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The importance of open and recursive circumscription
ARTIFICIAL INTELLIGENCE
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RESEARCH NOTE
The Importance of Open and Recursive Circumscription Philippe Besnard* and Yves Moinard*
IRISA, Campus de Beaulieu, 35402 Rennes-Cedex, France R o b e r t E. M e r c e r * *
University of Western Ontario, London, Ontario, Canada N6A 5B7
ABSTRACT
Circumscription is known to result in an inconsistency when applied to certain consistent theories. To counter this problem, closed nonrecursive circumscription, a restricted form of circumscription that has been proved not to affect the consistency of the theory over which circumscription is applied, has been proposed. We show that closed nonrecursive circumscription involves an excessive weakening of standard circumscription by establishing that closed nonrecursive circumscription is incomplete for some crucial theories over which standard circumscription is consistent and complete. First, we prove that closed circumscription cannot yield the desired uniqueness formula for the simplest of existential theories. Second, we prove that nonrecursive circumscription fails to be as strong as predicate completion for Horn clause theories. Third, we prove that the natural way to strengthen circumscription, that is, adding more variable predicates, may weaken nonrecursive circumscription.
I. Introduction By means of the so-called circumscription schema the circumscription of a predicate in a theory tends to reduce the relation denoted by that predicate to a minimum. Because circumscription results in an inconsistency when applied to certain consistent theories, Mott [12] suggests imposing two restrictions on the standard circumscription schema. Firstly, Mott proposes what we call closed circumscription where only those instances of the standard circumscription schema that are closed formulas are acceptable. Secondly, Mott proposes * This research was supported by CNRS PRC GRECO IA. **This research was supported by NSERC grant 0036853.
Artificial Intelligence 39 (1989) 251-262 0004-3702/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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what we call nonrecursive circumscription in which the only acceptable instances of the standard circumscription schema are those in which no substituting formula involves the predicates being circumscribed or allowed to vary. Mott proves that the combination of these restrictions is sufficient to ensure that the resulting circumscription (which is thus closed and nonrecursive) always yields a consistent set of theorems when applied to a consistent theory. The purpose of the present paper is to indicate how significantly Mott's circumscription schema reduces the class of theories over which circumscription is complete with respect to the standard model-theoretic approach to circumscription, namely minimal models. Indeed, the present paper exhibits very simple theories over which standard circumscription yields the desired theorems without inconsistency but over which Mott's circumscription is incomplete. In fact, we demonstrate that each restriction on its own is significantly weaker than standard circumscription for various theories for which standard circumscription already preserves consistency.
2. Circumscription McCarthy [10, 11] introduces circumscription, in order to express some commonsense problems formally. Let 3- be a finite set of first-order formulas in which the finite lists of predicates P and Q occur. P = (P~ . . . . , Pm) and Q=(Q1 .... ,Q,). The (first-order) circumscription of P in 3-, with the predicates of Q as variables, is accomplished by adding the following circumscription schema to 3-:
{ 3-[p, q] A Vx(p[x] ~ P(x))} ~ Vx(P(x) ~ p [ x ] ) , where p = (Pl . . . . , Pm), and q = ( q l , . • • , q,) are lists of first-order formulas in the language of 3-. 3-[p, q] is 3- except that each occurrence of Pi and Qj is replaced by Pi and q/, respectively. The notation Vx(p[x] ~ P(x)) is to be interpreted as: Vx I . . . Xm(Pl[Xl] D e l ( X l ) ) A ' ' " A (pm[Xm]~ em(Xm)). The notation Pi[Xi] m e a n s that pi may have free variables other than x i = (X~a,..., xiti), where lg is the arity of P~, for i = 1 . . . . . m. The notation p(x), which is used in Mott's restricted circumscription, is interpreted in a manner similar to the notation p[x] with the restriction that each p~ cannot have any extra free variables. We write Cireum(3- : P; Q) for the union of ff and these axioms. If Q is empty, we denote the circumscription of P in if, as introduced in [10], by Circum(3- :P). An instance of the circumscription schema is the result of substituting formulas for each of the p~ and q/. Under standard circumscription any formula can be substituted for the p~ and q~. Mott suggests restricting the formulas
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available for substitution in two ways. The first restriction replaces p[x] with p(x) in the circumscription schema, that is, no pi(xl) contains free variables other than those in Pi(x~). This version of circumscription we call closed circumscription (cf. "closed formula") since no instance of the circumscription schema will have unbound variables. The notation for closed circumscription will be Cireumcl. Circumscription without this restriction is open. Closed circumscription is used by Mott to prevent the following kind of inconsistent circumscription that can result from a consistent theory.
Example 2.1.
3xR(x) , Vx7 L(x, x ) , Vy3xL(y, x), VyVx(L(y, x) ~ R(x)) . The case to consider is Circum(8-1 : R ). If we were to substitute L(y, x) for r[x] we would get VyVx(L(y, x)~ L(y, x)) as the fourth sentence in fix[r]. This allows the derivation of the sentence VyVx(R(x)~ L(y, x)) which together with the first and second axioms of 31 produces an inconsistency. This cannot happen if extra free variables (like y) are not allowed in p or q. As shown in a later section, extra free variables are crucial, so rather than taking the closed circumscriptive approach, we suggest that this example simply indicates that no extra free variable in p or q should get bound in i f [ p , q]. C i r c u m ( ~ : R) is not inconsistent if L(z, x) is substituted for r. The (simplified) circumscription schema that results from this instance is
Vx[(~yL(y, x)) D L(z, x)] D Vx(R(x) D L(z, x)) . That is, if there exists a special element, denoted by z, which is an Lantecedent of every element having at least one L-antecedent, then every R-element has this z as an L-antecedent. Clearly, requiring the same z for every R-element is a specific instance of the expected result: Vx(R(x) 3yL(y, x)) (every R-element has an L-antecedent, not necessarily the same one for all R-elements). The choice of 3yL(y, x) for r gives an instance which allows this more general result to be derived, even from Circumcl(ff1 : R). It should be noted that it is the norm rather than the exception that classical mathematical systems prohibit the binding of free variables in substitutional frameworks. As pointed out by an anonymous referee, a strong argument that this normal method of substitution is implicitly intended in any definition of circumscription is that the soundness results for circumscription (see Theorem 2.3 below) no longer hold if the binding of free variables is allowed. In particular, there are R-minimal models for ~-1; however, Mott's nonclassical
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form of substitution generates a circumscription of R in 3-1 which has no models because it is inconsistent. The second restriction prevents any of the Pi or Qj from appearing in the formulas used to substitute for the pi and qj. This version of circumscription we call n o n r e c u r s i v e c i r c u m s c r i p t i o n . The notation used will be Cireum,~. This restriction prevents inconsistencies of the type first presented in [7].
Example 2.2. 3x(N(x) ^ VyN(y) D x ~ s(y)) , Vx(N(x) D N(s(x))) , V x V y ( s ( x ) = s ( y ) D x = y) .
Indeed, when circumscribing N, if we were to substitute 3 y ( N ( y ) ^ x = s ( y ) ) 1 for n [ x ] , the corresponding instance of the circumscription schema would give the sentence V x ( N ( x ) D 3 y ( N ( y ) ^ x = s ( y ) ) ) which together with the first axiom of 3-2 produces an inconsistency. Standard circumscription need not be closed or nonrecursive. Mott's closed and nonrecursive circumscription, denoted Cireumd~.r, preserves consistency [12]: Circumcl&n~(ff)~-/ only if ~ - ~ - 1 . That is, the closed nonrecursive circumscription over a consistent theory always yields a consistent theory. This is an important property if circumscription is blindly applied to a theory (unwittingly producing an inconsistent theory would be very troubling). However, an equally important property for circumscription is completeness: the circumscription of a theory corresponds to the minimal models of the theory (see below). In general, consistency and completeness for circumscription are incompatible since completeness requires that any consistent theory not having a minimal model (for example, 32) have an inconsistent circumscription. Although we must inevitably abandon completeness in the general setting [4], we should not overbalance the desirability for consistency guaranteed by a new form of circumscription with the needless abandoning of completeness where it can be accomplished. The next sections show that in this respect Mott's restricted form of circumscription is quite unsatisfactory. For convenience, the definitions of minimal model, and the soundness and completeness of circumscription are given next. Let ~t and X be two models of 3- over the language of 3-. We write :R < e;oW when A/and W are identical except that: 1Mott chooses ( N ( x ) ^ 3y(x = s(y)) for n, but the corresponding instance does not produce inconsistency. A model of 5r2 and this instance is: the structure is the standard natural n u m b e r s ;
N(x) ,~. ~ ~ ~ - {0}.
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the extensions of the Pi a r e such that Ieil ~ IPilx for 1 ~< i ~< m, with some Ie, l~ C Ie, l~, where IP, I~ denotes the extension of predicate Pi in the model ~t; there is no condition on the extensions of the Qj for 1 ~
-
Theorem 2.3 (Soundness of circumscription) [6, 8, 10]. Every model o f 9which is minimal for < e;o is a model o f Circum(9- : P; Q).
If 9- is such that the converse is also true, then we have the completeness o f circumscription. [13] is devoted to this problem. In order to obtain the converse of T h e o r e m 2.3 a different notion of minimality needs to be defined, one that corresponds precisely to what is expressed by the circumscription schema. The circumscription schema of P in 3- (for the sake of simplicity only one unary predicate is circumscribed and no predicate is allowed to vary) expresses that, in a model A/, if a set denoted by a formula p could be taken (according to the axioms of 9-) as a possible extension for P and if this set were contained in the actual extension of P in ~t then this set is indeed the extension of P itself. That is, the circumscription schema realizes the minimization of the extension of P in models of f f with respect to definable sets (given a model, a subset of the domain of the model is a definable set if it consists of exactly those individuals from the domain that satisfy some formula): in a model, only definable sets are taken into account regarding the determining of a minimal extension for P. In terms of an ordering over models of 9-, such a minimization ranging over definable sets is achieved by means of a relation, < 8P- T h e o r e m 2.4 states the formal equivalence between both approaches to such a minimization of P. In order to discuss definability with respect to the different kinds of circumscription introduced above, it is necessary to introduce ~-definability. Let ~ be a set of formulas dependent on the kind of circumscription at hand. The set IPil~ is ~-definable in N when there exists a first-order formula Fi(Xl, • • •, xt i, Yl . . . . . Ys,) in the set of formulas o~, where li is the arity of Pi, such that for some e~ . . . . . esi in the domain of N (or equivalently ~t), we have:
IP, I~ = ]Fi(x I . . . .
, xt,, e l , . . . ,
esi)lN •
This notion of definability (in models) is not to be confused with the notion of definability in theories as examined in [5]. As mentioned above the set ~ depends on the kind of circumscription concerned:
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- F o r standard circumscription, ~ denotes the set of all formulas in the language. 2 ~-definability in N is then definability with parameters in N. - F o r closed circumscription, ~ denotes, for each circumscribed or variable predicate, the set of all formulas which have no free variables (except the defining variables). ~-definability in N is then (classical) definability (without parameters) in N. - For nonrecursive circumscription, ~ denotes the set of all formulas with no occurrence of the circumscribed or variable predicate(s). ~-definability in Y is then definability with parameters in a restricted language in A;.
IOjl
If in addition t o t h e requirements for < P;Q each [eil~ and each is ~-definable in N, then we write ~ < e;Q N. If Q is empty, then we write < ~(~) (or < P for the relation between models used in Theorem 2.3). P The semantics induced by < e;Q may be called the standard semantics of circumscription, as it closely matches our intuitions regarding the minimization of predicates. The precise semantical characterization for circumscription is given by the more sophisticated relation < P:O : Theorem 2.4 [1]. The models o f Circum(~- : P; Q) are the models o f S minimal for < 8(~) p;Q , where ~ is the set o f all formulas in the language. Similar relationships hold for Circumcl(J- : P; Q) and Circum,r(9- : P; Q), where ~ is appropriately chosen. ~. ~ ( ~ )
For every theory, a model minimal for < e;~ is also minimal for - p ; e . If is such that the converse also holds, then we have the completeness of circumscription. In the next section we give an example of the incompleteness of closed circumscription for the simplest of existential theories. In this particular example closed circumscription generates a theory with models which are minimal for < p8(~) , but which are not minimal for < p. 3. The Need for Open Circumscription Example 3.1.
3xP(x) .
First we examine standard circumscription. Circum(~-3 : P) gives the expected uniqueness result: V x V y ( ( P ( x ) ^ P ( y ) ) D x = y) . 2In all the examples below, the language is taken to consist of the formulas whose only nonlogical symbols are " = " , the ones that occur in the theory, and an infinite set of variables.
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One proof goes as follows: p[x] = x = y. T h e corresponding instance of the circumscription schema is: [:Ix x = y A V x ( x = y ~ P(x))] ~ V x ( P ( x ) ~ x = y) . 3x x = y is easily provable, and so is P ( y ) ~ V x ( x = y ~ P(x)), thus we obtain: P(y)DVx(P(x)Dx=y), by universal generalization we get V y P ( y ) 3 V x ( P ( x ) ~ x = y) which is logically equivalent to the uniqueness result. So, in every model of 0-3 minimal for < e the extension of P is a singleton. Now we turn to closed circumscription. Theorem 3.2. Circumcl(0-3 : P ) ~- V x V y ( ( P ( x ) A P ( y ) ) ~ x = y). Proof. Let M be a model of 0-3 with domain D such that c a r d ( D ) = 2 and ]Pla~ = D. We prove that 0 and D are the only subsets of D which are off-definable in M without extra free variables. Notationally, if e E D then the other element of D is denoted e -1. If R k is the relation over D k (k I> 0) defined by a formula F with k free variables, then we can prove by induction on the length of the formula F that (e~ . . . . . ek) E R ~ if and only if (e~-1, . . . , e~-1) E R k. Indeed it is true for atomic formulas P ( x ) , x = x, x = y, y = x and it is easy to show that it is stable for 7 , A, and 3. Now, I~ cannot be taken as IPl~ for any model 2¢" whose domain is D (cf. 3 x P ( x ) ) . So Theorem 2.4 applies and M is a model of Circumd(0- 3 : P). But in M we have the true sentence 3 x 3 y ( P ( x ) A P ( y ) D x ~ y ) . The soundness theorem of first-order logic gives the theorem. [] Theorem 3.2 demonstrates that in order to obtain completeness for 0-3 it is necessary that substitution instances be allowed extra free variables. 4. The Need for Recursive Circumscription Let us now examine the completely skolemized version of Example 2.2. Example 4.1.
0-4: N ( 0 ) , Vx(N(x)
0
s(x)) ,
Vx(N(x) N(s(x))), V x V y ( s ( x ) = s( y) D x = y) .
Because 0-4 is a universal theory, circumscription is guaranteed to be consistent [3,7]. Taking n [ x ] - [ N ( x ) A x ~ s k ( x ) ] gives, in particular, the expected result: V x ( N ( x ) D x ~ s~(x) (for every k ~ 1)).
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T h e o r e m 4.2. CircUmnr(J-4 : N ) ~- V x ( N ( x ) ~ x ~ sk(x)), even f o r k = 1. Proof. Let us construct a model M from the structure of a nonstandard model X of PA (Peano arithmetic), to which is added an element, e, such that e = s(e). All standard numbers together with e are the only objects that satisfy N. Only one model of J-4 with a strictly smaller extension for N shares the same structure as M. In that model the extension of N consists of exactly all standard numbers. By T h e o r e m 2.4, ~ is a model of Circum,r(3- 4 : N ) if and only if the set of all standard numbers is not ~--definable in J / , where f f is the set of all formulas with no occurrence of N. So we consider a language from which N has been withdrawn. Now, the restriction of M to the domain of ?( is the reduct 3 of a nonstandard model of PA. It is well-known (see [15, p. 129] for instance) that the set of all standard numbers is not definable in such a model 2¢'. Together with the definition of a reduct, we get that the set of standard numbers is not ~ - d e f i n a b l e in N. So, by definition of a restriction of a model, the set of standard numbers is not ~ - d e f i n a b l e in M. Accordingly, M is a model of C i r c U m n r ( f f 4 : N). Now, AX falsifies V x N ( x ) ~ x ~ s(x). By the soundness theorem for first-order logic, the t h e o r e m follows. []
Along similar lines, an important property to consider is the relationship between circumscription and predicate completion for H o r n clause theories [14]. E x a m p l e 4.3.
N(0), Vx(N(x)
N(s(x))) .
Predicate completion gives N ( x ) <-->[x = 0 v 3 y ( N ( y ) A X = S( y ) ) ] .
C i r c u m ( f f 5 : N) derives the same result using the substitution n[x] ~ [x = 0 v 3 y ( N ( y ) A X = s ( y ) ) ] .
But it can be proved [2] in a m a n n e r similar to the proof of T h e o r e m 4.2 that this sentence cannot be obtained by nonrecursive circumscription. Theorem
4.4. CircUmnr(ff5 : N ) ~ Vx(N(x)<--> [x = 0 v 3 y ( N ( y ) A X = s(y))]) .
3Given a first-order model dX interpreting the symbols of some first-order language L, the first-order model obtained from d~ by interpreting only the symbols of a first-order language L' contained in L is a reduct of d/.
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Proof. There exists a model M of if5 which is a reduct of a nonstandard model of PA such that N is satisfied by all standard numbers as well as one final sequence of numbers in a nonstandard block. As in the proof of Theorem 4.2, the undefinability of the set of standard numbers is used so that Theorem 2.4 applies. The rest of the proof is unproblematic. [] Reiter [14] proves that for Horn clause theories standard circumscription implies predicate completion. Theorem 4.4 indicates that nonrecursive circumscription does not enjoy this important property. Although nonrecursive circumscription is not strictly stronger than predicate completion for Horn clause theories, it is easy to show that it is not strictly weaker either. Take for instance the theory {P(a) 3 P(b), P(b) ~ P(a)}. Using the substitution p[x] =- x # x we can derive Vx-a P(x). Predicate completion can derive neither 7 P ( a ) nor 7 P ( b ) . Example 3.1 shows a sentence which is not derivable using closed circumscription but which is derivable using open circumscription even if it is nonrecursive. Likewise, Examples 4.1 and 4.3 present sentences not derivable by nonrecursive circumscription but derivable with recursive circumscription even if it is closed. It should be noted that there are cases which can be proven to require a circumscription which is both open and recursive to derive a formula valid in all minimal models. One such case is the theory consisting of the single axiom VxVy(P(x, y) v P(y, x)). Indeed, the circumscription schema of predicate P in this theory can be instantiated using p[x, y] =- P(x, y) A (x, y) # (z, t) to derive the formula VxVy(P(x, y) A P(y, x) ~ x = y). P(x, y)A (X, y ) # (Z, t) is exactly the circumscription axiom of pointwise circumscription [9] in which the derivation is straightforward. Mott's circumscription being closed and nonrecursive stands at the extreme opposite of pointwise circumscription which involves one formula only in the form of an open and recursive instance.
5. Adding Variable Predicates May Weaken Nonrecursive Circumscription Here is another example. R is intended to designate the set of natural numbers. L is an auxiliary binary relation having the properties described by the theory.
Example 5.1. Vx3yL(x, y), :txL(x, 0), VxVy(L(x, y) ~ 3 z L ( z , s(y))) , VxVy(L(x, y) D R(y)) .
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Circum(9-6 : R ) F Vx(R(x) ~ 3yL(y, x)) as can be seen from the substitution r[x] =- 3yL(y, x). This is true even for Circumcl(9-6 : R) and CircUmnr(9-6 : R). However, if predicates are allowed to vary, we can get a weaker result as indicated by T h e o r e m 5.2.
Theorem 5.2. Circum,r(9- 6 : R; L) ~ Vx(R(x) D 3yL(y, x)). Proof. Let M be a model of 9-6 defined from the structure of a nonstandard model of PA in which L describes a function over all numbers onto the standard numbers, R being true for all standard numbers and exactly one nonstandard number. If ~f is to be a model of 9-6 that is identical to M except that ]R]~, C IR]~ and IL[~ may differ from ILIa, then Inl~ must consist exactly of all standard numbers. Since the ~-definability of [R]~v-is in question, neither L nor R occur in the formulas of ~. So, ~ is the reduct of a nonstandard model of PA and therefore, by the same arguments as in the proof of T h e o r e m 4.2, ~ is a model of Circumnr(9-6:R; L). But the sentence Vx(R(x)D ::lyL(y, x)) is false in ~/. Finally, apply the soundness theorem for first-order logic. [] The impact of T h e o r e m 5.2 is obvious when compared to the well-known T h e o r e m 5.3.
Theorem 5.3. Circum(9- :P U P'; Q u Q') ~- Circum(9- : P; Q), that is, standard circumscription becomes stronger as the number of variable or circumscribed predicates increases. The consequences of T h e o r e m 5.2 are quite revealing. Variable circumscription was introduced [11] in order to increase the power of circumscription, in those cases in which circumscription is too weak otherwise. In the case of nonrecursive circumscription, we start with a weak kind of circumscription and as predicates are allowed to vary, the circumscribed theory may be weaker. So, T h e o r e m 5.3 is no longer true for nonrecursive circumscription. Moreover, a similar p h e n o m e n o n occurs with joint (or parallel) circumscription. For standard circumscription, every sentence derivable by a union C i r c u m ( 9 - : P 1 ) U Circum(9- : P2) is derivable by the joint circumscription Circum(9- : (P1 U Pc)) [1]. This is no longer true for nonrecursive circumscription (choose if6 as 9-, with R as P1 and L as P2).
6. Conclusion Circumscription can produce inconsistent theories from consistent ones. One approach to counter this problem is to categorize those theories for which
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consistency is preserved [3, 7]. Another approach is to find a weaker form of circumscription that does not affect the consistency of the circumscribed theory. Closed and nonrecursive circumscription has been shown [12] to be a possible candidate. However, the method proposed by Mott to attain the goal of this second approach involves an excessive weakening of standard circumscription. Indeed, it has been shown in this paper that closed circumscription fails to yield the desired uniqueness formula for the simplest of existential theories. Also, nonrecursive circumscription has been shown not to imply predicate completion for Horn clause theories. Additionally, the natural way to strengthen circumscription, adding more variable predicates, may weaken nonrecursive circumscription. Certainly, nonrecursive circumscription is as strong as standard circumscription for theories with a domain closure axiom (in fact, standard circumscription and nonrecursive circumscription are both ideally well-behaved in this case since it follows from [13] that circumscription is complete and consistent for these theories). As well, closed circumscription seems as strong as standard circumscription for Horn clause theories. In conclusion, an acceptable solution to the inconsistency problem of circumscription in the manner of the second approach seems to lie between the general approaches like closed and nonrecursive circumscription and a complete version of circumscription (that, incidentally, cannot be defined [4]).
ACKNOWLEDGMENT We express our appreciation to the referees for their helpful suggestions.
REFERENCES 1. Besnard, Ph., An Introduction to Default Logic (Springer, Heidelberg, 1988). 2. Besnard, Ph., Houdebine, J. and Rolland, R., A formula circumscriptively both valid and unprovable, in: Proceedings ECAI-88 (1988) 516-518. 3. Bossu, G, and Siegel, E, Saturation, nonmonotonic reasoning and the closed-world assumption, Artificial Intelligence 25 (1985) 13-63. 4. Davis, M., The mathematics of non-monotonic reasoning, Artificial Intelligence 13 (1980) 73-80. 5. Doyle, J., Circumscription and implicit definability, Autom. Reasoning 1 (1985) 391-405. 6. Etherington, D.W., Reasoning with Incomplete Information (Pitman, London, 1988). 7. Etherington, D.W., Mercer, R.E. and Reiter, R., On the adequacy of predicate circumscription for closed-world reasoning, Comput. Intell. 1 (1985) 11-15. 8. Lifschitz, V., On the satisfiability of circumscription, Artificial Intelligence 28 (1986) 17-27. 9. Lifschitz, V., Pointwise circumscription: Preliminary report, in: Proceedings AAAI-86, Philadelphia, PA (1986) 406-410. 10. McCarthy, J., Circumscription: A form of non-monotonic reasoning, Artificial Intelligence 13 (1980) 27-39. 11. McCarthy, J., Application of circumscription to formalizing common-sense knowledge, Artificial Intelligence 28 (1986) 89-116.
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12. Mott, EL., A theorem on the consistency of circumscription, Artificial Intelligence 31 (1987) 87-98. 13. Perlis, D. and Minker, J., Completeness results for circumscription, Artificial Intelligence 28 (1986) 29-42. 14. Reiter, R., Circumscription implies predicate completion (sometimes), in: Proceedings AAAI82, Pittsburgh, PA (1982) 418-420. 15. van Dalen, D., Logic and Structure (Springer, Heidelberg, 1983).
Received June 1988; revised version received N o v e m b e r 1988
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RESEARCH NOTE
The Importance of Open and Recursive Circumscription Philippe Besnard* and Yves Moinard*
IRISA, Campus de Beaulieu, 35402 Rennes-Cedex, France R o b e r t E. M e r c e r * *
University of Western Ontario, London, Ontario, Canada N6A 5B7
ABSTRACT
Circumscription is known to result in an inconsistency when applied to certain consistent theories. To counter this problem, closed nonrecursive circumscription, a restricted form of circumscription that has been proved not to affect the consistency of the theory over which circumscription is applied, has been proposed. We show that closed nonrecursive circumscription involves an excessive weakening of standard circumscription by establishing that closed nonrecursive circumscription is incomplete for some crucial theories over which standard circumscription is consistent and complete. First, we prove that closed circumscription cannot yield the desired uniqueness formula for the simplest of existential theories. Second, we prove that nonrecursive circumscription fails to be as strong as predicate completion for Horn clause theories. Third, we prove that the natural way to strengthen circumscription, that is, adding more variable predicates, may weaken nonrecursive circumscription.
I. Introduction By means of the so-called circumscription schema the circumscription of a predicate in a theory tends to reduce the relation denoted by that predicate to a minimum. Because circumscription results in an inconsistency when applied to certain consistent theories, Mott [12] suggests imposing two restrictions on the standard circumscription schema. Firstly, Mott proposes what we call closed circumscription where only those instances of the standard circumscription schema that are closed formulas are acceptable. Secondly, Mott proposes * This research was supported by CNRS PRC GRECO IA. **This research was supported by NSERC grant 0036853.
Artificial Intelligence 39 (1989) 251-262 0004-3702/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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what we call nonrecursive circumscription in which the only acceptable instances of the standard circumscription schema are those in which no substituting formula involves the predicates being circumscribed or allowed to vary. Mott proves that the combination of these restrictions is sufficient to ensure that the resulting circumscription (which is thus closed and nonrecursive) always yields a consistent set of theorems when applied to a consistent theory. The purpose of the present paper is to indicate how significantly Mott's circumscription schema reduces the class of theories over which circumscription is complete with respect to the standard model-theoretic approach to circumscription, namely minimal models. Indeed, the present paper exhibits very simple theories over which standard circumscription yields the desired theorems without inconsistency but over which Mott's circumscription is incomplete. In fact, we demonstrate that each restriction on its own is significantly weaker than standard circumscription for various theories for which standard circumscription already preserves consistency.
2. Circumscription McCarthy [10, 11] introduces circumscription, in order to express some commonsense problems formally. Let 3- be a finite set of first-order formulas in which the finite lists of predicates P and Q occur. P = (P~ . . . . , Pm) and Q=(Q1 .... ,Q,). The (first-order) circumscription of P in 3-, with the predicates of Q as variables, is accomplished by adding the following circumscription schema to 3-:
{ 3-[p, q] A Vx(p[x] ~ P(x))} ~ Vx(P(x) ~ p [ x ] ) , where p = (Pl . . . . , Pm), and q = ( q l , . • • , q,) are lists of first-order formulas in the language of 3-. 3-[p, q] is 3- except that each occurrence of Pi and Qj is replaced by Pi and q/, respectively. The notation Vx(p[x] ~ P(x)) is to be interpreted as: Vx I . . . Xm(Pl[Xl] D e l ( X l ) ) A ' ' " A (pm[Xm]~ em(Xm)). The notation Pi[Xi] m e a n s that pi may have free variables other than x i = (X~a,..., xiti), where lg is the arity of P~, for i = 1 . . . . . m. The notation p(x), which is used in Mott's restricted circumscription, is interpreted in a manner similar to the notation p[x] with the restriction that each p~ cannot have any extra free variables. We write Cireum(3- : P; Q) for the union of ff and these axioms. If Q is empty, we denote the circumscription of P in if, as introduced in [10], by Circum(3- :P). An instance of the circumscription schema is the result of substituting formulas for each of the p~ and q/. Under standard circumscription any formula can be substituted for the p~ and q~. Mott suggests restricting the formulas
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available for substitution in two ways. The first restriction replaces p[x] with p(x) in the circumscription schema, that is, no pi(xl) contains free variables other than those in Pi(x~). This version of circumscription we call closed circumscription (cf. "closed formula") since no instance of the circumscription schema will have unbound variables. The notation for closed circumscription will be Cireumcl. Circumscription without this restriction is open. Closed circumscription is used by Mott to prevent the following kind of inconsistent circumscription that can result from a consistent theory.
Example 2.1.
3xR(x) , Vx7 L(x, x ) , Vy3xL(y, x), VyVx(L(y, x) ~ R(x)) . The case to consider is Circum(8-1 : R ). If we were to substitute L(y, x) for r[x] we would get VyVx(L(y, x)~ L(y, x)) as the fourth sentence in fix[r]. This allows the derivation of the sentence VyVx(R(x)~ L(y, x)) which together with the first and second axioms of 31 produces an inconsistency. This cannot happen if extra free variables (like y) are not allowed in p or q. As shown in a later section, extra free variables are crucial, so rather than taking the closed circumscriptive approach, we suggest that this example simply indicates that no extra free variable in p or q should get bound in i f [ p , q]. C i r c u m ( ~ : R) is not inconsistent if L(z, x) is substituted for r. The (simplified) circumscription schema that results from this instance is
Vx[(~yL(y, x)) D L(z, x)] D Vx(R(x) D L(z, x)) . That is, if there exists a special element, denoted by z, which is an Lantecedent of every element having at least one L-antecedent, then every R-element has this z as an L-antecedent. Clearly, requiring the same z for every R-element is a specific instance of the expected result: Vx(R(x) 3yL(y, x)) (every R-element has an L-antecedent, not necessarily the same one for all R-elements). The choice of 3yL(y, x) for r gives an instance which allows this more general result to be derived, even from Circumcl(ff1 : R). It should be noted that it is the norm rather than the exception that classical mathematical systems prohibit the binding of free variables in substitutional frameworks. As pointed out by an anonymous referee, a strong argument that this normal method of substitution is implicitly intended in any definition of circumscription is that the soundness results for circumscription (see Theorem 2.3 below) no longer hold if the binding of free variables is allowed. In particular, there are R-minimal models for ~-1; however, Mott's nonclassical
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form of substitution generates a circumscription of R in 3-1 which has no models because it is inconsistent. The second restriction prevents any of the Pi or Qj from appearing in the formulas used to substitute for the pi and qj. This version of circumscription we call n o n r e c u r s i v e c i r c u m s c r i p t i o n . The notation used will be Cireum,~. This restriction prevents inconsistencies of the type first presented in [7].
Example 2.2. 3x(N(x) ^ VyN(y) D x ~ s(y)) , Vx(N(x) D N(s(x))) , V x V y ( s ( x ) = s ( y ) D x = y) .
Indeed, when circumscribing N, if we were to substitute 3 y ( N ( y ) ^ x = s ( y ) ) 1 for n [ x ] , the corresponding instance of the circumscription schema would give the sentence V x ( N ( x ) D 3 y ( N ( y ) ^ x = s ( y ) ) ) which together with the first axiom of 3-2 produces an inconsistency. Standard circumscription need not be closed or nonrecursive. Mott's closed and nonrecursive circumscription, denoted Cireumd~.r, preserves consistency [12]: Circumcl&n~(ff)~-/ only if ~ - ~ - 1 . That is, the closed nonrecursive circumscription over a consistent theory always yields a consistent theory. This is an important property if circumscription is blindly applied to a theory (unwittingly producing an inconsistent theory would be very troubling). However, an equally important property for circumscription is completeness: the circumscription of a theory corresponds to the minimal models of the theory (see below). In general, consistency and completeness for circumscription are incompatible since completeness requires that any consistent theory not having a minimal model (for example, 32) have an inconsistent circumscription. Although we must inevitably abandon completeness in the general setting [4], we should not overbalance the desirability for consistency guaranteed by a new form of circumscription with the needless abandoning of completeness where it can be accomplished. The next sections show that in this respect Mott's restricted form of circumscription is quite unsatisfactory. For convenience, the definitions of minimal model, and the soundness and completeness of circumscription are given next. Let ~t and X be two models of 3- over the language of 3-. We write :R < e;oW when A/and W are identical except that: 1Mott chooses ( N ( x ) ^ 3y(x = s(y)) for n, but the corresponding instance does not produce inconsistency. A model of 5r2 and this instance is: the structure is the standard natural n u m b e r s ;
N(x) ,~. ~ ~ ~ - {0}.
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the extensions of the Pi a r e such that Ieil ~ IPilx for 1 ~< i ~< m, with some Ie, l~ C Ie, l~, where IP, I~ denotes the extension of predicate Pi in the model ~t; there is no condition on the extensions of the Qj for 1 ~
-
Theorem 2.3 (Soundness of circumscription) [6, 8, 10]. Every model o f 9which is minimal for < e;o is a model o f Circum(9- : P; Q).
If 9- is such that the converse is also true, then we have the completeness o f circumscription. [13] is devoted to this problem. In order to obtain the converse of T h e o r e m 2.3 a different notion of minimality needs to be defined, one that corresponds precisely to what is expressed by the circumscription schema. The circumscription schema of P in 3- (for the sake of simplicity only one unary predicate is circumscribed and no predicate is allowed to vary) expresses that, in a model A/, if a set denoted by a formula p could be taken (according to the axioms of 9-) as a possible extension for P and if this set were contained in the actual extension of P in ~t then this set is indeed the extension of P itself. That is, the circumscription schema realizes the minimization of the extension of P in models of f f with respect to definable sets (given a model, a subset of the domain of the model is a definable set if it consists of exactly those individuals from the domain that satisfy some formula): in a model, only definable sets are taken into account regarding the determining of a minimal extension for P. In terms of an ordering over models of 9-, such a minimization ranging over definable sets is achieved by means of a relation, < 8P- T h e o r e m 2.4 states the formal equivalence between both approaches to such a minimization of P. In order to discuss definability with respect to the different kinds of circumscription introduced above, it is necessary to introduce ~-definability. Let ~ be a set of formulas dependent on the kind of circumscription at hand. The set IPil~ is ~-definable in N when there exists a first-order formula Fi(Xl, • • •, xt i, Yl . . . . . Ys,) in the set of formulas o~, where li is the arity of Pi, such that for some e~ . . . . . esi in the domain of N (or equivalently ~t), we have:
IP, I~ = ]Fi(x I . . . .
, xt,, e l , . . . ,
esi)lN •
This notion of definability (in models) is not to be confused with the notion of definability in theories as examined in [5]. As mentioned above the set ~ depends on the kind of circumscription concerned:
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- F o r standard circumscription, ~ denotes the set of all formulas in the language. 2 ~-definability in N is then definability with parameters in N. - F o r closed circumscription, ~ denotes, for each circumscribed or variable predicate, the set of all formulas which have no free variables (except the defining variables). ~-definability in N is then (classical) definability (without parameters) in N. - For nonrecursive circumscription, ~ denotes the set of all formulas with no occurrence of the circumscribed or variable predicate(s). ~-definability in Y is then definability with parameters in a restricted language in A;.
IOjl
If in addition t o t h e requirements for < P;Q each [eil~ and each is ~-definable in N, then we write ~ < e;Q N. If Q is empty, then we write < ~(~) (or < P for the relation between models used in Theorem 2.3). P The semantics induced by < e;Q may be called the standard semantics of circumscription, as it closely matches our intuitions regarding the minimization of predicates. The precise semantical characterization for circumscription is given by the more sophisticated relation < P:O : Theorem 2.4 [1]. The models o f Circum(~- : P; Q) are the models o f S minimal for < 8(~) p;Q , where ~ is the set o f all formulas in the language. Similar relationships hold for Circumcl(J- : P; Q) and Circum,r(9- : P; Q), where ~ is appropriately chosen. ~. ~ ( ~ )
For every theory, a model minimal for < e;~ is also minimal for - p ; e . If is such that the converse also holds, then we have the completeness of circumscription. In the next section we give an example of the incompleteness of closed circumscription for the simplest of existential theories. In this particular example closed circumscription generates a theory with models which are minimal for < p8(~) , but which are not minimal for < p. 3. The Need for Open Circumscription Example 3.1.
3xP(x) .
First we examine standard circumscription. Circum(~-3 : P) gives the expected uniqueness result: V x V y ( ( P ( x ) ^ P ( y ) ) D x = y) . 2In all the examples below, the language is taken to consist of the formulas whose only nonlogical symbols are " = " , the ones that occur in the theory, and an infinite set of variables.
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One proof goes as follows: p[x] = x = y. T h e corresponding instance of the circumscription schema is: [:Ix x = y A V x ( x = y ~ P(x))] ~ V x ( P ( x ) ~ x = y) . 3x x = y is easily provable, and so is P ( y ) ~ V x ( x = y ~ P(x)), thus we obtain: P(y)DVx(P(x)Dx=y), by universal generalization we get V y P ( y ) 3 V x ( P ( x ) ~ x = y) which is logically equivalent to the uniqueness result. So, in every model of 0-3 minimal for < e the extension of P is a singleton. Now we turn to closed circumscription. Theorem 3.2. Circumcl(0-3 : P ) ~- V x V y ( ( P ( x ) A P ( y ) ) ~ x = y). Proof. Let M be a model of 0-3 with domain D such that c a r d ( D ) = 2 and ]Pla~ = D. We prove that 0 and D are the only subsets of D which are off-definable in M without extra free variables. Notationally, if e E D then the other element of D is denoted e -1. If R k is the relation over D k (k I> 0) defined by a formula F with k free variables, then we can prove by induction on the length of the formula F that (e~ . . . . . ek) E R ~ if and only if (e~-1, . . . , e~-1) E R k. Indeed it is true for atomic formulas P ( x ) , x = x, x = y, y = x and it is easy to show that it is stable for 7 , A, and 3. Now, I~ cannot be taken as IPl~ for any model 2¢" whose domain is D (cf. 3 x P ( x ) ) . So Theorem 2.4 applies and M is a model of Circumd(0- 3 : P). But in M we have the true sentence 3 x 3 y ( P ( x ) A P ( y ) D x ~ y ) . The soundness theorem of first-order logic gives the theorem. [] Theorem 3.2 demonstrates that in order to obtain completeness for 0-3 it is necessary that substitution instances be allowed extra free variables. 4. The Need for Recursive Circumscription Let us now examine the completely skolemized version of Example 2.2. Example 4.1.
0-4: N ( 0 ) , Vx(N(x)
0
s(x)) ,
Vx(N(x) N(s(x))), V x V y ( s ( x ) = s( y) D x = y) .
Because 0-4 is a universal theory, circumscription is guaranteed to be consistent [3,7]. Taking n [ x ] - [ N ( x ) A x ~ s k ( x ) ] gives, in particular, the expected result: V x ( N ( x ) D x ~ s~(x) (for every k ~ 1)).
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T h e o r e m 4.2. CircUmnr(J-4 : N ) ~- V x ( N ( x ) ~ x ~ sk(x)), even f o r k = 1. Proof. Let us construct a model M from the structure of a nonstandard model X of PA (Peano arithmetic), to which is added an element, e, such that e = s(e). All standard numbers together with e are the only objects that satisfy N. Only one model of J-4 with a strictly smaller extension for N shares the same structure as M. In that model the extension of N consists of exactly all standard numbers. By T h e o r e m 2.4, ~ is a model of Circum,r(3- 4 : N ) if and only if the set of all standard numbers is not ~--definable in J / , where f f is the set of all formulas with no occurrence of N. So we consider a language from which N has been withdrawn. Now, the restriction of M to the domain of ?( is the reduct 3 of a nonstandard model of PA. It is well-known (see [15, p. 129] for instance) that the set of all standard numbers is not definable in such a model 2¢'. Together with the definition of a reduct, we get that the set of standard numbers is not ~ - d e f i n a b l e in N. So, by definition of a restriction of a model, the set of standard numbers is not ~ - d e f i n a b l e in M. Accordingly, M is a model of C i r c U m n r ( f f 4 : N). Now, AX falsifies V x N ( x ) ~ x ~ s(x). By the soundness theorem for first-order logic, the t h e o r e m follows. []
Along similar lines, an important property to consider is the relationship between circumscription and predicate completion for H o r n clause theories [14]. E x a m p l e 4.3.
N(0), Vx(N(x)
N(s(x))) .
Predicate completion gives N ( x ) <-->[x = 0 v 3 y ( N ( y ) A X = S( y ) ) ] .
C i r c u m ( f f 5 : N) derives the same result using the substitution n[x] ~ [x = 0 v 3 y ( N ( y ) A X = s ( y ) ) ] .
But it can be proved [2] in a m a n n e r similar to the proof of T h e o r e m 4.2 that this sentence cannot be obtained by nonrecursive circumscription. Theorem
4.4. CircUmnr(ff5 : N ) ~ Vx(N(x)<--> [x = 0 v 3 y ( N ( y ) A X = s(y))]) .
3Given a first-order model dX interpreting the symbols of some first-order language L, the first-order model obtained from d~ by interpreting only the symbols of a first-order language L' contained in L is a reduct of d/.
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Proof. There exists a model M of if5 which is a reduct of a nonstandard model of PA such that N is satisfied by all standard numbers as well as one final sequence of numbers in a nonstandard block. As in the proof of Theorem 4.2, the undefinability of the set of standard numbers is used so that Theorem 2.4 applies. The rest of the proof is unproblematic. [] Reiter [14] proves that for Horn clause theories standard circumscription implies predicate completion. Theorem 4.4 indicates that nonrecursive circumscription does not enjoy this important property. Although nonrecursive circumscription is not strictly stronger than predicate completion for Horn clause theories, it is easy to show that it is not strictly weaker either. Take for instance the theory {P(a) 3 P(b), P(b) ~ P(a)}. Using the substitution p[x] =- x # x we can derive Vx-a P(x). Predicate completion can derive neither 7 P ( a ) nor 7 P ( b ) . Example 3.1 shows a sentence which is not derivable using closed circumscription but which is derivable using open circumscription even if it is nonrecursive. Likewise, Examples 4.1 and 4.3 present sentences not derivable by nonrecursive circumscription but derivable with recursive circumscription even if it is closed. It should be noted that there are cases which can be proven to require a circumscription which is both open and recursive to derive a formula valid in all minimal models. One such case is the theory consisting of the single axiom VxVy(P(x, y) v P(y, x)). Indeed, the circumscription schema of predicate P in this theory can be instantiated using p[x, y] =- P(x, y) A (x, y) # (z, t) to derive the formula VxVy(P(x, y) A P(y, x) ~ x = y). P(x, y)A (X, y ) # (Z, t) is exactly the circumscription axiom of pointwise circumscription [9] in which the derivation is straightforward. Mott's circumscription being closed and nonrecursive stands at the extreme opposite of pointwise circumscription which involves one formula only in the form of an open and recursive instance.
5. Adding Variable Predicates May Weaken Nonrecursive Circumscription Here is another example. R is intended to designate the set of natural numbers. L is an auxiliary binary relation having the properties described by the theory.
Example 5.1. Vx3yL(x, y), :txL(x, 0), VxVy(L(x, y) ~ 3 z L ( z , s(y))) , VxVy(L(x, y) D R(y)) .
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Circum(9-6 : R ) F Vx(R(x) ~ 3yL(y, x)) as can be seen from the substitution r[x] =- 3yL(y, x). This is true even for Circumcl(9-6 : R) and CircUmnr(9-6 : R). However, if predicates are allowed to vary, we can get a weaker result as indicated by T h e o r e m 5.2.
Theorem 5.2. Circum,r(9- 6 : R; L) ~ Vx(R(x) D 3yL(y, x)). Proof. Let M be a model of 9-6 defined from the structure of a nonstandard model of PA in which L describes a function over all numbers onto the standard numbers, R being true for all standard numbers and exactly one nonstandard number. If ~f is to be a model of 9-6 that is identical to M except that ]R]~, C IR]~ and IL[~ may differ from ILIa, then Inl~ must consist exactly of all standard numbers. Since the ~-definability of [R]~v-is in question, neither L nor R occur in the formulas of ~. So, ~ is the reduct of a nonstandard model of PA and therefore, by the same arguments as in the proof of T h e o r e m 4.2, ~ is a model of Circumnr(9-6:R; L). But the sentence Vx(R(x)D ::lyL(y, x)) is false in ~/. Finally, apply the soundness theorem for first-order logic. [] The impact of T h e o r e m 5.2 is obvious when compared to the well-known T h e o r e m 5.3.
Theorem 5.3. Circum(9- :P U P'; Q u Q') ~- Circum(9- : P; Q), that is, standard circumscription becomes stronger as the number of variable or circumscribed predicates increases. The consequences of T h e o r e m 5.2 are quite revealing. Variable circumscription was introduced [11] in order to increase the power of circumscription, in those cases in which circumscription is too weak otherwise. In the case of nonrecursive circumscription, we start with a weak kind of circumscription and as predicates are allowed to vary, the circumscribed theory may be weaker. So, T h e o r e m 5.3 is no longer true for nonrecursive circumscription. Moreover, a similar p h e n o m e n o n occurs with joint (or parallel) circumscription. For standard circumscription, every sentence derivable by a union C i r c u m ( 9 - : P 1 ) U Circum(9- : P2) is derivable by the joint circumscription Circum(9- : (P1 U Pc)) [1]. This is no longer true for nonrecursive circumscription (choose if6 as 9-, with R as P1 and L as P2).
6. Conclusion Circumscription can produce inconsistent theories from consistent ones. One approach to counter this problem is to categorize those theories for which
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consistency is preserved [3, 7]. Another approach is to find a weaker form of circumscription that does not affect the consistency of the circumscribed theory. Closed and nonrecursive circumscription has been shown [12] to be a possible candidate. However, the method proposed by Mott to attain the goal of this second approach involves an excessive weakening of standard circumscription. Indeed, it has been shown in this paper that closed circumscription fails to yield the desired uniqueness formula for the simplest of existential theories. Also, nonrecursive circumscription has been shown not to imply predicate completion for Horn clause theories. Additionally, the natural way to strengthen circumscription, adding more variable predicates, may weaken nonrecursive circumscription. Certainly, nonrecursive circumscription is as strong as standard circumscription for theories with a domain closure axiom (in fact, standard circumscription and nonrecursive circumscription are both ideally well-behaved in this case since it follows from [13] that circumscription is complete and consistent for these theories). As well, closed circumscription seems as strong as standard circumscription for Horn clause theories. In conclusion, an acceptable solution to the inconsistency problem of circumscription in the manner of the second approach seems to lie between the general approaches like closed and nonrecursive circumscription and a complete version of circumscription (that, incidentally, cannot be defined [4]).
ACKNOWLEDGMENT We express our appreciation to the referees for their helpful suggestions.
REFERENCES 1. Besnard, Ph., An Introduction to Default Logic (Springer, Heidelberg, 1988). 2. Besnard, Ph., Houdebine, J. and Rolland, R., A formula circumscriptively both valid and unprovable, in: Proceedings ECAI-88 (1988) 516-518. 3. Bossu, G, and Siegel, E, Saturation, nonmonotonic reasoning and the closed-world assumption, Artificial Intelligence 25 (1985) 13-63. 4. Davis, M., The mathematics of non-monotonic reasoning, Artificial Intelligence 13 (1980) 73-80. 5. Doyle, J., Circumscription and implicit definability, Autom. Reasoning 1 (1985) 391-405. 6. Etherington, D.W., Reasoning with Incomplete Information (Pitman, London, 1988). 7. Etherington, D.W., Mercer, R.E. and Reiter, R., On the adequacy of predicate circumscription for closed-world reasoning, Comput. Intell. 1 (1985) 11-15. 8. Lifschitz, V., On the satisfiability of circumscription, Artificial Intelligence 28 (1986) 17-27. 9. Lifschitz, V., Pointwise circumscription: Preliminary report, in: Proceedings AAAI-86, Philadelphia, PA (1986) 406-410. 10. McCarthy, J., Circumscription: A form of non-monotonic reasoning, Artificial Intelligence 13 (1980) 27-39. 11. McCarthy, J., Application of circumscription to formalizing common-sense knowledge, Artificial Intelligence 28 (1986) 89-116.
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12. Mott, EL., A theorem on the consistency of circumscription, Artificial Intelligence 31 (1987) 87-98. 13. Perlis, D. and Minker, J., Completeness results for circumscription, Artificial Intelligence 28 (1986) 29-42. 14. Reiter, R., Circumscription implies predicate completion (sometimes), in: Proceedings AAAI82, Pittsburgh, PA (1982) 418-420. 15. van Dalen, D., Logic and Structure (Springer, Heidelberg, 1983).
Received June 1988; revised version received N o v e m b e r 1988