Marco Cadoli �, Francesco M. Donini �, Marco Schaerf �,Riccardo Silvestri yOn Compact Representations ofPropositional Circumscription

Technical Report 14.95, July 1995.Dipartimento di Informatica e Sistemistica. Universit�a di Roma \La Sapienza"Roma, Italy(RAP. 14.95 LUGLIO 1995)Accepted for publication on Theoretical Computer Science� Dipartimento di Informatica e SistemisticaUniversit�a di Roma \La Sapienza"Via Salaria 113, I-00198 Roma, Italye-mail: <lastname>@dis.uniroma1.ity Dipartimento di Scienze dell'InformazioneUniversit�a di Roma \La Sapienza"Via Salaria 113, I-00198, Roma, Italyemail: silver@dsi.uniroma1.it

Sommario1La circumscription �e una nota tecnica per il ragionamento di senso comune,utilizzata nei campi dell'Intelligenza Arti�ciale, delle Basi di Dati e della Pro-grammazione Logica. Nel presente lavoro studiamo la taglia delle rappresentazioni(formule, strutture di dati) equivalenti alla circumscription di una formula propo-sizionale T , prendendo in considerazione tre di�erenti de�nizioni di equivalenza.Vengono mostrate condizioni necessarie e su�cienti per l'esistenza di rappresen-tazioni di taglia polinomiale equivalenti alla circumscription di T nei tre casi. Tuttequeste condizioni implicano il collasso della gerarchia polinomiale. In particolareproviamo che, a meno che la gerarchia polinomiale collassi al secondo livello, la tagliadella formula proposizionale T 0 pi�u corta che sia equivalente alla circumscription diT cresce, al crescere della taglia di T , pi�u rapidamente di qualsiasi polinomio. Vieneinoltre analizzata la signi�cativit�a di tale risultato nell'ambito del ragionamento dimondo chiuso.

1Questo lavoro �e la versione estesa e rivista di un articolo presentato alla conferenza STACS-95[CDS95b].

Abstract2Circumscription is a popular common-sense reasoning technique, used in the�elds of Arti�cial Intelligence, Databases and Logic Programming. In this paperwe investigate the size of representations (formulae, data structures) equivalent tothe circumscription of a propositional formula T , taking into account three di�erentde�nitions of equivalence. We �nd necessary and su�cient conditions for the ex-istence of polynomial-size representations (formulae, data structures) equivalent tothe circumscription of T in the three cases. All such conditions imply the collapseof the polynomial hierarchy. In particular, we prove that { unless the polynomialhierarchy collapses at the second level { the size of the shortest propositional formulaT 0 logically equivalent to the circumscription of T grows faster than any polynomialas the size of T increases. The signi�cance of this results in the related �eld ofclosed-world reasoning is then analyzed.

2This is an extended and revised version of a paper presented at STACS-95 [CDS95b].

Contents1 Introduction 51.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Preliminaries 93 Compact Representations 113.1 Model equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Logical equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Query equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Analysis of the Results 194.1 Closed-World Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Large Instances of EGCWA . . . . . . . . . . . . . . . . . . . . . . . 214.3 Generality of Main Result . . . . . . . . . . . . . . . . . . . . . . . . 235 Conclusions and open problems 24

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1 IntroductionReasoning with selected (or intended) models of a logical formula is a commonreasoning technique used in Databases, Logic Programming, Knowledge Represen-tation and Arti�cial Intelligence (AI). One of the most popular criteria for selectingintended models is minimality wrt the set of true atoms. The idea behind mini-mality is to assume that a fact is false whenever possible. Such a criterion allowsone to represent only true statements of a theory, saving the explicit representationof all false ones. For propositional theories, the explicit (and �nite) representationis always possible; but how large is its size, compared with the size of the implicitrepresentation?In this paper we address the following problem:Is it the case that for each propositional formula T , there is a \com-pact" representation of the minimal models of T ? By compact we meanpolynomially-sized wrt the size of T , for some �xed polynomial.We consider three di�erent formal notions of \representation of the minimal models",and give a negative answer to this problem for all of them (provided the polynomialhierarchy does not collapse).1.1 MotivationA well-established formalization of minimality is circumscription, which has been in-troduced in the AI literature [McC80, McC86] for capturing some important aspectsof common-sense reasoning, and was shown to be strictly related to closed-world rea-soning in Databases. From a formal point of view, circumscription is a fragment ofsecond-order logic, as circumscription of a �rst-order formula yields a second-orderuniversal formula. The propositional version has also been de�ned: Circumscriptionof a propositional formula yields a universally quanti�ed boolean formula.Several studies about computational properties of circumscription appeared inthe literature. Several aspects, such as time complexity of inference, model checkingand model �nding have been studied. Noticeably, those studies proved that reason-ing with circumscriptive formulae is harder than reasoning with formulae of classi-cal logic. As an example, inference in propositional circumscription is �p2-complete[EG93], while the same problem is coNP-complete in classical propositional logic.Another interesting computational aspect that has been addressed is collapsi-bility. The question can be stated as follows: Given a �rst-order formula T , is itscircumscription (denoted by CIRC(T )) { which is a second-order formula { equiv-alent to some �nite �rst-order formula? The answer in general is no [KP90, Kri88],5

but there are syntactically restricted classes of formulae in which this is true [KP90,Lif85, Rab89].In principle, two distinct notions of equivalence can be analyzed: logical equiv-alence, i.e., the two formulae have exactly the same models, and query equivalence,i.e., the two formulae have exactly the same theorems. Clearly, logical equivalenceimplies query equivalence, but the converse does not necessarily hold.As for the propositional case, collapsibility to a logically equivalent formula is nota problem at all: Given a propositional formula T we can easily write a propositionalformula T 0 that is equivalent to CIRC(T ). A trivial way to do that is to make a dis-junction of all the minimal models of T , as they are exactly the models of CIRC(T ).It is easy to see that such a process may generate an exponential-size representationof CIRC(T ), as T can have exponentially many minimal models. A smarter methodwould be to compute the extended generalized closed world assumption EGCWA(T )of T , which is equivalent to CIRC(T ) [GPP89, YH85]. Syntactically, EGCWA(T )is T plus a set of clauses, which constrain the models exactly to the minimal ones.Nevertheless the size of EGCWA(T ) may be exponential, as discussed in Section 4.In this paper we prove that, unless the polynomial hierarchy collapses at a su�cientlylow level, as the size of T increases, the size of the explicit representation of CIRC(T )grows faster than any polynomial. This result has several consequences. Supposeyou have a knowledge base T and you want to pose it several di�erent queries undercircumscription. An (apparently) reasonable approach is to rewrite (o�-line) theknowledge base into a propositional one T 0, equivalent to CIRC(T ), and then query(on-line) T 0. This approach seems to move the complexity from on-line to o�-line.Our result shows that, in general, this approach is not feasible and it does not makeon-line reasoning any quicker, due to the super-polynomial increase in the size ofthe knowledge base.While this is a negative result on circumscription, it has a positive side. In fact,our result also implies that circumscription is able to represent information in a verycompact fashion: Imagine you have a certain amount of propositional knowledge tobe represented; you may go for classical semantics or for circumscriptive semantics.Let A and B be the formulae you obtain, respectively (obviously A � CIRC(B)must hold). There are cases where the size of A is signi�cantly bigger than the sizeof B.1.2 Related workThe problem of collapsibility of the circumscription of �rst-order formulae has re-ceived considerable attention in the literature [Lif85, Rab89, GO92, DLS95]. Theissue of the size of the resulting formulae is addressed in [KP90], where it is notedthat computing the �rst-order sentence equivalent to the circumscription of a �rst-6

order existential formula T is possible, but its size is exponential wrt T . The questionwhether this is inherent to existential �rst-order formulae is left as an open problem.A pragmatic approach to the problem is taken in [NNS95], where an algorithm forcomputing all minimal models of a deductive database is proposed. The underlyingidea of the method is to store the set of models, once they are computed. Thealgorithm for computing the minimal models is based on a translation of the databaseinto an integer programming problem. Our results suggest that, even for such asophisticated technique, there must be cases in which the space needed to store themodels is super-polynomial.Related work appears also in AI: A popular idea in this �eld is that of prepro-cessing a logical formula T to obtain a data structure in which fast algorithms foranswering T j= Q can be used. In general, one wants a vivid form of knowledgewhere reasoning is polynomially tractable [Lev86]. An example of this kind is re-ported in [MT93]. In the paper they analyze the possibility of speeding up queryanswering in propositional logic (i.e., checking whether T j= Q holds) through aprevious o�-line transformation of the theory T . Abstractly, they want to transforma coNP-complete problem into a polynomial one (obviously not in polynomial time).In the same spirit, our problem can be seen as an attempt at transforminga reasoning problem into a simpler one via o�-line reasoning. In fact, if we areable to construct a polynomial-size T 0 equivalent to CIRC(T ) then inference un-der circumscription (which is �p2-complete [EG93]) is transformed into inferencein propositional logic (which is coNP-complete). Similarly, model checking in cir-cumscription (i.e., given a propositional theory T and an interpretation M decidewhether M j= CIRC(T ), a coNP-complete problem [Cad92]), is transformed intomodel checking in propositional logic (which is solvable in polynomial time). There-fore it is unlikely that the transformation from T to T 0 can be accomplished inpolynomial time. In fact in this paper we do not impose any restriction on the timeneeded for the construction of T 0, which could even be a non-recursive process.This technique has also been used in related �elds. As an example, in [CDLS95]the issue of the size of a formula which is the result of a revision of a propositionalknowledge base is analyzed. In particular, such a size can be proven to be eitherpolynomial or super-polynomial, depending on factors such as the semantics adoptedfor belief revision, or the notion of equivalence which is taken into account. In[CDS94, CDS95a], the issue of \compiling" polynomially intractable non-monotonicinference problems into polynomial-time solvable problems is addressed.The idea of \compiling" a propositional formula into another formula (calledHorn least upper bound) which is not a faithful representation of the original one isproposed in [SK91, KS92]. By using non-uniform complexity classes, the authors areable to exhibit the proof that the Horn least upper bound may have super-polynomial7

size wrt the original formula.1.3 Main resultsWe focus on the size of representations of the circumscription CIRC(T ) of a propo-sitional formula T . Two distinct notions of representation are considered: model-based (i.e., preserving the set of models), and query-based (i.e., preserving the setof theorems). As far as the former is concerned, it is useful to distinguish betweenpropositional formulae having the same set of models, and generic data structures(e.g. boolean circuits) that allow to do model checking. As for the latter, such adistinction is not necessary, as we will see in Section 3.3.Three notions of equivalence, which are now informally described, are thereforeconsidered:logical equivalence: Propositional formula T 0 is logically equivalent to CIRC(T )i� they have exactly the same models, i.e., for each truth assignment M ,M j= CIRC(T ) i� M j= T 0;model equivalence: Data structure D is model equivalent to CIRC(T ) i� thereexists a polynomial-time algorithm ASK such that it is possible to do modelchecking for CIRC(T ) by using D, i.e., for each truth assignment M , M j=CIRC(T ) i� ASK(D;M) returns \yes";query equivalence: Propositional formula T 0 is query equivalent to CIRC(T ) i�they have exactly the same theorems on the common language, i.e., for eachformula Q in which only symbols of T occur, fQ j CIRC(T ) j= Qg = fQ j T 0 j=Qg.The three notions are partially ordered with respect to their strength: A formula T 0satisfying logical equivalence is also a data structure satisfying model equivalence anda formula satisfying query equivalence. Analogously, a data structure D satisfyingmodel equivalence satis�es query equivalence as well. The other direction doesnot necessarily hold. Intuitively, model equivalence allows to do model checkingby using circuits, while query equivalence gives the possibility of introducing newpropositional atoms.Logical equivalence is the strongest notion, but the other ones might have a prac-tical interest in �elds such as Automated Theorem Proving or Deductive Databases.We �nd necessary and su�cient conditions for the existence of polynomial-sizerepresentations equivalent to CIRC(T ) in the three di�erent cases. All such condi-tions imply the collapse of the polynomial hierarchy. In particular, we prove that {8

unless the polynomial hierarchy collapses at the second level { the size of the short-est propositional formula T 0 logically equivalent to CIRC(T ) grows faster than anypolynomial as the size of T increases. The major tools we use are:1. the notion of non-uniform computation [KL80, Joh90];2. results proving that { in propositional circumscription { logical entailment is�p2-complete [EG93] and model checking is coNP-complete [Cad92];3. results relating inclusion of uniform complexity classes into non-uniform com-plexity classes to the collapse of the polynomial hierarchy [KL80, Yap83].1.4 Structure of the paperThe structure of the paper is the following: In the next section we recall somede�nitions about propositional circumscription and non-uniform computation; then,in Section 3 we prove our main results. In Section 4 we discuss these results, andanalyze their signi�cance in the related �eld of closed-world reasoning. In the lastsection we draw some conclusions and address open problems.2 PreliminariesThe alphabet of a propositional formula is the set of all propositional atoms occurringin it. An interpretation of a formula is a truth assignment to the atoms of itsalphabet. A model M of a formula T is an interpretation that satis�es T (writtenM j= T ). Interpretations and models of propositional formulae will be denoted assets of atoms (those which are mapped into 1). Given a propositional formula T , wedenote with M(T ) the set of its models. Following Lifschitz [Lif85], we de�ne:De�nition 1 Let M 2 M(T ). M is called a minimal model of T if there is nomodel N of T such that N �M (i.e., N 6�M and N �M .)De�nition 2 Let T be a propositional formula and X = fx1; : : : ; xng its alphabet.The circumscription CIRC(T ) is the following quanti�ed boolean formulaT ^ (8Y:T [Y ]! :(Y < X)) (1)where Y = fy1; : : : ; yng is a list of atoms disjoint from X, T [Y ] is T with all theoccurrences of atoms of X substituted by the corresponding ones in Y . The meaningof Y < X is de�ned in terms of the relation �. In particular Y < X is (Y �X) ^ :(X � Y ), and Y � X stands for the conjunction of the formulaeyi ! xi (1 � i � n):9

Proposition 1 (Lifschitz [Lif85, Proposition 1]) A model M of T is minimali� it is a model of CIRC(T ), i.e., i� M j= CIRC(T ).More sophisticated de�nitions of minimal models and circumscription have been de-�ned (e.g., not all atoms are minimized [GPP89, De�nition 3.1]). As such de�nitionsare extensions of basic circumscription, the results we present in this paper hold forthem.Throughout this paper, the symbol jxj denotes the size of x and also the cardi-nality of x, when x is a set. Furthermore, h�; �i denotes a pairing function over binary�nite strings with the standard nice computability and invertibility properties, andsuch that 8x; y:jhx; yij = 2jxj+ 2jyj.As already pointed out, our proof uses the notion of non-uniform computations.We now brie y recall the de�nitions needed in the sequel (Cf. [KL80]).De�nition 3 An advice function is a function A : N! f0; 1g�, where N is the setof non-negative integers. The advice function A is polynomial if jA(n)j � p(n) forsome polynomial p and all non-negative integers n.De�nition 4 Let C be any class of languages. C=poly is the class of all languagesof the form fx j hA(jxj); xi 2 Lg where L 2 C and A is a polynomial advice function.Any class C/poly is also known as non-uniform C. Non-uniformity is due to thepresence of the advice. Notice that the advice depends only on the size of the input,not on the input itself. Throughout the paper, we will be interested in several non-uniform classes. More precisely, we use the classes P/poly, NC1/poly, NP/poly andcoNP/poly. Following Johnson [Joh90], the class NC1 is de�ned as:De�nition 5 The class NC1consists of all languages recognizable by log-space uni-form families of Boolean circuits having polynomial size and depth O(log n).It is important to point out that both P/poly and NC1/poly are closed un-der complementation, while NP/poly and coNP/poly are classes of complementarylanguages.Relations between non-uniform and uniform complexity classes, were studied inthe literature by several researchers (e.g. [KL80, Yap83]). The relevant results, forour work, can be summarized as follows:NP � NC1/poly ) �p2 = PH [KL80]NP � P/poly ) �p2 = PH [KL80]NP � coNP/poly ) �p3 = PH [Yap83]10

That is, if NP is included in any of the three classes NC1/poly, P/poly or coNP/polythen the polynomial hierarchy collapses at a low level (either the second or the thirdone).Clearly, NP � NC1=poly ) NP � P=poly ) NP � coNP=poly, but the inverseimplications are not known to hold. In particular, condition \NP � NC1=poly"seems to be much stronger than \NP � P=poly" in that NC1 is a class that isbelieved much smaller than P; indeed NC1 � LOGSPACE. Also, while it is notcurrently known whether P is a proper subclass of PSPACE, this is known to betrue for NC1.3 Compact RepresentationsIn this section we address the three forms of equivalence previously de�ned, eachone in a separate subsection. We prove that the existence of an equivalent com-pact representation of the circumscription of a propositional formula corresponds tothe inclusion of NP in a non-uniform class, di�erent for any notion of equivalenceconsidered.Many of the proofs share a common property, which we state and prove as ageneral lemma below. Intuitively, the lemma tells us that there is a class of proposi-tional formulae Tn;m, such that the validity of any quanti�ed boolean formula F withn universally quanti�ed variables followed by m existentially quanti�ed variables isequivalent to entailment under circumscription between Tn;m and a particular queryQF . The formal statements follow.Given two sets of propositional atoms X = fx1; : : : ; xng, Y = fy1; : : : ; ymg, anda 3CNF formula E containing literals on the alphabet X [Y , we call a 89-QBF thefollowing quanti�ed boolean formula F :F = 8x1; : : : ; xn9y1; : : : ; ym:E (2)We call E the matrix of F .Lemma 2 Let X and Y be two alphabets of n and m atoms, respectively, Z analphabet one-to-one with X (i.e., Z has n atoms), and W a fourth alphabet with16(n + m)3 + 1 atoms. There exists a 5CNF formula Tn;m over X [ Y [W [ Z(depending only on n and m, of polynomial size wrt n + m) such that given any89-QBF F of the form (2), there exists a clause QF , containing all atoms in Y [W ,such that F is valid i� CIRC(Tn;m) j= QF .Proof. The proof is inspired by a reduction given in [EG93], showing that inferencein a 3CNF-theory under circumscription is �p2-hard. The key di�erence is that wenow need to encode every possible 89-QBF of the form (2) in our theory Tn;m.11

Let C be a set of new atoms, one for each three-literals clause over X [ Y , i.e.,C = fci j i is a three-literals clause of X [ Y g. Moreover, let D be a set of newatoms in one-to-one correspondence with atoms of C, and let Z be as above. Finally,let W be the set of atoms C [D[fug, where u is a distinguished atom. Notice thatjW j is 2(2(n+m))3 + 1 = 16(n+m)3 + 1.We want to impose non-equivalence between atoms inX and their correspondingatoms in Z, and the same for C and D. We call �n;m the 2CNF formula made upby the following clauses:1. for each atom xi of X, there are two clauses xi _ zi, and :xi_:zi in �n;m (2nclauses),2. for each atom ci of C, there are two clauses ci _ di, and :ci _ :di in �n;m(2(2(n +m))3 clauses),Now we want to encode every possible 3CNF formula over X [ Y , using the atomsin C as \enabling gates". We call �n;m the 4CNF formula containing, for eachthree-literals clause i over X [ Y , a clause i _ :ci ((2(n +m))3 clauses).We de�ne T (omitting subscripts n;m from now on, for readability) as:� ^ ((u ^ y1 ^ � � � ^ yn) _ (:u ^ �)) (3)Notice that the size of T is O((n+m)3), and T can be rewritten as an equivalent5CNF formula. Moreover, T does not depend on a speci�c 89-QBF F , but only onX and Y .Given the 89-QBF F with matrix E, we denoteCE = fci 2 C j i is a clause of Egand similarly for DE . Moreover, CE = C � CE, and DE = D � DE . GivenM 2M(�), we denote E(M) =M [ Y [ fug.We �rst note the following properties of T .Lemma 3 Let T be as in (3). Then:1. T is satis�able.2. If M 2M(�) then E(M) 2M(T ).3. Every model M 2M(T ) de�nes a unique matrix E = f i j ci 2Mg.4. For every model M 2 M(T ) such that M \ C = CE, M satis�es � i� Msatis�es E. 12

Proof.1. Let M = D [X. Obviously M j= � and M j= :u ^ �. Therefore M j= T .2. M already satis�es �; moreover, Y [ fug satis�es the �rst disjunct in (u ^y1 ^ � � � ^ yn) _ (:u ^ �).3. Obvious.4. M satis�es � i� it satis�es all the clauses i _ :ci of �. Since M \ C = CEand each clause of � contains a di�erent :ci, each clause is satis�ed i� eitherit contains a :ci such that ci 2 CE or the remaining part of the clause ( i) issatis�ed. But the conjunction of these remaining clauses is exactly E.We de�ne the query QF as:QF = :u _ :y1 _ � � � _ :ym _ ( _ci2CE :ci) _ ( _ci2CE ci) _ ( _di2DE di) _ ( _di2DE :di)Observe that QF contains all atoms of Y [W . We prove the lemma by showingthat F is valid i� CIRC(T ) j= QF .If part. Suppose that QF is true in all minimal models of T . Consider any modelM 2 M(�) such that M \ C = CE , that is, M contains exactly the atoms of Ccorresponding to clauses of E. Note that the model E(M) = M [ Y [ fug cannotbe a minimal model, because it does not satisfy QF . Hence for any such M thereexists a di�erent extension E1(M), with E1(M) � E(M), that is a minimal model ofT satisfying QF . Since E1(M) satis�es QF , it must satisfy :u. As a consequence,E1(M) j= � (Cf. (3)). But any model satisfying � and whose intersection with Cequals CE satis�es also E (Cf. Lemma 3 point 4). Hence for any M 2 M(�) suchthat M \C = CE, there is an extension E1(M) satisfying E. Since each M 2M(�)contains a truth assignment to variables in X, it follows that for each assignment tovariables in X there is an assignment to variables in Y (namely, E1(M) \ Y ) suchthat E is satis�ed. Therefore, F is valid.Only if part. Assume that there exists a minimal model M of T such that M j=:QF . Observe that M j= u ^ y1 ^ � � � ^ yn, M \ C = CE , M \ D = DE . LetM� =M \ (X [Z [C [D). Obviously, M is an extension of M�. The minimalityof M implies that M 6j= �, otherwise M � fug would be a model of T . SinceM \C = CE and M 6j= �, by Lemma 3 (point 4) we also have that M 6j= E. Againfrom the minimality of M , it follows that no other extension M 0 of M� satis�es T .13

As a consequence, there exists an assignment to the variables in X (i.e., M� \X)for which there is no assignment to the variables in Y that makes E true. ThereforeF is not valid.3.1 Model equivalenceIn this section we prove that, unless the polynomial hierarchy collapses at the secondlevel, there is no polynomial in jT j bounding the size of the shortest data structurerepresenting exactly the minimal models of T . We recall the notion of model equiv-alence, �xing it to polynomial-size data structures:Let p be a �xed polynomial; for any propositional formula T we want to �nd adata structure DT with the following characteristics:1. jDT j < p(jT j);2. there exists a relation ASK(�; �), such that given any interpretation M of T ,ASK(DT ;M) is true i�M 6j= CIRC(T ) (i.e., ASK computes the complementof model checking);3. deciding the relation ASK(�; �) is a problem in P, where the inputs are itsarguments.Intuitively, this means that we are trying to \compile" CIRC(T ) in such a waythat the NP-complete problem of deciding M 6j= CIRC(T ) becomes a problem inP. Note that a way of doing that would be to rewrite CIRC(T ) into an equivalentpropositional formula T 0 of size bounded by p(jT j), where ASK corresponds to thecomplement of classical model checking, i.e., ASK(T 0;M) = true i�M 6j= T 0 (whichcan be checked in time polynomial wrt the size of M and T 0). However, we are nowlooking not just for a formula, but for any data structure (i.e., any circuit).We are able to show that it is very unlikely that such a polynomial p and datastructure DT may exist. As a consequence, T 0 does not exist either. In order toprove this, we resort to the notion of non-uniform computation. In what follows, arelation R such that deciding R is a problem in P will be called a P-relation.Theorem 4 Let p be any polynomial and let ASK(�; �) be a P-relation. If for eachCNF formula T there is a data structure DT such that jDT j < p(jT j) and for anyinterpretation M , M 6j= CIRC(T ) if and only if ASK(DT ;M) is true, then NP �P=poly.Proof. Consider Lemma 2, with X = ; (i.e., n = 0, and Z = ;, too). In this case thelemma says that there exists a 5CNF formula T0;m over Y [W such that given any14

89-QBF F using the atoms in Y for existentially quanti�ed variables, there exists aclause QF , containing all atoms in Y [W , such that F is valid i� CIRC(T0;m) j= QF .Observe that CIRC(T0;m) 6j= QF i� there exists a minimal model MF satisfying:QF . Since QF contains all atoms in T0;m, :QF uniquely identi�es the model MFof CIRC(T0;m) falsifying QF . Hence, F is valid i� MF 6j= CIRC(T0;m).Let us assume that there exists a polynomial p with the properties claimed inthe statement of the theorem. Then, for each T0;m there exists a data structureDT0;m , with jDT0;m j < p(jT0;mj), and a P-relation ASK(�; �) such that given anyinterpretation M of T0;m, ASK(DT0;m ;M) is true i� M 6j= CIRC(T0;m). From theabove particular case of Lemma 2, it follows that for any 89-QBF F with X = ;and jY j = m, F is valid i� ASK(DT0;m ;MF ) is true. From DT0;m one can de�nea polynomial advice function which depends only on m for deciding validity of89-QBF-formulae with X = ;. Since deciding the validity of such formulae is anNP-complete problem (it is just the well-known 3SAT problem), NP � P=poly.Non-uniform complexity classes for proving lower bounds on the size of formulae inknowledge representation were used �rst in [KS92].The above theorem shows the unfeasibility, under certain conditions, of compilingthe original circumscription so that the compiled version is more e�ective whenperforming model checking. Notice that no bound is imposed on the time spent inthe compilation process.One may wonder if a stronger unconditioned theorem holds, namely, a theoremsaying that there is no bounding polynomial at all, without referring to NP �P=poly. We are not able to prove such a theorem. However, we can prove that suchan unconditioned theorem would imply NP 6� P=poly, by proving the converse ofTheorem 4. Note that, since P � P=poly, if NP 6� P=poly then NP 6= P. Henceproving the unconditioned non-existence of a compact data structure would be aresult at least as strong as proving NP 6= P.Theorem 5 If NP � P=poly then there exists a P-relation ASK(�; �) and a polyno-mial p such that for each propositional formula T there is a data structure DT suchthat jDT j � p(jT j) and for any interpretation M of T , M 6j= CIRC(T ) if and onlyif ASK(DT ;M) is true.Proof. For any propositional formula T , let nT be the number of propositional atomsof T . If M is an interpretation of T we denote by M also the encoding of M as abinary string of length nT . Moreover, let T denote also an encoding of the formulaT . Let L = fhT;Mi j M is an interpretation of T and M 6j= CIRC(T )g. Clearly,L 2 NP. Since NP � P=poly, it follows that there exist R 2 P, an advice function15

A, and a polynomial q such that 8n jA(n)j � q(n) and8T;M hT;Mi 2 L () hA(jhT;Mij); hT;Mii 2 R:For any T , let DT = hT;A(2jT j+2nT )i. It is clear that there is a polynomial p suchthat 8T jDT j � p(jT j). De�ne ASK(DT ;M) as hA(2jT j + 2nT ); hT;Mii 2 R. Notethat fromDT it is possible to compute in polynomial time both T and A(2jT j+2nT ).Hence, ASK(DT ;M) is computable in polynomial time. Moreover, ASK(DT ;M)is true i� M 6j= CIRC(T ). In fact, for any interpretation M of T ,ASK(DT ;M) is true () hA(2jT j + 2nT ); hT;Mii 2 R (def. of ASK)() hA(jhT;Mij); hT;Mii 2 R (def. of jhT;Mij)() hT;Mi 2 L (def. of R and A).3.2 Logical equivalenceTheorem 4 shows that, in general, compact representations of CIRC(T ) by datastructures do not exist, unless the polynomial hierarchy collapses to the secondlevel. Moreover, Theorem 5 says that this result cannot be improved, in the sense ofproving the non existence of compact representations unconditionally, unless we areable to settle some very hard conjecture like P 6= NP. Nevertheless, we could stillhope to unconditionally prove the non-existence, in general, of compact representa-tions by a very specialized kind of data structures such as propositional formulae.However, the next result shows that this is at least as hard as to solve another oldconjecture, namely NC1 6= NP.Theorem 6 There exists a polynomial p such that for each propositional formulaT there is a formula T 0, over the same alphabet of T , which is logically equivalentto CIRC(T ) and whose size is bounded by p(jT j), if and only if NP � NC1=poly.Proof. Only if part. We need the following notations. For any integer k > 0, letVk = fv1; : : : ; vkg be a �xed alphabet of propositional atoms. For any binary stringv of length k, let Mv = fvi j ith symbol of v is 1g.We will prove that for any L 2 NP a polynomial q exists such that, for anyk, there is a propositional formula k over the alphabet Vk, of size at most q(k),such that v 2 L if and only if Mv j= k, for all v of length k. From a result ofSpira [Spi71], showing that every polynomial-size formula can be converted into anequivalent boolean circuit of logarithmic depth, it follows that L 2 NC1=poly, andthus NP � NC1=poly. 16

Let L be any language in NP. By the Cook-Levin theorem, for any k, there isa 3CNF formula 'k, over an alphabet Y with Vk � Y , whose size is polynomiallybounded in k, and such that, for every v of length k,v 2 L () 9N � Y � Vk : Mv [N j= 'k:Let T0;m be the formula constructed in the proof of Lemma 2 with respect to thealphabets X = ; and Y with jY j = m. Recall that the alphabet of T0;m is, in thiscase, the set Y [ C [ D [ fug, where C and D are two sets of new atoms whichare both in one-to-one correspondence with the set of all the three-literals clausesof Y , and u is a distinguished new atom. By Lemma 2 it follows that for any 3CNFformula E over the alphabet Y , E is satis�able if and only if ME 6j= CIRC(T0;m),where ME = Y [ CE [DE [ fug, CE = fc 2 C j c corresponds to a clause of Eg,and DE = fd 2 D j d does not correspond to a clause of Eg.For any v of length k, let Ev = �1 ^ � � � ^ �k, where �i is equal to vi _ vi _ viif the ith symbol of v is 1, and is equal to :vi _ :vi _ :vi otherwise. Observe thatMv is the unique model of Ev. Let 'v = Ev ^ 'k. It holds that v 2 L if andonly if 'v is satis�able. Therefore, v 2 L if and only if M'v 6j= CIRC(T0;m). Byhypothesis there exists a propositional formula T 0, over the same alphabet of T0;m,which is equivalent to CIRC(T0;m) and whose size is bounded by p(jT0;mj). Forevery i = 1; : : : ; k, let ai and ai be the atoms of C that correspond to the clausesvi _ vi _ vi and :vi _ :vi _ :vi respectively, and similarly for bi and bi as atomsof D. Moreover, let C1 = fai; ai j i = 1; : : : ; kg and D1 = fbi; bi j i = 1; : : : ; kg.Let k be the formula over the alphabet Vk obtained from :T 0 by substituting truefor every occurrence of atoms in M'v � (C1 [D1), false for every occurrence of theremaining atoms that do not belong to (C1 [D1), vi for every occurrence of atomsai and bi, and :vi for every occurrence of atoms ai and bi. It is immediate to verifythat Mv j= k if and only if M'v j= :T 0. Hence, v 2 L if and only if Mv j= k.If part. For any propositional formula T , let nT be the number of propositionalatoms of T . If M is an interpretation of T we denote by M also the encoding ofM as a binary string of length nT . Moreover, T will also denote an encoding of theformula T .Since L = fhT;Mi j M is an interpretation of T and M 6j= CIRC(T )g belongsto NP and NP � NC1=poly, there exists a family fCng of boolean circuits such that,for any n, Cn has n inputs, computes L \ f0; 1gn, and has depth at most k log nfor some constant k. Let T be any propositional formula and let m = 2jT j + 2nT .Circuit Cm, on input hT;Mi, outputs 1 if and only if M 6j= CIRC(T ). It is easy tosee that Cm can be converted to a circuit C with nT inputs, depth at most that ofCm, and such that C on input M outputs 1 if and only if M j= CIRC(T ). Sinceany circuit of depth d can be converted to a formula of size at most 2d, there is a17

formula T 0, over the same alphabet of T , of size at most 22kjT jk, which is equivalentto CIRC(T ).3.3 Query equivalenceTheorem 6 states that the size of any formula T 0 such that T 0 � CIRC(T ) growsfaster than any polynomial as the size of T increases. What happens if we giveup logical equivalence and go for the weaker \query equivalence"? Using a similartechnique, we are able to prove that a polynomial-sized T 0 query-equivalent to Texists if and only if NP � coNP=poly.Theorem 7 There exists a polynomial p such that for each propositional formulaT there is a formula T 0 over an extended alphabet, whose size is bounded by p(jT j),such that fQ j T 0 j= Qg = fQ j CIRC(T ) j= Qg where Q is any formula over thealphabet of T , if and only if NP � coNP=poly.Proof. We exploit the fact that NP � coNP=poly if and only if �p2 � coNP=poly(see [Yap83]), and refer to the latter inclusion in what follows.Only if part. We show that the �p2-complete problem of deciding validity of 89-QBF formulae belongs to coNP=poly. By Lemma 2, for any n;m, there is a formulaTn;m such that for any 89-QBF F of the form (2) it holdsF is valid () CIRC(Tn;m) j= QF :By hypothesis, in correspondence with Tn;m, there exists a polynomial-size proposi-tional formula T 0n;m such that fQ j T 0n;m j= Qg = fQ j CIRC(Tn;m) j= Qg where Qis any formula over the alphabet of Tn;m. This implies that, for any 89-QBF formulaF with n 8-quanti�ers and m 9-quanti�ers,F is valid () hT 0n;m; F i 2 R:where R = fhT; F i j T j= QF g is clearly a language in coNP. It is very easy to verifythat the above implies that the 89-QBF problem belongs to coNP=poly, and hence�p2 � coNP=poly.If part. For any propositional formula T , let nT be the number of propositionalatoms of T . If M is an interpretation of T we denote by M also the encoding ofM as a binary string of length nT . Moreover, T will also denote an encoding of theformula T .We need the following notations. For any integer n > 0, let Zn = fz1; : : : ; zngbe a �xed alphabet of propositional atoms. For any binary string z of length n, letMz = fzi j ith symbol of z is 1g. 18

Observe that �p2 � coNP=poly implies that coNP � NP=poly. Thus, sinceL = fhT;Mi jM is an interpretation of T andM j= CIRC(T )g belongs to coNP andcoNP � NP=poly, there exists an R 2 NP, an advice function A and a polynomialq, such that jA(n)j � q(n) for all n, and8T;M hT;Mi 2 L () hA(jhT;Mij); hT;Mii 2 R:By the Cook-Levin theorem, for any n, there is a 3CNF formula 'n, over the alphabetZn [ Yn, whose size is bounded by r(n) for some �xed polynomial r, and such that,for every z of length n,z 2 R () 9N � Yn : Mz [N j= 'n:Let T be any propositional formula, let X be the alphabet of T , and let m =2jA(2jT j + 2nT )j + 2(2jT j + 2nT ). Note that jhA(jhT;Mij); hT;Miij = m for anyinterpretation M of T , and m � 2q(4jT j) + 8jT j. Since T and A(2jT j + 2nT ) are�xed, we can easily convert 'm into a formula T 0 over the alphabet X [ Ym, whosesize is at most that of 'm, and such that, for any interpretation M of T ,M j= CIRC(T ) () 9N � Ym :M [N j= T 0:Thus, the size of T 0 is at most r(2q(4jT j) + 8jT j) and, for any formula Q over thealphabet X of T , T 0 j= Q if and only if CIRC(T ) j= Q.Note that, in the case of the query equivalence, the existence of compact represen-tations by data structures is equivalent to the existence of compact representationsby propositional formulae. This is in contrast with the case of the model and logicalequivalences in which the two kinds of compact representations do not seem equiv-alent. In fact, the former is possible if and only if NP � P=poly, while the latter ispossible if and only if the stronger condition NP � NC1=poly holds. This leads us toconjecture that there may exist subclasses of formulae whose circumscriptions admitcompact representations by data structures but not by propositional formulae.4 Analysis of the ResultsIn this section we analyze the generality of our results and their impact on a topicstrictly related to circumscription, namely closed-world reasoning. Closed-worldreasoning is a collection of ideas and de�nitions developed in the Database �eld foraddressing the issue of reasoning using lack of information. Motivations for closed-world reasoning are very close in spirit to those behind circumscription. The maindi�erence is that, while the circumscription of a propositional formula T is de�ned19

as a second-order formula (cf. formula (1)), making the closure of T amounts toadding to T new propositional formulae according to some criterion (cf. formula(4)). Despite these syntactical di�erences, the two approaches are strictly related atthe semantic level.We �rst recall two di�erent proposals of Closed-World Assumption (CWA): Gen-eralized CWA (GCWA) and Extended Generalized CWA (EGCWA). Then we showhow the proof of our main theorem can be used to de�ne theories whose closureunder EGCWA has super-polynomial size. Finally, we discuss the generality of ourtechnique (\is it always possible to exploit intractability results to show incompress-ibility?") and take GCWA as an example of a closure operator which is compressible.The reason why compressibility of GCWA is interesting is that the two closure op-erators have similar time complexity: if T; q are propositional formulae and M isan interpretation, testing GCWA(T ) j= q and EGCWA(T ) j= q are both �p2-hardproblems, and testingM j= GCWA(T ) andM j= EGCWA(T ) are both coNP-hardproblems.4.1 Closed-World ReasoningGeneralized Closed World Assumption GCWA(T ) of a propositional formula T[Min82] is de�ned as follows (K is an atom and B is a clause { possibly empty{ in which only positive literals occur):T [ f:K j 8B: T 6j= B ) T 6j= B _Kg: (4)All models of CIRC(T ) are models of GCWA(T ), but not the other way around[Min82, Theorem 2].A semantically more clear formalism for treatment of incomplete information isExtended Generalized Closed World Assumption EGCWA(T ) [YH85]. Its de�nitionis like (4), except thatK is now an arbitrary conjunction of atoms. Such conjunctionsare called \free-for-negation" for T . Observe that in a reasonable representation ofEGCWA(T ), only minimal conjunctions of atoms need to be added to T , wherea free-for-negation conjunction K is minimal i� any subconjunction of K is notfree-for-negation. The models of EGCWA(T ) are exactly the models of CIRC(T )[YH85], therefore Theorem 6 says that the size of EGCWA(T ) is not likely to bepolynomial in jT j, as jT j increases.It is worth noting that EGCWA(T ) might be a much smarter representation ofCIRC(T ) than listing all minimalmodels of T . As an example, let a1; : : : ; an; b1; : : : ; bnbe distinct atoms and T be (a1_b1)^� � �^(an_bn). EGCWA(T ) is T ^(:a1_:b1)^� � � ^ (:an_:bn). The simple-minded representation of CIRC(T ) is the disjunctionof all possible conjunctions x1^� � � ^xn^:y1^� � � ^:yn, where for all i (1 � i � n),20

xi is a member of fai; big, and yi is the other member. The latter representationhas clearly exponential size.4.2 Large Instances of EGCWAWe now are able to reveal an in�nite set of T 's, where { even when consideringminimal free-for-negation conjunctions { the size of EGCWA(T ) is superpolynomial.Such formulae are inspired by the one built in the proof of Lemma 2. This is, tothe best of our knowledge, the �rst example proving that such a smart techniquefor representing propositional circumscription outputs, in the worst case, a theoryof superpolynomial size.We use the alphabets of atoms X = fx1; : : : ; xng, C = fc+1 ; c�1 ; : : : ; c+n ; c�n g,D = fd+1 ; d�1 ; : : : ; d+n ; d�n g and a new distinct atom u. We de�ne a propositionalformula Tn over these alphabets, with the help of two formulae �n;�n, which areanalogous to �n;m;�n;m of Lemma 2. To simplify notation, we use a 6= b as ashorthand for (a _ b) ^ (:a _ :b). Let�n =^fxi _ :c+i j1 � i � ng ^^f:xi _ :c�i j1 � i � ng�n =^fc+i 6= d+i j1 � i � ng ^^fc�i 6= d�i j1 � i � ngTn = �n ^ [(u ^ x1 ^ � � � ^ xn) _ �n]Notice that the size of X [ C [D is 5n, and the size of Tn is O(n). Given a subsetE of X, we de�ne CE = fc+i jxi 2 Eg [ fc�i jxi 62 Eg, and similarly DE = fd+i jxi 2Eg [ fd�i jxi 62 Eg. Moreover, let DE = D �DE = fd+i jc+i 62 CEg [ fd�i jc�i 62 CEg.Lemma 8 Let Tn be as above, and for any E � X let ME = E [ CE [DE. ME isa minimal model of Tn.Proof. Since E [ CE satis�es �n, and CE [DE satis�es �n, ME is a model of Tn.Suppose M �ME is also a model of Tn, and let MC =M \ C, MD =M \D, andMX =M \X. Since M satis�es �n, if MC � CE then MD � DE , and vice versa ifMD � DE then MC � CE . Hence, MC = CE and MD = DE . Therefore, it shouldbe MX � E, so let xi 2 E �MX . By de�nition of CE, c+i 2 CE , hence the clausexi _ :c+i is not satis�ed by M , hence �n is not satis�ed. Since also the conjunctionu^ x1 ^ � � � ^ xn is not satis�ed, we conclude that any such M is not a model of Tn,therefore ME is minimal.We exploit the previous property in the proof of our next theorem. To simplifynotation, we denote by DE ^ u the formula obtained as a conjunction of all atomsin the set DE and u. 21

Theorem 9 Let E be any subset of X; then DE ^ u is a minimal free-for-negationformula for Tn.Proof. First of all, we show by contradiction that DE ^ u is free-for-negation:Assume there exists a minimal model M of Tn satisfying DE ^ u. Now, M mustalso satisfy x1; : : : ; xn, because otherwise M satis�es �n, and M � fug would be amodel, contradicting minimality of M . Therefore M � DE [ fx1; : : : ; xng [ fug.Let MC =M \ C, MD =M \D, so that M can be partitioned as MC [MD [fx1; : : : ; xng [ fug. Since M satis�es DE ^ u, then MD � DE. Then MC � CE ,since M satis�es also �n. Now let N = E [MC [MD. We show that N satis�esTn. First, N satis�es �n because it gives the same interpretation as M to literals inC and D. We now show that N satis�es each clause of �n.1. Let xi 2 E. Then the clause xi _ :c+i is satis�ed by N . By de�nition of CE ,c�i 62 CE . Since MC � CE, also c�i 62 MC . Hence the clause :xi _ :c�i issatis�ed too.2. Let xi 62 E. Then the clause :xi _ :c�i is satis�ed. By de�nition of CE, thistime c+i 62 CE , so c+i 62MC . Hence the clause xi _ :c+i is satis�ed.Since N � M , M is not minimal, contradicting the hypothesis. We conclude thatDE ^ u is free-for-negation.We now show that if we remove one conjunct from DE ^u, the resulting formulais not free-for-negation, thus showing that DE ^ u is a minimal free-for-negationformula.First observe that if we remove u, then DE (considered as a conjunction) is notfree-for-negation because ME satis�es it, and by Lemma 8 ME is a minimal model.Secondly, we prove that if we take out a literal d�i 2 D from DE ^ u the resultingformula is not free-for-negation. In fact, letM = (DE�fd�i g)[CE[fc�i g[fug[Xbe an interpretation satisfying the smaller conjunction. It holds that M satis�es�n, hence M is also a model of Tn because it satis�es u ^ x1 ^ � � � ^ xn. We nowshow that M is also a minimal model of Tn, by proving that for any model N suchthat N � M , it results N = M . Since N satis�es �n, if N \ C � M \ C thenN \ D � M \ D. Hence to be N � M , it must be N \ C = M \ C, and alsoN \D = M \ D. Therefore N and M can di�er at most on X [ fug. But noticethat both c+i and c�i belong to M , hence they belong to N too. Observe that �ncontains the two clauses xi _:c+i and :xi _:c�i , which cannot both be satis�ed byN , for any possible interpretation of xi. Hence N cannot satisfy �n. Therefore tosatisfy Tn, N must satisfy u ^ x1 ^ � � � ^ xn. But this implies N =M .Since there are exponentially many subsets of X, there are also exponentiallymany distinct free-for-negation conjuncts. So EGCWA(Tn) contains at least 2n22

clauses :K, each clause having n+1 disjuncts. Therefore jEGCWA(Tn)j is (n2n),while jTnj is O(n). Observe also that Tn could be rewritten as a 3CNF-formula (bydistributing the conjunction u^x1 ^ � � � ^xn over �n) having O(n2) clauses. Hence,even when Tn is in 3CNF, the above line of reasoning yields a super-polynomiallower bound for the size of EGCWA(Tn).4.3 Generality of Main ResultIn Theorems 4, 6 and 7 we used the reduction of NP-hard problems { decidingwhetherM 6j= CIRC(T ) or not { and �p2-hard ones { deciding whether CIRC(T ) j=Q or not { to show that a polynomial-size representation of CIRC(T ) is unlikely toexist, regardless of the e�ort we spend for doing the \compilation" of CIRC(T ).The technique employed readily applies to a much wider spectrum of reasoningproblems in knowledge bases. Using well-known reductions of circumscription intoother reasoning problems, we were able to extend our result to the explicit represen-tations of disjunctive databases under the stable [GL88] or well-founded semantics[vGRS91] as extended by Przymusinski [Prz91], skeptical reasoning in default logic[Rei80] and autoepistemic logics [MT91]. Furthermore, in [CDS95a] we apply thismethod to skeptical and credulous reasoning in (fragments of) default logic, whilein [CDLS95] we analyze the space complexity of most operators for belief revisionand update introduced in the literature.In this paper we have shown that the existence of a polynomial-size represen-tation of CIRC(T ) is unlikely to exist, regardless of time needed and equivalencecriterion adopted. In [CDLS95] we presented operators which do not admit a com-pact representation if we require model-equivalence, but there exist some if we onlygo for query-equivalence. Therefore, the three equivalence criteria have di�erentimpacts on the existence of compact representations and it might very well be thecase that, for restricted languages, circumscription admits a compact representationwrt some equivalence criteria but not wrt all of them.However, the technique is not applicable to all reductions of NP-hard problemsin knowledge representation. As an example, we now show that model checkingunder GCWA is coNP-hard, but the closure of a theory under GCWA has always arepresentation of polynomial size.The reduction for GCWA rephrases the one showing that M 6j= CIRC(T ) isNP-hard. Given any formula F on alphabet X = fx1; : : : ; xng, and another atomu 62 X, de�ne T = (F ^:u)_ (u^x1^ � � � ^xn). Let M = fug[X. It can be shownthat F is satis�able i� M 6j= GCWA(T ). Hence model checking under GCWA iscoNP-hard.Nevertheless, there exists a simple polynomial-size explicit representation ofGCWA(T ): for every atom K, simply decide if :K must be added or not to T ,23

T 0 logically equivalent , NP � NC1/poly Th. 6D model equivalent , NP � P/poly Th. 4, 5T 0 query equivalent , NP � coNP/poly Th. 7Table 1: Necessary and su�cient conditions for the existence of polynomially-sizedrepresentations equivalent to CIRC(T )and if so add it. Hence, if T is �xed then GCWA(T ) can be \compiled", once andfor all. Observe that this does not prove NP � P/poly, since the compilation ofGCWA(T ) depends on T itself, and not only on its size.5 Conclusions and open problemsIn this paper we have investigated the size of representations equivalent to the cir-cumscription CIRC(T ) of a propositional formula T , taking into account threedi�erent de�nitions of equivalence. We have found that necessary and su�cient con-ditions for the existence of polynomial-size representations equivalent to CIRC(T )in the three cases is inclusion of NP into three di�erent non-uniform complexityclasses. As such conditions imply the collapse of the polynomial hierarchy at a lowlevel, it is likely that the size of a propositional representation of CIRC(T ) growsfaster than any polynomial as the size of T increases.We want to point out that we identi�ed the exact conditions under which com-pact representations exist. These results cannot be easily strengthened as provingthe existence of compact representations implies a collapse in the polynomial hier-archy, while, for example, proving their non-existence under the model equivalencecriterium implies that P 6= NP.This result has a negative side: It is unfeasible to (o�-line) compile a knowledgebase so that (on-line) reasoning under circumscription becomes easier. On the otherside, our results imply that circumscription allows more compact representation ofknowledge. As a consequence, circumscription may be used to produce a compactrepresentation of some boolean functions whose propositional representation is in-herently super-polynomial wrt the number of boolean variables.The results presented in this paper are summarized in Table 1.Some interesting questions that we did not consider in the present work arebrie y listed: 24

1. Are there syntactically restricted classes of formulae for which, e.g., polynomially-sized query equivalent formulae exist, while logically equivalent formulae donot?2. Why have formalisms with similar time complexity (e.g., GCWA and EGCWA)di�erent compactability properties?3. The degree of undecidability of in�nitary propositional (sentential) circum-scription has been analyzed in [PW90], where it is proven that the inferenceproblem is more di�cult than the corresponding problem in in�nitary propo-sitional logic. What is the impact of such a result from the point of view ofthe size of the representation?AcknowledgmentsThe authors are grateful to Pierluigi Crescenzi for an interesting discussion on thenon-uniform polynomial hierarchy and to Phokion Kolaitis for suggesting us to in-vestigate \inverse" results, such as Theorem 5.References[Cad92] M. Cadoli. The complexity of model checking for circumscriptive formu-lae. Information Processing Letters, 44:113{118, 1992.[CDLS95] M. Cadoli, F. M. Donini, P. Liberatore, and M. Schaerf. The sizeof a revised knowledge base. In Proceedings of the Fourteenth ACMSIGACT SIGMOD SIGART Symposium on Principles of Database Sys-tems (PODS-95), pages 151{162, 1995.[CDS94] M. Cadoli, F. M. Donini, and M. Schaerf. Is intractability of non-monotonic reasoning a real drawback? In Proceedings of the TwelfthNational Conference on Arti�cial Intelligence (AAAI-94), pages 946{951,1994. Extended version as RAP.09.95 DIS, Univ. of Roma \La Sapienza",July 1995.[CDS95a] M. Cadoli, F. M. Donini, and M. Schaerf. Is intractability of non-monotonic reasoning a real drawback? Technical Report RAP.09.95, Di-partimento di Informatica e Sistemistica, Universit�a di Roma \La Sapien-za", July 1995. To Appear in Arti�cial Intelligence Journal.25

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