula 18, which we know M’ satisfies. We conclude that there is no

homomorphism from M to M’, and that the infinite encoding of the node-path

problem has no preferred models under structural circumscription.

Not surprisingly, the construction of Theorem 12 breaks down in the same

manner for this example as for the other encoding of the node-path problem.

If ordinary circumscription rather than structural circumscription is applied

to this example, T remains consistent: all models of T are preferred. Because

left and right are held fixed, and ordinary circumscription also fixes the

universe and function interpretations, there is no room for improvement in any

model of T. To cause inconsistency with ordinazy circumscription, one can

introduce a unary predicate blae on paths, and add to formula 19 the require-

ment that p be a blue path, Then, one can make any single path of a model

non-blue, but infinitely many paths must remain blue.

In the other sections of this paper, we disallowed infinite theories, since the

usual AI applications of common-sense reasoning require a finite knowledge

base. This section has shown that infinite theories may be troublesome in other

aspects as well: An infinite theory with no alternations of quantifiers at all mayfail to have preferred models.

7. Otlzer Work otz Conznzon-Sense Reasotzing and Equalip

The original inspiration for structural circumscription came from Goguen and

Burstall’s [1985] concept of an initial model in category theory. Certain

theories T have a model 1, called the ilzitial model of T, such that 1 has

exactly one homomorphism h: 1 ~ M to every model M of T. The initial

model of T is unique up to isomorphism, and is the right choice for the

preferred model of T in a number of applications, including equational logic,

sets of horn clauses, and some programming languages.

Circunlscriptiotl with Homomorphisms 865

Initial model semantics does not work for common-sense reasoning, how-

ever, because a theory has at most one initial model (up to isomorphism), and

in common-sense reasoning we often expect to have many different preferred

models. For example, given the theory T containing just P(a) V H b), there is

no single model that our intuition prefers, though we do expect T to have at

least one preferred model. But T has no initial model at all. Similarly, the

theory containing just 3W.X # y has no initial model. More generally, theories

that include disjunction or existentially quantified variables often do not have

initial models. Nonetheless, the preferred models of structural circumscription

may be seen as a generalization of initial models.

Ordinary circumscription’s difficulties with respect to equality were first

described in Etherington et al. [1985], which points out that circumscribing the

equality predicate can add no new information to a theory. This inability is

perhaps best understood in terms of the model theory for circumscription.

Ordinary circumscription can only compare models that have the same under-

lying universe and extensions for constants and functions. Whenever models

differ on sentences involving equality, they have different underlying universes,

which means that ordinary circumscription will be unable to prefer one over

the other. For example, one cannot use ordinary circumscription to conclude

that Tweety flies, if it is known that Blutto is an ostrich, because one cannot

establish that Tweety # Blutto. lJ This problem with equality has been held as

a black mark against circumscription, since other common-sense reasoning

approaches such as default logic [Reiter 1980] and autoepistemic logic [Moore

1985] deal satisfactorily with equality.

In the past, many authors have sidestepped the problem with equality by

adopting a herbrand universe assumption. However, this is too strong a step for

common-sense reasoning, since it states that different terms must denote

different objects, so that, for example, tweety cannot be the same individual as

fiiendOf( syk]ester). We desire a mechanism that will allow us to conclude that

tweety = ,ft’iefzdof(,yk’ester ) if this is entailed by T.

Several authors have proposed responses to this difficulty, all using ordinary

circumscription. These solutions modify the theory to be circumscribed, by

adding names as a distinguished class of elements. However, as shown below,

these approaches will miss many highly desirable defduh conclusions regarding

equality of unnamed universe elements.

McCarthy [1986] takes the constants and ground terms of the theory to

denote, not elements, but names for elements. For each function symbol f, a

special constant ‘f is added. All constants (including the added function

names) are then made distinct by adding axioms of the form

C,+cj.

If n is the number of constants in the theory, a naive approach will require

O(nz) such axioms to ensure that all constants are distinct; more compact

encodings are possible. Ground terms are made distinct by axioms of the form:

Vxy.[f(x) =f(y)l --+ [x =Yl (22)

and

‘dx.fnmze(f(x)) = ‘f. (23)

14 The problem persists whether one treats equality as a built-in predicate, as we do, or defines a

predicate E to seine as the equality predicate, and applies ordinary circumscription to E.

866 P. K. R~THMANN ET AL.

Axiom 22 ensures that unless the arguments to a function are identical, the

function values will be distinct. Axiom 23 uses an added function jkzme, which

ensures that results from one function will be distinct from all others, by

labeling the results of functions with the function name.

So far, these axioms encode a herbrand universe assumption, which is too

strong. Therefore, a new binary predicate e is added, which is meant to

recapture the idea of equality. If e( x, JI) is true, x and y are considered

equivalent in some sense. Any sentences in the original theory with equality

are recast in terms of e, and axioms are added to make sure that e is an

equivalence relation, that e is preserved by functions, and that the truth or

falsity of a predicate is not changed by substituting equivalent arguments.

However, if e(.-c, y) is true, it does not mean that .1- and y are equal in the

traditional sense of denoting the same universe element. In particular, we

cannot expect that x and y can be substituted one for the other in any

second-order sentence whatsoever, preserving truth, though truth will be

preserved for first-order sentences. When e is minimized, one obtains unique-

name axioms for the e predicate.

A problem with this formulation is that, although it is intended to introduce

“names themselves as the only objects, ” the axioms do not ensure that there

are no unnamed elements. In some ways this is fortunate, since theories with

existentially quantified variables may require the existence of unnamed ele-

ments. However, since the circumscription is in terms of the names for

elements, unnamed elements are unrestricted by the circumscription. For

example, consider a variant of Example 1, in which it is known that normal

birds fly and that some (unnamed) bird does not fly. If we know that Tweety is

a bird, McCarthy’s approach will be unable to conclude that Tweety flies,

because the unnamed nonflyer might be Tweety. Structural circumscription

gives the conclusion that Tweety flies.

As another example, consider the empty theory with signature containing the

constant O and successor function S’. Although 3x. S(.~ ) = x is inconsistent with

a herbrand universe assumption, Axioms 22 and 23 do not rule out the

existence of such an x. All they require is that x not be denoted by a term.l~

Since S(x) = x ~ e(S(x), x), McCarthy’s circumscription will not imply the

sentence Vx. 1 e(S(x), x), a desirable unique-name-like axiom. Thus, problems

with unnamed elements restrict the kinds of universal sentences derivable from

this formulation. As mentioned in the blue integers example, structural circum-

scription will produce this axiom.

Lifschitz [1984] describes a different approach, using true equality rather

than a surrogate predicate. He uses a form of circumscription in which

functions are allowed to vary, as well as predicates. This allows one to reachdefault conclusions, both positive and negative, about equality, if such conclu-

sions help to minimize the extensions of minimized predicates. Lifschitz’

technique is presented in the limited context of encoding the unique names

hypothesis for a finite set of constants. As such, he allows only a finite set of

constant symbols, and no function or predicate symbols other than = and those

technical symbols (such as Ab) which are used in the construction itself.

Like McCarthy, Lifschitz introduces a set of constant symbols for names, as

well as axioms to ensure that all such names are distinct. For each constant

]5 We thank Jun Arima for pointing out this problem to us.


symbol C, of the original theory, he adds a name symbol ‘C,. He also adds a set

of axioms of the form ‘Cl # ‘C,, for all i #j. The original constants are meant

to be the denotations of the name symbols, and to formalize this relationship,

he defines a function denot and for each C,, adds an axiom denof( ‘C, ) = C,.

Finally, he adds an axiom saying that a pair of names with equal denotations is

abnormal:

Vly.[isname(x ) A isname( y) A denot( x) = denot( y)] ~ Ab(x, y),

where isname is an expression defined to be true precisely if the argument is

one of ‘Cl, ‘Cz, ‘Cj, . . . . ‘C,,. Thus augmented, the theory is circumscribed,

minimizing Ab and allowing the constants Cl from the original theory, as well

as the function denot, to vary. 16 Lifschitz proves that under this circumscrip-

tion, equality of the constants from the original theory is minimized.

An advantage of Lifschitz’s approach is that the equality is true equality, and

not a surrogate equivalence relation. On the other hand, some universal

theories become inconsistent under ordina~ circumscription when functions

are allowed to vary, so it is not clear whether this method of producing unique

name axioms might not also make T inconsistent. Structural circumscription

has the advantage of preserving consistency of universal theories, as well as

producing default conclusions about equality of unnamed as well as named

universe elements.

Lifschitz’s approach is explored in further detail by Arima [1988], who

generalizes Lifschitz’s method to signatures that contain functions other than

constants. Here also, there are two classes of elements, names and denotations.

To avoid Lifschitz’s reliance on a finite set of ground terms, Arima includes

axioms that assign a unique Godel number to each ground term, thus providing

a herbrand universe assumption for the names. He also uses a denotation

function from the names onto a defined class of elements. Arima intends his

approach to be suitable for signatures in which every universe element is

denoted by some term; when this is not the case, that is, there are unnamed

universe elements present, many natural default conclusions will be missed in

the circumscription, as described below.

Due to the presence of ghosts, structural circumscription does not give all

universal conclusions that can be obtained using ordinary circumscription.

However, the ghost elements in structural circumscription are better behaved

than the unnamed elements in default logic, autoepistemic logic, or McCarthy’s,

Lifschitz’s, or Arima’s approaches, because ghosts are guaranteed to be “like”

desirable elements of models: Ghosts map onto the desirable elements under

homomorphism. Under autoepistemic logic and default logic, due to the usual

means of constructing extensions using ground terms, one cannot in general

draw any conclusions about the behavior of unnamed elements, other than that

they satisfy the original theory. The same holds in McCarthy’s, Lifschitz’s, and

Arima’s approaches, due to the reliance on names during circumscription. For

example, consider a revised version of the Tweety theory in Example 1, that

includes the additional sentence

Vx.oswtclz(x) - (bird(x) A nji’ies(x)).

1“ Under structural circumscription, one can also allow functions to vary, a la Lifschitz [1985], by

eliminating the requirement in tbe definition of horn and reL)erseHom that the varying functions

be preserved. We have not explored this extension.

S68 P. K. RATHMANN ET AI..

Ab and = are minimized at level 1, and bird at level 2.17 Given the information

that Tweety is a bird, and no information about Blutto, structural circumscrip-

tion will conclude that Tweety flies, that there may be other birds but none of

them are named, and that all birds fly. Default logic and autoepistemic logic,

and McCarthy’s and Arima’s approaches (we are not sure how to apply

Lifschitz’s circumscription to this type of theory), will not conclude that all

birds fly. Ordinary circumscription will conclude that all birds fly and that

Tweety is the only bird. Told also that Blutto is an ostrich, and that ostriches

are nonflying birds, structural circumscription will no longer conclude that all

birds fly, but will conclude that Tweety does fly. Similarly, given a theory whose

signature has no ground terms, that is, in which elements are described by

existential statements rather than by name, structural circumscription will

conclude that the descriptions are of different elements, whenever possible: the

other nonmonotonic reasoning approaches will not. For example, told that

there exists a bird and there exists a bachelor, where both predicates are

minimized, only structural circumscription will conclude that the bird and the

bachelor are different individuals. Since applications cannot always assign a

name to all universe elements, we suspect that guaranteeing “good” behavior

of unnamed and existentially identified elements, a la structural circumscrip-

tion, will be helpful in applications.

The dependence of default logic on ground terms has been noted by authors

who have proposed mechanisms to remove the dependency and allow default

conclusions about unnamed universe elements, by giving a meaning to open

defaults (defaults that may contain free variables). In the original proposal for

default logic. Reiter [1980] suggests an approach to open defaults that essen-

tially skolemizes’s the default theory, creating a set of defaults in which every

possible universe element has a name. Reiter notes, however, that this ap-

proach is not quite satisfactory, as it produces some unexpected and unintuitive

default conclusions. Our model-theoretic bias also makes us a bit uncomfort-

able with this approach, because while Skolemization preserves derivability, it

may make radical changes in the models of a theory. For example, we have

seen that both ordina~ and structural circumscription do not always have a

preferred model for a theory T; but if T is then skolemized, both kinds of

circumscription will have preferred models, as T will then be a universal

theory.

As another proposed solution to the problem, Lifschitz [1990] introduces a

mechanism by which extension construction operations, such as the application

of default rules, can be applied directly to universe elements, rather than only

to terms. Lifschitz does this by requiring the user to supply both the default

theory and a listing of the elements of the universe, and only consideringmodels that have that particular universe; this is a stronger restriction than one

would ideally like to make. The remainder of our comments on default and

autoepistemic logic assume that some means is being employed to allow

defaults to be applied to unnamed universe elements.

Etherington et al. [1 991] also address several of the problems mentioned

above, and devise a solution by restricting the scope of the circumscription to a

‘? This minlcs the hierarchy of abnormality wherein Ixrds are abnormal animals because they fly,

a!ld ostrlchcs tire abnormal birds because they do not fly.

‘~ See Enderton [1972] for a dmwssion of Skolemlzation.

Circumscription with Honzomo~hisms 869

subset of the universe of the models under consideration. Etherington et al.

[1991] suggest that the elements included in the scope should be those

individuals of current interest to the user. They apply this technique to

ordinary circumscription, default logic, and autoepistemic logic; one could also

add a scoping mechanism to structural circumscription, by only requiring that

homomorphisms between models preserve predicates and functions for uni-

verse elements that are included in the scope. With either structural or

ordinary circumscription, the use of scoping introduces the problem of deter-

mining what elements to include in the scope; a general approach to determin-

ing scope has not yet been developed. On the other hand, a finite scope

guarantees that circumscription will preserve consistency [Etherington et al.

1991], a very desirable property. We have not investigated the effect of adding

a scoping mechanism to structural circumscription, and so will compare here

the features of plain structural circumscription and scoped ordina~ circum-

scription. To make the discussion clearer, we continue using the example in the

previous paragraph about Tweety and Blutto.

With scoped ordinary circumscription, when the scope is just Tweety, one

obtains the conclusion that Tweety flies, if Tweety is known not to be the only

individual: with scoping, named or unnamed counterexamples outside the

scope, like Blutto the ostrich, no longer prevent desirable default conclusions,

as long as the theory guarantees that there are enough universe elements

outside the scope for all exceptional properties to be assigned to them rather

than to elements inside the scope. In addition, scoped circumscription avoids

the conclusion that Tweety is the only bird. Structural circumscription also

offers robustness in the fkce of counterexamples, and avoids concluding that

Tweety is the only bird; it does this by ascribing exceptional properties to

unnamed, anonymous universe elements when possible. When Tweety and

Blutto are both in the scope or ordinary circumscription, the conclusion that

Tweety flies will be blocked, because Tweety may be the same bird as Blutto;

structural circumscription does not have this problem, because of its ability to

conclude unique name axioms by default.

Etherington et al. [1991] point out that traditional nonmonotonic reasoning

approaches conclude that Blutto is the only ostrich and Tweety and Blutto the

only birds, because that way the remaining individuals in preferred models are

more normal. With scoped ordina~ circumscription and a scope of just Tweety,

preferred models may contain nonflying birds other than Blutto. Even if Blutto

is not known to be an ostrich, scoped preferred models can still contain

nonflying birds. More generally, with scoping, one no longer reaches the

conclusion that individuals are normal whenever possible, and so exceptional

classes may be nonempty, even if no individual is specified to belong to them in

the theory. Under structural circumscription, there may be unnamed birds,

unnamed ostriches, and other unnamed nonflying birds in preferred models;

such individuals are ghosts of Tweety or of Blutto. With structural circumscrip-

tion, all abnormal individuals must either be required to exist by the theory or

else be ghosts of abnormal individuals that are required to exist by the theory,

so exceptional classes such as the ostriches are empty by default, unless thetheory entails that some individual such as Blutto has the exceptional proper-

ties. Furthermore, structural circumscription disallows new combinations of

exceptional properties, as in the cat-bachelor example used above; scoped

ordinary circumscription would allow bachelor cats in preferred models.


Lastly, scoping offers a solution to the lottery paradox. Under traditional

approaches to nonmonotonic reasoning, one cannot conclude by default that a

particular ticket sold for a lottery will not be a winning ticket; such a

conclusion would lead to inconsistency, since some ticket must win, and we

could conclude by default that no particular ticket could win. Scoped ordinary

circumscription allows one to conclude that the tickets placed in the scope will

not win, as long as one ticket remains outside the scope. 1y Whether structural

circumscription can reach this conclusion depends on the encoding of the

lottery example. If all tickets are named, then structural circumscription

concludes that any ticket might win. If not all tickets are named, then

structural circumscription will conclude by default that, for example,

ticket( Marianne) will not win. Thus, structural circumscription handles situa-

tions like the lottery paradox by ascribing exceptional properties to unnamed

universe elements, in preference to named elements. This confirms a conjec-

ture of Perlis [1986] that the lottery paradox (called the cou~zterexarnpleproblern

in Etherington et al. [1991]) can be handled in a manner that also deals with

the unique names problem.

We have seen that a minimal universe assumption can be implemented in

many cases using structural circumscription, by minimizing the predicate + .

Ordinary circumscription cannot be used directly to produce a minimal uni-

verse assumption [Etherington 1988], though one can use a closely related

technique, called domain circumscription [Davis 1980; McCarthy 1977, 1980].

to do so. The implementation via structural circumscription has the advantage

of not introducing inconsistency when there are isomorphic models having

minimal universes.~o For example, if T is the two sentences 3X Vy.x + S(y)

and V.xy. S(x) = S(y) ~ x = y, then domain circumscription produces inconsis-

tency, because one can always “place zero later.” Under structural circumscrip-

tion, minimizing # gives the standard model (up to isomorphism) of the

natural numbers.

8. Conclusiotls

In this paper, we have investigated the advantages in default reasoning of the

use of a preference ordering on models that is based on homomorphisms

between models, rather than the model/submodel relationship usual in cir-

cumscription.

Our first and most obvious conclusion is that the use of homomorphisms

allows us to reach desirable default conclusions about equality that are

unobtainable using a model /submodel relationship. Structural circumscription

easily generates unique-name axioms, and also allows one to conclude bydefault that unnamed individuals are distinct. Other forms of default reason-

ing, such as simple versions of default and autoepistemic logic, also easily

1“ This example underlines the impact of the choice of scope, since if all but one lottery ticket arc

included in the scope, it would be unreasonable to conclude that none of the tickets in the scope

will win. (Similar considerations apply with respect to ticket names under structural circumscrip-

tion. ) A solid theory of scope is needed for one to be able to use ~coping with confidence.

2(1Moinard and Rolland [1991] showed that the last step of the process of minimizing the domain

vm ordinary circumscription won’t make the result inconsistent, if previous steps in the process

have not already introduced inconsistency. In other words, lt is the mmunization of the new

predicate U, introduced to represent the universe of a model, that causes inconsistency under

ordma~ circumscription.

Circumscription with Homomo@zisnzs 871

generate default unique-name axioms, but cannot be used to conclude by

default that two existentially-specified unnamed universe elements are distinct.

When the language of the theo~ does not include any constants, the ability of

structural circumscription to reach default conclusions regarding the equality

of unnamed universe elements is crucial; in this situation, simple approaches to

default and autoepistemic logic, and previously proposed circumscriptive ap-

proaches, will not give the desired default conclusions about equality. The

“scoping” mechanism [Etherington et al. 1991] that has been proposed for

traditional nonmonotonic reasoning approaches to overcome this shortcoming

will need a comprehensive theory of scope to make it safely usable; the

semantics for open defaults that have been proposed to overcome the same

deficiency are not fully satisfactory, either.

Second, the subset inclusion tests used in ordina~ circumscription do not

extend well to the case of infinite predicate extensions or infinite universes;

consider, for example, the blue integers problem (Example 4 of Section 6, or

the discussion of the definition of “minimal universe” in Section 4.3). The

difficulty is that when predicate extensions are infinite, there may well be an

infinite descending chain of submodels, each isomorphic to the one before, and

each with a smaller predicate extension when measured by set inclusion.

Homomorphisms handle these cases easily, as two isomorphic models are

always equally preferred.

Third, it may be argued that homomorphisms in general are a better means

of gauging similarity between models than the model\ submodel relationship.

Homomorphisms are often thought of as behavior-preserving mappings, so that

if two models are related by a homomorphism, they share similar behavior.

Although ordinary circumscription does locate those cases of similar behavior

that are manifested in a model/submodel relationship, it misses many other

cases of similar behavior by considering too many pairs of models incompara-

ble. For example, ordinary circumscription considers two models incomparable

if they differ on whether Tweety = Blutto, and so ordinary circumscription

cannot be used to prefer models satisfying the unique name axiom Tweety +

Blutto to models where ‘rweety = Blutto. As mentioned above, homomor-

phisms also appear to be more robust in handling infinite predicate extensions.

Fourth, structural circumscription extends naturally to the case where one

wishes to assume by default that the universe is “as small as possible,” that is,

that only those elements that must exist do exist. Traditionally, this assumption

is implemented by an additional round of a different type of circumscription,

that checks to see whether the universe of a model could possibly be subsetted.

The use of a subset test when the universe is infinite suffers from all theproblems mentioned above in connection with infinite predicate extensions.

Homomorphisms appear to be much better behaved, though we have not been

able to obtain as precise a characterization of the behavior of structural

circumscription in this area as we would like. By including “ # “ among the

minimized predicates (and thus minimizing the universe), structural circum-

scription will conclude by default many universally quantified sentences that

are not obtainable using default or autoepistemic logic, such as drawing thedefault conclusion VX.S(X) # x from the empty theory. When the predicate

‘< # “ is not minimized, however, ordinary circumscription gives many default

universal sentences that are not obtained with structural circumscription, and

which may be desirable in some applications. These conclusions (such as

872 P. ~. RATHM~NN ET AL.

VX.XM(X) = (.x = Mutto) in the Tweety theory of Example 1) are blocked by

the presence of ghost universe elements in preferred models under structural

circumscription.

Lastly, structural circumscription appears to be no more likely than other

types of circumscription to introduce inconsistency into the circumscribed

theory by having no models preferred overall. We have given examples of

theories where ordinary circumscription gives models that are preferred over-

all, and structural circumscription does not (and also theories illustrating the

reverse); but in all the examples of this nature with which we are acquainted, a

conservative extension of the original theory (adding one new predicate) will

leave ordinary circumscription with no models preferred overall. Structural

circumscription never prefers one isomorphic model over another, a feature

lacking in ordinary circumscription.

ACKNOWLEDGMENTS. We would like to thank Moshe Vardi, Joseph Goguen,

Phokion Kolaitis, Vladimir Lifschitz, Jun Arima, Yoav Shoham, Ken Forbus,

Peter Wegner, Don Perlis, Fangzhen Lin, and the anonymous referees for their

helpful advice. Joe Goguen invented the concept of “initial model” in category

theory; he also suggested that we investigate free extensions over a variety, to

produce the circumscription semantics where certain predicates are allowed to

vary. Yoav Shoham suggested an investigation of the tradeoffs between deriva-

tion of universal sentences and minimization of equality. Phokion Kolaitis

helped us to understand our model theory when we found ourselves confused.

Moshe Vardi, Vladimir Lifschitz and Ken Forbus generously volunteered to

read our early manuscripts, and gave valuable comments.

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Circumscription with homomorphisms: solving the equality and counterexample problems

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