I N F S Y S

R E S E A R C H

R E P O R T

Institut fur Informationssysteme

Abtg. Wissensbasierte Systeme

Technische Universitat Wien

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A-1040 Wien, Austria

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INSTITUT FUR INFORMATIONSSYSTEME

ABTEILUNG WISSENSBASIERTESYSTEME

ON COMPUTING SOLUTIONS TO BELIEF

CHANGE SCENARIOS

James P. DELGRANDE Torsten SCHAUBHans TOMPITS Stefan WOLTRAN

INFSYS RESEARCHREPORT1843-03-03

M ARZ 2003

INFSYS RESEARCH REPORT

INFSYS RESEARCHREPORT1843-03-03, MARZ 2003

ON COMPUTING SOLUTIONS TO BELIEF CHANGE SCENARIOS

James P. Delgrande1 Torsten Schaub2,Hans Tompits3, and Stefan Woltran3

Abstract. Belief change scenarioshave been introduced as a framework for expressing differentforms of belief change. The essential idea is that for a belief change scenario(K;R;C) the setof formulasK, representing the knowledge base, is modified so that the sets of formulasR andCare respectively true in, and consistent with, the result. By restricting the form of a belief changescenario one obtains specific belief change operators including belief revision, contraction, update,and merging. In this paper, we show how the belief revision and belief contraction operators can beaxiomatised by means of quantified Boolean formulas. Basically for both the general approach andfor specific operators, we give a quantified Boolean formula such that satisfying truth assignmentsto the free variables correspond to belief changeextensionsin the original approach. Hence, inthis paper we reduce the problem of determining the results of a belief change operation to that ofsatisfiability. This approach has several benefits. First, it furnishes an axiomatic specification ofbelief change with respect to belief change scenarios. Thisalternative characterisation then leads tofurther insight into the belief change framework. Second, this axiomatisation allows us to identifystrict complexity bounds for the considered reasoning tasks. Finally, we obtain an implementationof different forms of belief change by appeal to existing solvers for quantified Boolean formulas.

1School of Computing Science Simon Fraser University Burnaby, B.C., Canada V5A 1S6 E-mail: jim@cs.sfu.ca2Institut fur Informatik, Universitat Potsdam Postfach 90 03 27, D–14439 Potsdam, Germany. E-mail: [sarsakov,

torsten]@cs.uni-potsdam.de3Institut fur Informationssysteme, Abteilung Wissensbasierte Systeme 184/3, Technische Universitat Wien, Fa-

voritenstraße 9-11, A–1040 Wien, Austria. E-mail: [tompits, stefan]@kr.tuwien.ac.at

Acknowledgements: This work was partially supported by the Austrian Science Fund Project under grantP15068 as well as a Canadian NSERC Research Grant.

A preliminary version of this paper appeared in theProceedings of the Sixth European Conference on Sym-bolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’2001).

Copyright c 2003 by the authors

Contents

1 Introduction 1

2 Belief Change and Belief Change Scenarios 22.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 22.2 General Approaches to Belief Change . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 22.3 Belief Change via Belief Change Scenarios . . . . . . . . . . . .. . . . . . . . . . . . . . 32.4 Formal Elements of the Belief Change Framework . . . . . . . .. . . . . . . . . . . . . . 4

3 Quantified Boolean Formulas 6

4 Reductions 84.1 Encodings of the Basic Tasks . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 94.2 Expressing Changed Knowledge Bases . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 12

5 Complexity Results 14

6 Implementation 16

7 Conclusion 16

A Proofs 17

INFSYS RR 1843-03-03 1

1 Introduction

In [7] a consistency-based framework for expressing belief change operators is developed. The basic ideawith respect to belief revision is that, given a knowledge baseK and a sentence� for revision,K and� areexpressed in disjoint languages; the languages are coerced(via a maximisation process) to agree on truthvalues of atoms wherever consistently possible; and the result is then re-expressed in the original language.Informally, in the maximisation step, models ofK are syntactically forced to correlate with those of�insofar as consistently possible. The inherent nondeterminism of the maximisation process gives rise totwo notions of revision. Inchoice revisionone suchbelief change extensionis selected as the revised state.In general (skeptical) revision, the revised state consists of the intersection of all such extensions. Beliefcontraction is defined similarly.

In this paper, we discuss a method to implement this approachto belief change, based on reductionsto quantified Boolean formulas. By a quantified Boolean formula (or QBF for short) one understands aformula which is constructed like an ordinary propositional formula, except that quantifiers ranging overpropositional variables may also occur. Quantified Booleanformulas thus belong to the language of second-order logic. As well, they allow a compact representation ofa large class of properties. The latter pointis reflected by the fact that the evaluation problem of QBFs—i.e., the problem of determining the truth ofa given QBF—is PSPACE-complete, whilst the evaluation problem of QBFs having prenex normal formwith i� 1 alternating (groups of) quantifiers is complete for thei-th level of the polynomial hierarchy[34;39].

The general mechanism of our approach is to translate (in polynomial time) a given reasoning taskinto the evaluation problem for QBFs and then use a QBF evaluator to compute the resultant instances. Theexistence of efficient QBF solvers, such as the systems developed by Cadoliet al. [4], Giunchigliaet al.[18],Rintanen[32], or Feldmannet al. [15], makes such a rapid prototyping approach practicably applicable.

A similar approach for solving various reasoning tasks belonging to the area of nonmonotonic reasoninghas been realized in the systemQUIP [12; 11; 27]. This prototype implementation currently handles thecomputation of the main reasoning tasks for logic-based abduction, default logic, several types of modalnonmonotonic logics, and equilibrium logic, a generalisation of the stable model semantics for logic pro-grams. We have implemented the translations for belief change problems by incorporating them into thesystemQUIP.

Reduction methods to QBFs naturally generalise similar approaches for problems inNP; these latterproblems can in turn be solved by translating them (in polynomial time) toSAT, the satisfiability problemof classical propositional logic (see e.g.,[20] for such an application in Artificial Intelligence). Besidesthe implementation of different nonmonotonic reasoning tasks as realized by the systemQUIP, successfulapplications based on reductions to QBFs have also been applied to conditional planning[31].

There are several reasons why we are interested in a reformulation of the approach of[7] using QBFs.First, it provides a straightforward implementation of thegeneral framework by appeal to extant QBFsolvers. Second, in the original approach several steps were expressed at the metalevel. In particular, thereis a metatheoretic step in which pairs of atoms are asserted to be equivalent wherever consistent. Here incontrast, we obtain an object-level representation of the approach. In fact, we provide an axiomatisation ofthe original belief change method in terms of QBFs by constructing suitable translation schemas such thatthere is a one-to-one correspondence between the satisfying assignments to the free propositional atoms ofthe QBFs and the belief change extensions obtained in the original framework. This in turn leads to furtherinsight into the original approach. Finally, the expression of belief change problems in terms of QBFs givesa direct way to estimate the computational complexity of theconsidered reasoning tasks. More specifically,

2 INFSYS RR 1843-03-03

by using the respective QBF encodings, we show that reasoning from choice revision is complete for�P2 ,and, dually, reasoning from skeptical revision is completefor �P2 . Additionally, we also discuss the com-plexity of other decision problems associated with belief change. In this regard we generalise and improveon the results in[7].

In the next section we briefly introduce notions of belief change as well as those aspects of belief changescenarios that interest us. In Section 3, we similarly introduce quantified Boolean formulas. Section 4 givesthe polynomial-time constructible reductions of the relevant reasoning tasks into QBFs. Section 5 discussescomplexity issues, while Section 6 briefly sketches an implementation of the reductions. Section 7 suppliessome concluding remarks.

2 Belief Change and Belief Change Scenarios

2.1 Basic Notation

We deal with propositional languages and use the logical symbols>, ?, :, _ , ^ , ! , and� to constructformulas in the standard way. We writeLP to denote a language over an alphabetP of propositional vari-ablesor atoms. Formulas are denoted by Greek lower-case letters (possibly with subscripts). Disjunctionsof form

Wi2I i are assumed to stand for the logical constant? wheneverI = ;. A literal, L, is eitheran atomp (a positive literal) or a negated atom:p (a negative literal). The set of all atoms occurring in aformula� is denoted byVar(�). Similarly, for a setS of formulas,Var(S) is the set of all atoms occurringin elements ofS, i.e.,Var(S) = S�2S Var(�).

The (propositional) derivability operator,, is defined in the usual way, and likewise its semantic coun-terpartj=. Thedeductive closureof a setS � LP of formulas is given byCnP(S) = f� 2 LP j S ` �g.We say thatS is deductively closediff S = CnP(S). Furthermore,S is consistentproviding? =2 CnP(S).If the language is clear from the context, we usually drop theindex “P” from CnP (�) and simply writeCn(�). Knowledge bases, or, equivalently,belief sets, are initially identified with deductively-closed sets offormulas; later we relax this restriction. We useK;K1; : : : to denote knowledge bases.

Given an alphabetP, we define a disjoint alphabetP 0 asP 0 = fp0 j p 2 Pg: Then, for� 2 LP , wedefine�0 as the result of replacing in� each atomp fromP by the corresponding atomp0 inP 0 (so implicitlythere is an isomorphism betweenP andP 0). This is defined analogously for sets of formulas.

2.2 General Approaches to Belief Change

A common approach in belief revision and other belief changefunctions is to provide a set ofrationality pos-tulatesthat constrain the results of any such function. TheAGM approachof Alchourron, Gardenfors, andMakinson[1; 17] provides the best-known set of such postulates. Belief states are modelled by deductively-closed sets of sentences, calledbelief sets, where the underlying logic includes classical propositional logic.K + �, the expansionof K by �, is defined to beCn(K [ f�g). K? is the inconsistent belief set (i.e.K? = LP).

A revision function, _+, is a mapping from2LP �LP to 2LP satisfying the following postulates.(K _+1) K _+� is a belief set.(K _+2) � 2 K _+�:(K _+3) K _+� � K + �:

INFSYS RR 1843-03-03 3(K _+4) If :� 62 K; thenK + � � K _+�:(K _+5) K _+� = K? iff j= :�:(K _+6) If j= � � �; thenK _+� = K _+�:(K _+7) K _+(� ^ �) � (K _+�) + �:(K _+8) If :� 62 K _+�; then(K _+�) + � � K _+(� ^ �):Informally, these postulates state that the result of revising K by � is a belief set in which� is believed;whenever the result is consistent, revision consists of theexpansion ofK by �; the only time thatK? isobtained is when� is inconsistent; and revision is independent of the syntactic form of K and�. The lasttwo postulates assert that in revising by a conjunction, an expansion with a conjunct is employed whereconsistent.

Contractionis the dual notion of revision, in which beliefs are retracted but no new beliefs are added.Postulates governing a contraction function_� are similarly given. The intuition underlying revision andcontraction is that an agent receives new information concerning a static world or domain.[19] explores thedistinct notions of beliefupdateanderasurein which an agent changes its beliefs in response to changes inits external environment. As well, belief setmerging(or arbitration), in which the contents of two beliefsets are combined, is addressed in[23].

There has also been work on specific revision operators basedon thedistancebetween models of aknowledge base and a sentence to be incorporated in the knowledge base[3; 36; 5; 33; 38; 16]. For example,in [5] the revision operator measures the distance between interpretations by the number of propositionalvariables on which the interpretations differ. It is shown that this operator satisfies the AGM postulates.

Another direction in belief revision is to assume that revision is not carried out on a belief set itself, butrather on a finite subset of the theory. Belief change operations would take place with respect to thisbeliefbase, while the underlying belief set would correspond to the deductive closure of this base. The notion ofbelief base revision is proposed in[24], and independently, with respect to database systems in[14]. Theseapproaches are fully explored in[25]. While conceptually simple, revision in these approaches frequentlyrelies on arbitrary syntactic distinctions.

There has been some work with respect to implementations. For example, the aforecited distance-basedapproaches admit straightforward implementations. Otherwise,[37] provides an example of a computationalmodel for belief base revision. (See also[35; 2] for other approaches.)

2.3 Belief Change via Belief Change Scenarios

In [7] a consistency-based framework for expressing a suite of belief change operators is developed. Theintent was to specify an approach that has good formal properties, but that particularly lent itself to imple-mentation. The approach is discussed formally in the next section; here we give an informal introduction tothe approach to revision. As a starting point, it is clear that the syntactic form of a sentence does not give afirm indication as to which sentences should be included in a revisionK _+�. Alternately one can considerinterpretations, and look at the models ofK and�. Informally, if K[f�g is unsatisfiable, a model ofK _+�should contain models of�, but in a sense retaining aspects of models ofK that do not conflict with thoseof �.

We accomplish this by first expressingK and� in different languages, in essence replacing everyoccurrence of an atomic sentencep in K by a new atomic sentencep0 yielding knowledge baseK 0, and

4 INFSYS RR 1843-03-03

leaving� unchanged. Under this relabelling, the models ofK 0 and� are independent andK 0 [ f�g issatisfiable (assuming that each ofK, � are satisfiable). The models ofK 0 and� are linked by asserting thatthe languages are (with respect to truth conditions) the same wherever consistently possible. That is, foreveryp 2 P, we assert thatp � p0 wherever consistently possible. We obtain a set of such equivalences,call it EQ, such thatK 0 [ f�g [ EQ is consistent. A model ofK 0 [ f�g [ EQ then will be a model of�where the truth values of atomic sentences inK 0 and� are linked wherever possible. A candidate “choice”revision ofK by � then consists ofK 0 [ f�g [EQ re-expressed in the original language. General revisioncorresponds to the intersection of all candidate choice revisions.

For example, consider whereK = Cn(f(p _ q) ^ rg) and � = (:p _ :q) ^ :r:Renaming the atoms inK gives K 0 = Cn(f(p0 _ q0) ^ r0g). Clearly K 0 [ f�g is consistent, eventhoughK [ f�g is not. In the step to link the respective interpretations ofK 0 and � we obtain thatCn(K 0 [ f�g [ fp0 � p; q0 � qg) is consistent, butCn(K 0 [ f�g [ fp0 � p; q0 � q; r0 � rg) is not. Hencewe takeEQ = fp0 � p; q0 � qg. IntersectingCn(K 0 [ f�g [EQ) with the original language yieldsCn(f(p � :q) ^ :rg) as the revised knowledge base.

The general framework allows the expression of contractionand integrity constraints, as well as update,erasure, and merging operations. Significantly, the approach is independent of how the knowledge baseand formula for revision are represented. In particular, the original and revised knowledge base can berepresented by a formula whose deductive closure gives the corresponding belief set. As well, the scopeof a revision (for example) can be restricted to just those propositions common to the knowledge base andsentence for revision. The approach (essentially) satisfies the AGM postulates[1], with the exception of therevision postulate(K _+8) and the contraction postulate(K _�8). The approach to belief change is foundedon the same intuitions asconsistency-basedreasoning methodologies in Artificial Intelligence. Examples ofsuch systems include Theorist[29], diagnosis from first principles[30], and the assumption-based approachto truth maintenance[6].

2.4 Formal Elements of the Belief Change Framework

Following Delgrande and Schaub[7; 8], we define abelief change scenarioin languageLP as a tripleB = (K;R;C), whereK;R;C are sets of formulas inLP . Informally,K is a knowledge base that will bechanged such that the setR will be implied by the result, and the setC will be consistent with the result.For a base approach to revision we takeC = ; and for a base approach to contraction we takeR = ;.

We extend our notationVar(�) to belief change scenarios in the obvious way, i.e., forB = (K;R;C),we defineVar(B) = Var (K[R[C). In the definition below, “maximal” is with respect to set containment(rather than set cardinality). The following definition is central[7; 8]:

Definition 2.1 Let B = (K;R;C) be a belief change scenario inLP . DefineEQ as a maximal set ofequivalencesEQ � fp � p0 j p 2 Pg such thatK 0 [ EQ [R [ C 6` ?:Then,Cn(K 0 [ EQ [R) \ LPis a (consistent) belief change extensionofB.

If there is no such setEQ thenB is inconsistentandLP is defined to be the sole(inconsistent) beliefchange extensionofB.

INFSYS RR 1843-03-03 5K 0 � EQ K _+�p0 ^ q0 :q fp � p0g p ^ :q:p0 � q0 :q f p � p0; q � q0 g p ^ :qp0 _ q0 :p _ :q f p � p0; q � q0 g p � :qp0 ^ q0 :p _ :q fp � p0g; fq � q0g p � :qTable 1: (Skeptical) revision examples.

So, a (consistent) belief change extension ofB is a modification ofK in whichR is true, and in whichC is consistent. We say thatEQ determinesthe belief change extensionCn(K 0 [ EQ [R) \ LP of B.Clearly, for a given belief change scenario there may be morethan one belief change extension.

Definition 2.1 provides a very general framework for specifying belief change. In what follows, wegive specific definitions for the belief change operationsrevisionandcontraction. In these definitions wemake use of the notion of aselection function, , that for any setI 6= ; has as value some element ofI.These primitive functions can be regarded as inducing selection functions 0 on belief change scenarios, suchthat 0(B) has as value some belief change extension ofB = (K;R;C). This is a slight generalisation ofselection functions as found in the AGM approach[17].

Definition 2.2 (Revision) LetK be a knowledge base and� a formula, and let(Ei)i2I be the family of allbelief change extensions of(K; f�g; ;). Then,

1. K _+ � = Ei is achoice revisionofK by� with respect to some selection function with (I) = i;and

2. K _+� = Ti2I Ei is the(skeptical) revisionofK by�.

Table 1 gives examples of (skeptical) revision. The first column gives the original knowledge base, but withatoms already renamed. The second column gives the revisionformula, while the third gives theEQ set(s)and the last column gives the results of the revision. For thefirst and last column, we give a formula whosedeductive closure gives the corresponding belief set.

In detail, for the last example, we wish to determinefp ^ qg _+(:p _ :q) :We find maximal setsEQ � fp � p0; q � q0g such thatfp0 ^ q0g [ EQ [ f:p _ :qg [ ; 6` ?:We get two such sets of equivalences, viz.EQ1 = fp � p0g andEQ2 = fq � q0g. Accordingly, we obtainfp ^ qg _+(:p _ :q) = Ti=1;2Cn(fp0 ^ q0g [ EQ i [ f:p _ :qg) \ LP :In addition to(:p _ :q), we get(p _ q), jointly implying (p � :q).

Contraction is defined similarly to revision.

Definition 2.3 (Contraction) LetK be a knowledge base and� a formula, and let(Ei)i2I be the family ofall belief change extensions of(K; ;; f:�g). Then,

6 INFSYS RR 1843-03-03K 0 � EQ K _��p0 ^ q0 q fp � p0g pp0 ^ q0 ^ r0 p _ q fr � r0g rp0 _ q0 p ^ q f p � p0; q � q0 g p _ qp0 ^ q0 p ^ q fp � p0g; fq � q0g p _ qTable 2: (Skeptical) contraction examples.

1. K _� � = Ei is a choice contractionof K by � with respect to some selection function with (I) = i; and

2. K _�� = Ti2I Ei is the(skeptical) contractionofK by�.

Table 2 gives examples of (skeptical) contraction, using the same format and conventions as Table 1.In detail, for the first example we wish to determinefp ^ qg _�q :

We compute the belief change extensions of(fp ^ qg; ;; f:qg). We rename the propositions infp^ qg andlook for maximal subsetsEQ of fp � p0; q � q0g such thatfp0 ^ q0g [ EQ [ ; [ f:qg 6` ?:We obtainEQ = fp � p0g; yieldingfp ^ qg _�q = Cn(fp0 ^ q0g [ fp � p0g [ ;) \ LP= Cn(fpg):3 Quantified Boolean Formulas

Quantified Boolean formulas (QBFs) generalise ordinary propositional formulas by the addition of quan-tifiers over propositional variables. Informally, a QBF of form 8p9q� means that for all truth assign-ments ofp there is a truth assignment ofq such that� is true. For instance, it is easily seen that the QBF9p9q ((p ! q) ^ 8r(r ! q)) evaluates to true. QBFs are denoted by Greek upper-case letters.

The precise semantical meaning of QBFs is defined as follows.First, some ancillary notation. Anoccurrence of a propositional variablep in a QBF� is free iff it does not appear in the scope of a quantifierQp (Q 2 f8;9g), otherwise the occurrence ofp is bound. If � contains no free variable occurrences,then� is closed, otherwise� is open. Furthermore,�[p1=�1; : : : ; pn=�n℄ denotes the result of uniformlysubstituting each free occurrence of a variablepi in � by a formula�i, for 1 � i � n.

By aninterpretation, M , we understand a set of atoms. Informally, an atomp is true underM iff p 2M .In general, the truth value,�M (�), of a QBF� under an interpretationM is recursively defined as follows:

1. if � = >, then�M (�) = 1;

2. if � = p is an atom, then�M(�) = 1 if p 2M , and�M (�) = 0 otherwise;

3. if � = :, then�M (�) = 1� �M ();

INFSYS RR 1843-03-03 7

4. if � = (�1 ^ �2), then�M (�) = min(f�M (�1); �M (�2)g);5. if � = 8p, then�M (�) = �M ([p=>℄ ^ [p=?℄);

The truth conditions for?, _ , ! ,�, and9 follow from the above in the usual way. Note that9 is definedhere similarly as in first-order logic, i.e.,9p = :8p:, for each formula. Hence, the truth value for9is given by�M (9p) = �M ([p=>℄ _ [p=?℄):

We say that� is true underM iff �M (�) = 1, otherwise� is false underM . If �M (�) = 1, thenMis amodelof �. The set of all models of� is denoted byMod (�). If Mod(�) 6= ;, then� is said to besatisfiable. If � is true under every interpretation, then� is valid. As usual, we writej= � to express that�is valid.

It is easily seen that the truth value of a QBF� under interpretationM depends only on the free variablesin �. Hence, without loss of generality, for determining the truth value of QBFs, we may restrict ourattention to interpretations which contain only atoms occurring free in the given QBF. In particular, closedQBFs are either true under every interpretation or false under every interpretation, i.e., they are either validor unsatisfiable, so for closed QBFs there is no need to refer to particular interpretations. As well, if aclosed QBF� is valid, we say that� evaluates to true, and, correspondingly, if� is unsatisfiable, we saythat� evaluates to false. Two sets of formulas (i.e., ordinary propositional formulas or QBFs) arelogicallyequivalentiff they possess the same models. Thus, formulas� and are logically equivalent iff� � isvalid.

In the sequel, we use the following abbreviations in the context of QBFs: LetS = f�1; : : : ; �ng andT = f 1; : : : ; ng be indexed sets of formulas. Then,S � T abbreviatesVni=1(�i ! i), andS � T

is an abbreviation forVni=1(�i � i). Furthermore, for an indexed setV = fp1; : : : ; png of propositional

variables and a quantifierQ 2 f8;9g, we letQV � stand for the formulaQp1Qp2 � � �Qpn�. Finally, a finitesetT = f�1; : : : ; �ng of formulas is usually identified with the conjunction

Vni=1 �i of its elements.The operator� is a fundamental tool for expressing tests on sets of formulas in terms of QBFs. In

particular, we use� in conjunction with the following task:

Given finite setsS andT of formulas, compute all subsetsR � S such thatT [R is consistent.

This problem can be encoded by a QBF in the following way:

Proposition 3.1 LetS = f�1; : : : ; �ng andT be finite sets of formulas, letV be the set of atoms occurringin S[T , and letG = fg1; : : : ; gng be a set of new variables not occurring inS or T . Furthermore, considersomeR � S and someM � G such that�i 2 R iff gi 2M , for 1 � i � n.

Then,T [R is consistent iffM is a model of the QBFC[T; S℄ = 9V (T ^ (G � S)):Note thatC[T; S℄ is an open QBF havingG as its set of free variables. These variables facilitate the

selection of those elements ofS which determine the setsR such thatT [ R is consistent. Moreover,C[T; S℄ is designed to expressall potential subsetsR � S such thatT [ R is consistent. To express, forexample, allmaximalsuch subsets, some additional elements are required. The computation of maximalsets satisfying certain criteria, using QBFs, is discussedin Section 4.

Finally, we note some useful relations concerning the shifting and renaming of quantifiers, parallelingsimilar results from standard first-order logic.

8 INFSYS RR 1843-03-03

Proposition 3.2 Let p, q be atoms,Q 2 f8;9g, and let�, be QBFs such that does not contain freeoccurrences ofp. Then,

1. j= (:9p�) � 8p(:�);2. j= (:8p�) � 9p(:�);3. j= ( Æ Qp�) � Qp( Æ �) for Æ 2 f^ ; _ ; !g; and

4. j= (Qq) � (Qp[q=p℄).We say that QBF� is in prenex formif � = Q1p1 : : :Qnpn�, whereQi 2 f8;9g, for 1 � i � n, and�

is some propositional formula. Proposition 3.2 implies that any QBF can be effectively transformed intoa logical equivalent QBF� in prenex form.

4 Reductions

In this section, we present efficient (polynomial-time constructible) reductions of the relevant reasoningtasks in the context of belief change scenarios into QBFs. More specifically, these reductions are constructedin such a way that there is a one-to-one correspondence between belief change extensions and models of thetranslated QBFs. Based on these reductions, in Section 5 we analyse the computational complexity of theconsidered reasoning tasks.

Concerning the specific tasks, we deal with the following decision problems and their correspondingsearch problems:

EXT : Decide whether a given belief change scenarioB has some consistent belief change extension.

CHOICE : Given a belief change scenarioB and some formula�, decide whether� is contained in at leastone consistent belief change extension ofB.

SKEPTICAL : Given a belief change scenarioB and some formula�, decide whether� is contained in allbelief change extensions ofB.

Note thatEXT andCHOICE are specified with respect to consistent belief change extensions. Droppingthe consistency condition inEXT would result in a trivial decision problem, because, according to Defi-nition 2.1, any belief change scenario possesses always at least one belief change extension. ConcerningCHOICE, although here, like forEXT, we are primarily interested in consistent belief change extensions, laterwe relax this condition and deal also with the inconsistent case. ForSKEPTICAL, however, the consistencyrequirement is actually irrelevant, because it holds that aformula� is contained in all belief change exten-sions of a given belief change scenarioB iff it is contained in all consistent belief change extensions ofB.In general,EXT is arguably less interesting thanCHOICE or SKEPTICAL, given that it depends only on theconsistency of the constituents of the given belief change scenario—however, the relevance of this task liesin fact in the correspondingsearch problem, i.e., in the actual computation of the consistent belief changeextensions.

INFSYS RR 1843-03-03 9

4.1 Encodings of the Basic Tasks

From now on we assume that, for any belief change scenarioB = (K;R;C), its constituentsK, R, andC are finite; thus, these sets are also represented as the conjunction of their elements. Furthermore, forour subsequent encodings it is convenient to use the following alternative characterisation of belief changeextensions, which is a straightforward consequence of results given in[8]. Basically, this characterisationshows that sets of equivalences can be restricted to subsetsof fp � p0 j p 2 Var(B)g.Proposition 4.1 For any belief change scenarioB = (K;R;C) in LP , there is a one-to-one correspon-dence between the determining setsEQ � fp � p0 j p 2 Pg of the belief change extensions ofB and setsEQ℄ � fp � p0 j p 2 Var(B)g satisfying the following conditions:(a) K 0 [ EQ℄ [R [ C 6` ?; and(b) for eachp 2 Var(B) with (p � p0) =2 EQ℄, we haveK 0 [ EQ℄ [ fp � p0g [R [C ` ?.

In particular, for belief change extensionE ofB with determining setEQ � fp � p0 j p 2 Pg, it holdsthatE = Cn(K 0 [ EQ℄ [R) \ LP ; whereEQ℄ satisfies Conditions(a) and(b), andEQ℄ = EQ \ fp �p0 j p 2 Var(B)g.

Thus, with a slight abuse of notation, we also refer to setsEQ � fp � p0 j p 2 Var(B)g satisfying theabove Conditions(a) and(b) as a determining set of belief change extensionCn(K 0 [ EQ [R) \ LP :

We proceed with the following basic QBF module:

Definition 4.1 LetB = (K;R;C) be a belief change scenario overLP , let V = Var(B) be the set ofvariables occurring inB, and letVeq = fpeq j p 2 V g be a set of new variables. Then,M[B℄ = K 0 ^ (Veq � (V � V 0)) ^ R:

The computation of belief change extensions can be expressed in terms of QBFs as follows:

Theorem 4.2 Let B = (K;R;C) be a belief change scenario inLP , let V = Var (B) be the atomsoccurring inB, and letVeq = fpeq j p 2 V g be a set of variables disjoint fromV andV 0. Furthermore, letEQ � fp � p0 j p 2 Var (B)g be a set of equivalences and letM � Veq be defined such thatpeq 2 M iff(p � p0) 2 EQ .

Then,Cn(K 0 [ EQ [R) \ LP is a belief change extension ofB iff M is a model of the QBFText[B℄ = 9V 9V 0(M[B℄ ^ C) ^p2V �:peq ! :9V 9V 0((p � p0) ^ M[B℄ ^ C)�:Note thatVeq constitutes the set of free variables ofText[B℄. Intuitively, Veq guesses a setEQ of

equivalences determining a belief change extension ofB. The first conjunct ofText[B℄ checks consistency,and the second conjunct checks whetherEQ is maximal with respect to set containment.

For an illustration of the translationText[�℄, consider the belief change scenarioB = (fp ^ qg; f:p _:qg; ;) from Section 2.4. The free variables ofText[B℄ are given byfpeq ; qeqg, so we get the following fourinterpretations serving as potential models ofText[B℄:M1 = fg; M3 = fqeqg;M2 = fpeqg; M4 = fpeq ; qeqg:

10 INFSYS RR 1843-03-03

SinceB has two belief change extensions, generated byEQ1 = fp � p0g andEQ2 = fq � q0g(cf. Table 1), we expectM2 andM3 to be models ofText[B℄. Let us first look at the left conjunct9V 9V 0(M[B℄ ^ C) of Text[B℄. ForB as above, we obtain9V 9V 0(M[B℄ ^ C) =9pqp0q0�(p0 ^ q0) ^ (peq ! (p � p0)) ^ (qeq ! (q � q0)) ^ (:p _ :q)�: (1)

This QBF has three models, viz.M1, M2, andM3. InterpretationM1 is a model because both conjuncts(peq ! (p � p0)) and(qeq ! (q � q0)) of (1) evaluate to true (given thatpeq ; qeq =2 M1), and since theremaining formula(p0 ^ q0) ^ (:p _ :q) is consistent. ForM2, we similarly get that(qeq ! (q � q0))is true and that(p0 ^ q0) ^ (peq ! (p � p0)) ^ (:p _ :q) is consistent, sincefp0; q0; pg is a model of(p0 ^ q0) ^ (p � p0) ^ (:p _ :q). M3 is a model by analogous arguments. However,M4 is not a modelof (1). This is because, underM4, formula (1) can be reduced to(p0 ^ q0) ^ (p � p0) ^ (q � q0) ^ (:p _ :q); (2)

which is not satisfiable.Hence, we are left with three possible models ofText[B℄, viz.M1,M2, andM3.Now we investigate the remaining conjuncts ofText[B℄, which are given by�1 = h:peq ! :9pqp0q0�(p � p0) ^ (p0 ^ q0) ^ (peq ! (p � p0)) ^^ (qeq ! (q � q0)) ^ (:p _ :q)�i

and �2 = h:qeq ! :9pqp0q0�(q � q0) ^ (p0 ^ q0) ^ (peq ! (p � p0)) ^^ (qeq ! (q � q0)) ^ (:p _ :q)�i:First, consider interpretationM2. Sincepeq 2 M2, conjunct�1 evaluates to true, and it remains to

analyse�2. The latter formula evaluates to true if�(q � q0) ^ (p0 ^ q0) ^ (peq ! (p � p0)) ^ (qeq ! (q � q0)) ^ (:p _ :q)� (3)

is not satisfiable. However, givenM2, (3) reduces to (2), which is indeed unsatisfiable. Hence,M2 is a modelof �2 and thus also a model ofText[B℄. By a similar argument it follows thatM3 is a model of�1 ^ �2.It remains to see thatM1 is not a model of�1 ^ �2. In fact, it holds that�M1(�1) = �M1(�2) = 0. Weshow the case of�1 (the case of�2 follows analogously). SinceM1 = fg, �1 is false underM1 iff9pqp0q0�(p � p0) ^ (p0 ^ q0) ^ (peq ! (p � p0)) ^ (qeq ! (q � q0)) ^ (:p _ :q)�is true underM1. Given that bothpeq andqeq are false underM1, the previous condition holds iff�(p � p0) ^ (p0 ^ q0) ^ (:p _ :q)� (4)

is satisfiable. Clearly, this is the case, sincefp; p0; q0g is a satisfying truth assignment for (4). Thus,M1 isnot a model of�1. This shows thatM1 is not a model ofText[B℄.

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Concerning a QBF encoding forEXT, from Theorem 4.2 it immediately follows that a belief changescenarioB = (K;R;C) with V = Var(B) has a consistent belief change extension iff9VeqText[B℄evaluates to true. However, this encoding is in some sense not optimal because it is possible to characteriseEXT in terms of a simpler QBF, corresponding to an ordinary satisfiability problem, by observing thatB hasa consistent belief change extension iffK 0 [R [ C is consistent. Consequently, we can state the followingresult:

Theorem 4.3 LetB = (K;R;C) be a belief change scenario inLP and letV = Var(B).Then,B has a consistent belief change extension iff9V 9V 0(K 0 ^ R ^ C) evaluates to true.

Next, we discuss the translations of the reasoning tasksCHOICE and SKEPTICAL. We begin with theencodings of the corresponding search problems.

Theorem 4.4 LetB = (K;R;C) be a belief change scenario inLP and let� be a formula. Furthermore,let V = Var (B), let W = Var (B) [ Var (�), and letVeq = fpeq j p 2 V g be a set of globally newvariables. Finally, consider the QBFM[B℄ = K 0 ^ (Veq � (V � V 0)) ^ R:from Definition 4.1.

Then, forEQ � fp � p0 j p 2 Var (B)g andM � Veq such that(p � p0) 2 EQ iff peq 2 M , thefollowing properties hold:

1. Cn(K 0 [ EQ [R) \ LP is a belief change extension ofB containing� iff M is a model of the QBFT hoi e[B;�℄ = Text[B℄ ^ 8W�(9W 0M[B℄) ! ��;2. Cn(K 0 [ EQ [R) \ LP is a belief change extension ofB not containing� iff M is a model of the

QBF Tskept[B;�℄ = Text[B℄ ^ :8W�(9W 0M[B℄) ! ��:Intuitively, the two encodingsT hoi e[B;�℄ andTskept[B;�℄ are realised by (i) checking whether a se-

lected set of equivalences determines a consistent belief change extensionE ofB, and (ii) checking whetherE contains a given formula�, or checking whetherE does not contain�. Task (i) is modeled using thebasic encodingText[B℄, and Task (ii) is captured by a suitable QBF expressing derivability (in case ofT hoi e[B;�℄) or non-derivability (in case ofTskept[B;�℄) of � from the selected belief change extension.Observe that the selection process is facilitated in terms of the members fromVeq , which represent the freevariables ofT hoi e[B;�℄ andTskept[B;�℄.

Concerning the decision problemsCHOICEandSKEPTICAL, QBF encodings for these tasks are obtainedfrom T hoi e[B;�℄ andTskept[B;�℄ as follows.CHOICE is expressed by the closed QBF9VeqT hoi e[B;�℄,which states that there is some some setM � Veq corresponding to a setEQ of equivalences such thatEQ determines a consistent belief change extensionE entailing�, andSKEPTICAL is realised by the closedQBF :9VeqTskept[B;�℄, which expresses that there is no setM � Veq corresponding to a setEQ ofequivalences such thatEQ determines a consistent belief change extensionE not entailing�. Thus, weobtain the following corollary:

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Corollary 4.5 LetB be a belief change scenario and� a formula. Then,

1. � is contained in at least one consistent belief change extension ofB iff 9Veq T hoi e[B;�℄ evaluatesto true; and

2. � is contained in all belief change extensions ofB iff :9Veq Tskept[B;�℄ evaluates to true.

In contrast to taskEXT, which can be expressed by a QBF containing only one sort of quantifier, here weobtain encodingspossessing both existential quantifiers as well as universal ones. As we show in Section 5,this quantifier alternation is in some sense unavoidable andreflects the inherent complexity ofCHOICE andSKEPTICAL.

We remark that discarding the consistency condition ofCHOICE can be easily incorporated into thetranslation9Veq T hoi e[�; �℄. Indeed, it is a simple matter to check that there is a (possibly inconsistent)belief change extension ofB = (K;R;C) containing formula� iff the QBF(9V 9V 0(K 0 ^ R ^ C)) ! (9Veq T hoi e[B;�℄)evaluates to true.

Finally, observe that Theorems 4.2, 4.3, and 4.4, as well as Corollary 4.5, provide encodings of rea-soning tasks forarbitrary belief change scenarios. In particular, they subsume the characterisation of thecorresponding reasoning tasks associated with revision and contraction, as illustrated by the revision exam-ple discussed previously. For convenience, we list the tasks for revision:

REXT : Given a knowledge baseK and some formula�, decide whether a consistent belief change exten-sion ofB = (K; f�g; ;) exists.

RCHOICE : Given a knowledge baseK and formulas� and�, decide whether there is some consistentchoice revisionK _+ � containing�.

RSKEPTICAL : Given a knowledge baseK and formulas� and�, decide whether� is contained in theskeptical revisionK _+�.

The corresponding tasks for belief contraction, denoted byCEXT, CCHOICE, and CSKEPTICAL, are definedaccordingly.

4.2 Expressing Changed Knowledge Bases

Another interesting issue in the context of belief change isto determine the actual form of a given knowledgebase after a revision or contraction operation has been applied to it. This task has already been analysed in[7;8], and, as we show in the following, it can also be described in terms of QBFs. Before going into details,we briefly summarize the relevant results from[7; 8], starting with some notation.

Given a belief change scenarioB in languageLP along with a set of equivalencesEQ i � fp � p0 j p 2Pg, definePEQi = fp 2 P j p � p0 2 EQ ig; andPEQi = P n PEQi :Then, for� 2 LP , let d�ei be the result of replacing in� eachp 2 PEQ i by :p. Furthermore, for a set offunctions�i = f�ki j �ki : PEQi ! f>;?gg;

INFSYS RR 1843-03-03 13

let b� ki be the result of replacing in� eachp 2 PEQi by �ki .Then, the result of applying revision or contraction to a given knowledge base is described by the fol-

lowing formulas[7; 8]:

Proposition 4.6 LetK be a finite knowledge base and� some formula. Then,

1. K _+� is logically equivalent toWi2IdKei ^ �; and

2. K _�� is logically equivalent toWi2I;�ki 2�ibK ki ;

for (EQ i)i2I being the family of sets of equivalences determining the belief change extensions of(K; f�g; ;)and(K; ;; f:�g), respectively.

We can express the models of revision and contraction by QBFsusing the following construction.

Definition 4.2 Let B be a belief change scenario inLP with V = Var(B), and letVeq = fpeq j p 2Var(B)g be a set of new variables. Then,Tm [B℄ = 9Veq(Text[B℄ ^ 9V 0M[B℄):Observe thatTm [B℄ is an open QBF havingV as its set of free variables.We obtain the following result:

Theorem 4.7 LetK be a finite knowledge base,� some formula, andV = Var(K [ f�g). Then,

1. M � V is a model ofK _+� iff M is a model ofTm [(K; f�g; ;)℄, and

2. M � V is a model ofK _�� iff M is a model ofTm [(K; ;; f:�g)℄.By collecting all the models ofK _+� orK _�� into a single formula, we thus obtain a disjunctive normal

form ofK after revision or contraction with�.To formally express this, given a belief change scenarioB with V = Var(B), defineF [B℄ = _M2Mod(Tm [B℄)�M ^ ^p2V nM :p�:

Then, we get the following characterisation, representingan alternative to Proposition 4.6:

Corollary 4.8 LetK and� be as in Theorem 4.7. Then,

1. K _+� is logically equivalent toF [(K; f�g; ;)℄, and

2. K _�� is logically equivalent toF [(K; ;; f:�g)℄.For illustration of transformationTm [�℄, consider again the belief change scenarioB = (fp ^ qg; f:p _:qg; ;). As already discussed in Section 2.4,B possesses the skeptical revisionCn(p � :q), which can be

rewritten asCn((p ^ :q) _ (:p ^ q)). Thus, in virtue of Theorem 4.7, we expectfpg andfqg to be themodels ofTm [B℄ = 9Veq�Text[B℄ ^ 9V 0M[B℄�:

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We first compute the models of the QBFText[B℄ ^ 9V 0M[B℄: (5)

The free variables of (5) are given byVeq [ V = fpeq ; qeqg [ fp; qg. As argued previously,Text[B℄ is trueunder interpretationsfpeqg andfqeqg. In fact, sinceText[B℄ has no free variables fromV , it holds thatfpeqg[U andfqeqg[U are also models ofText[B℄, for any setU � V . Consider now the second conjunctof (5), which is given by9p0q0�(p0 ^ q0) ^ (peq ! (p � p0)) ^ (qeq ! (q � q0))�:This formula is obviously true under the following interpretations:M1 = fpeq ; pg; M2 = fqeq ; qg;as well as under every interpretation in which bothpeq andqeq are false. Thus, onlyM1 andM2 satisfy bothconjuncts of (5). Taking the existential closure of QBF (5) with respect to the variablespeq andqeq , we getthatfpg andfqg are the only models ofTm [B℄.5 Complexity Results

In this section, we analyse the computational complexity ofthe tasks considered so far. Additionally, wealso deal with the complexity of checking whether a given setof equivalences determines some consistentbelief change extension of a given belief change scenario.

A particular advantage of our reduction approach is that upper complexity bounds are derived directlyfrom the respective QBF encodings. This is due to the fact that our QBF translations are polynomial in thesize of a given belief change scenario, and that the complexity of evaluating a given QBF� is determinedby the quantifier order of�. For each of the upper bounds obtained in this fashion, we show also that theyarestrict, i.e., they possess a matching lower bound. The results presented here strengthen the complexityanalysis given in[7].

In what follows, we assume that the reader is familiar with the basic concepts of complexity theory(see, e.g.,[26]). For convenience, we briefly recapitulate the definitions and some elementary propertiesof the complexity classes considered in the following. As usual, for any complexity classC, by co-C weunderstand the class of all problems which are complementary to the problems inC.

Four complexity classes are relevant here, viz.NP, DP , �P2 , and�P2 . The classNP consists of alldecision problems which can be solved with a nondeterministic Turing machine working in polynomial time;DP is defined as the class of all problems which can be described as the conjunction of two (independent)problems fromNP and co-NP; �P2 is the class of all problems solvable on a nondeterministic Turingmachine in polynomial time having access to an oracle for problems inNP; and�P2 = co-�P2 .

Observe thatNP, �P2 , and�P2 are part of the polynomial hierarchy, which is given by the followingsequence of objects: the initial elements are�P0 = �P0 = �P0 = P;and, fori > 0,�Pi = P�Pi�1 ; �Pi = NP�Pi�1 ; �Pi = co-NP�Pi�1 :

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Here,P is the class of all problems solvable on a deterministic Turing machine in polynomial time, and,for complexity classesC andA, the notationCA stands for therelativized versionof C, consisting of allproblems which can be decided by Turing machines of the same sort and time bound as inC, only thatthe machines have access to an oracle for problems inA. It holds that�P1 = NP, �P2 = NPNP, and�P2 = co-NPNP. A problem is said to be at thek-th levelof the polynomial hierarchy iff it is in�Pk+1 andeither�Pk -hard or�Pk -hard.

The classDP can be described as part of a family of complexity classesDPk , k � 1, whereDP = DP1and eachDPk consists of all problems expressible as the conjunction of aproblem in�Pk and a problem in�Pk . Notice that, for allk � 1, �Pk � DPk � �Pk+1 holds; both inclusions are widely conjectured to be strict.Moreover, any problem inDPk can be solved with two�Pk oracle calls, and is thus intuitively easier than aproblem complete for�Pk .

In the same way as the satisfiability problem of classical propositional logic is the “prototypical” problemof NP, i.e., being anNP-complete problem, the satisfiability problem of QBFs possessingk � 1 quantifieralternations is the “prototypical” problem of thek-th level of the polynomial hierarchy. More specifically,the following property holds:

Proposition 5.1 ([39]) Given a propositional formula� whose atoms are partitioned intoi � 1 setsV1; : : : ; Vi, deciding whether9V18V29V3 : : :QVi� evaluates to true, whereQ = 9 if i is odd andQ = 8 ifi is even, is�Pi -complete. Moreover, the problem remains�Pi -hard even if� is in conjunctive normal formandi is odd, or if� is in disjunctive normal form andi is even.

From this result it follows that the evaluation problem of QBFs of form8V19V28V3 : : :QVi� is �Pi -complete, whereQ = 8 if i is odd andQ = 9 if i is even. As well, the problem remains�Pi -hard even if�is in disjunctive normal form andi is odd, or if� is in conjunctive normal form andi is even.

Given the above characterisations, we can estimate upper complexity bounds for the decision problemsdiscussed in Section 4 by simply inspecting the quantifier order of the respective QBF encodings. This canbe argued as follows. First of all, by applying the transformation rules described in Proposition 3.2, each ofthe above QBF encodings can be transformed in polynomial time into a closed QBF in prenex form. Then,by invoking Proposition 5.1 and observing that completeness of a decision problemD for a complexity classC implies membership ofD in C, the quantifier order of the resultant QBFs determines in which class ofthe polynomial hierarchy the corresponding decision problem lies.

Applying this method to the decision problemsEXT, CHOICE, andSKEPTICAL, we get the followingresults. To begin with, according to Theorem 4.3, we have that EXT lies inNP. Hence, REXT and CEXT

are also inNP because they are just special cases ofEXT. Furthermore, the encoding9Veq T hoi e[B;�℄for CHOICE can be transformed into a QBF of prenex form9W18W2 , and, dually, the encoding:9Veq Tskept[B;�℄ for SKEPTICAL can be transformed into a QBF of prenex form8Z19Z2'. Thus,CHOICE

is in�P2 , andSKEPTICAL is in�P2 . Similar to the case ofEXT, �P2 is also an upper bound for RCHOICE andCCHOICE, and�P2 is an upper bound for RSKEPTICAL and CSKEPTICAL.

Concerning lower complexity bounds, it turns out that all ofthe above given estimations arestrict, i.e.,the considered decision problems are hard for the respective complexity classes. Summarizing, we can statethe following results:

Theorem 5.2 The decision problemsEXT, CHOICE, andSKEPTICAL, as well as its variants for revision andcontraction, enjoy the following completeness properties:

1. EXT, REXT, andCEXT areNP-complete;

16 INFSYS RR 1843-03-03

2. CHOICE, RCHOICE, andCCHOICE are�P2 -complete; and

3. SKEPTICAL, RSKEPTICAL, andCSKEPTICAL are�P2 -complete.

Thus, the completeness results forCHOICE andSKEPTICAL, as well as for their specialisations for revi-sion and contraction, imply that, unless the polynomial hierarchy collapses, it is not possible to efficientlyrepresent these tasks in terms QBFs having a prenex form withonly one sort of quantifier, i.e.,these taskscannot be polynomially reduced to standard propositional logic. Hence, under the above proviso, the en-codings described in Corollary 4.5 cannot be simplified further to avoid an inherent quantifier alternation.

Rounding off our complexity analysis, we deal with the problem of checking whether a given set ofequivalences determines some consistent belief change extension of a given belief change scenario.

Theorem 5.3 Given a belief change scenarioB and a setEQ � fp � p0 j p 2 Var(B)g, checking whetherEQ determines some consistent belief change extension ofB isDP -complete.

6 Implementation

Our methodology for expressing reasoning tasks associatedwith belief change scenarios in terms of quanti-fied Boolean formulas is motivated by the availability of several practicably efficient QBF-solvers. Amongthe different tools, there is a propositional theorem-prover, boole, based onbinary decision diagrams(the system can be downloaded from the Web at http://www.cs.cmu.edu/�modelcheck/bdd.html), a sys-tem using a generalised resolution principle[21], several provers implementing an extended Davis-Putnamprocedure[4; 18; 22; 15; 32], as well as a distributed algorithm running on a PC-cluster[15].

The translations discussed in the previous section have been implemented as a special module of thereasoning systemQUIP [12; 11; 10; 27], a prototype tool for solving various nonmonotonic reasoning tasksbased on reductions to QBFs. Among others,QUIP handles tasks for logic-based abduction, default logic,several types of modal nonmonotonic logics, and the stable model semantics for logic programs.

The general architecture ofQUIP is depicted in Figure 1.QUIP consists of three parts, namely thefilter program, a QBF-evaluator, and the interpreterint. The input filter translates the given problemdescription (in our case, a belief change scenario and a specified reasoning task) into the correspondingquantified Boolean formula, which is then sent to the QBF-evaluator. The current version ofQUIP providesinterfaces to most of the sequential QBF-solvers mentionedabove. For the solvers requiring prenex normalform, the QBFs are translated into structure preserving normal form[9; 28]. The result of the QBF-evaluatoris interpreted byint. Depending on the capabilities of the employed QBF-evaluator, int provides anexplanation in terms of the underlying problem instance (e.g., listing all consistent definitional extensions ofa given belief change scenario). This task relies on a protocol mapping of internal variables of the generatedQBF into concepts of the problem description which is provided byfilter.

The systemQUIP has been implemented in C using standard tools like LEX and YACC (comprisinga total of 2000 lines of code, excluding the used QBF-solver); it runs currently in a Unix environment(Sun/Solaris and Linux), but is easily portable to other operating systems.

7 Conclusion

We have shown how belief revision and belief contraction, asdefined using belief change scenarios, canbe axiomatised by means of quantified Boolean formulas. The general mechanism of our approach is to

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filter QSAT intQBF SOP

protocol mapping

Figure 1: Architecture to use different QBF-solvers.

translate (in polynomial time) a reasoning problem, expressed in terms of belief change scenarios, into theevaluation problem for QBFs. Following this we use a QBF-evaluator to compute the resultant instances.

The approach has several benefits. First, the given axiomatics provides us with further insight about howbelief revision and contraction work within belief change scenarios. As well, this axiomatisation allows usto furnish upper bounds for precise complexity results, going beyond those presented in[7]. Last but notleast we obtain a straightforward implementation technique of belief change in belief change scenarios byappeal to the existing systemQUIP [12; 11]. The implemented operators possess good formal properties,in that most (except for the last) AGM postulates obtain. In particular, the postulate of irrelevance of syntaxis retained, and so the results of a belief change operation is independent of the syntactic expression ofits arguments. While the interesting decision problems involving reasoning lie at the second level of thepolynomial hierarchy, it remains to be seen whether the implementation may nonetheless prove practical forlarge-scale applications.

A Proofs

Proof. According to the semantics of the existential quantifier9, and since for any setU of formulas,U isconsistent iffU is satisfiable, it obviously holds thatT [ R is consistent iff9V (T ^ R) evaluates to true.By observing thatR andM are chosen in such a way that�i 2 R iff gi 2M , for 1 � i � n, it follows that9V (T ^ R) can be rewritten as9V �T ^ ^gi2M �i�: (6)

But (6) evaluates to true exactly if9V �T ^ ni=1(gi ! �i)� (7)

is true underM , because eachgi ! �i is equivalent to�i if gi 2 M , andgi ! �i is trivially true underM if gi =2M . Finally, it remains to observe that QBF (7) coincides withC[T; S℄, by definition ofG � S. �Proof. Let us writeText[B℄ as�1 ^ �2, where�1 = 9V 9V 0(M[B℄ ^ C); and�2 = p2V �:peq ! :9V 9V 0((p � p0) ^ M[B℄ ^ C))�:

Consider Conditions (a) and (b) of Proposition 4.1. We show that Condition (a) holds iffM is a modelof �1, and that (b) holds iffM is a model of�2.

18 INFSYS RR 1843-03-03

To begin with, since, by hypothesis,EQ andM satisfy the condition that(p � p0) 2 EQ iff peq 2 M ,we can apply Proposition 3.1 and obtain thatK 0 [EQ [R [ C is satisfiable iffM is a model of9V 9V 0�K 0 ^ R ^ C ^ (Veq � (V � V 0))�;which is obviously equivalent to�1. It remains to show that Condition (b) holds iffM is a model of�2.

Consider somep 2 Var(B) such that(p � p0) =2 EQ andK 0 [EQ [ fp � p0g [R[C ` ?. InvokingProposition 3.1 again, it follows that the last condition holds exactly if the QBF:9V 9V 0�K 0 ^ (p � p0) ^ R ^ C ^ (Veq � (V � V 0))�is true underM . In general, if we perform this test for eachp 2 Var (B) with (p � p0) =2 EQ, we get thatCondition (b) is equivalent to the condition that the QBF^p2V;(p�p0)=2EQ :9V 9V 0�K 0 ^ (p � p0) ^ R ^ C ^ (Veq � (V � V 0))� (8)

is true underM . Observing that, for anyp 2 V , (p � p0) 2 EQ iff peq 2 M , it follows thatM is a modelof (8) iff it is a model ofp2V :peq ! :9V 9V 0�K 0 ^ (p � p0) ^ R ^ C ^ (Veq � (V � V 0))�: (9)

Since (9) is logically equivalent to�2, we conclude that Condition (b) holds iffM is a model of�2. �Proof. According to Theorem 4.2,Cn(K 0 [ EQ [R) \ LP is a consistent belief change extension ofBiff M is a model ofText[B℄. Thus, for proving Part 1 of the theorem, it suffices to show that the followingcondition holds:

(�) � 2 Cn(K 0 [ EQ [R) iff M is a model of8W�(9W 0M[B℄) ! ��:Furthermore, sinceM is a model of8W ((9W 0M[B℄) ! �) precisely if M is not a model of:8W ((9W 0M[B℄) ! �); we get that Condition (�) implies that� =2 Cn(K 0 [ EQ [R) iff M is amodel of:8W ((9W 0M[B℄) ! �); which in turn proves Part 2 of the theorem. It remains to show that (�)holds.

Since� 2 Cn(K 0 [ EQ [R) iff K 0 [ EQ [ R [ f:�g is unsatisfiable, Proposition 3.1 implies that� 2 Cn(K 0 [ EQ [R) iff M is a model of:9W9W 0�K 0 ^ R ^ :� ^ (Veq � (V � V 0))�: (10)

Given that� does not contain any primed atoms, we can rewrite (10) by moving:� outside the scope of thequantifier9W 0, thus obtaining:9W�9W 0(K 0 ^ R ^ (Veq � (V � V 0))) ^ :��;which is in turn equivalent to:9W�(9W 0M[B℄) ^ :��: (11)

INFSYS RR 1843-03-03 19

But (11) is clearly equivalent to8W�(9W 0M[B℄) ! ��; (12)

and therefore we obtain that� 2 Cn(K 0 [ EQ [R) iff M is a model of (12). �Proof. LetK be a finite knowledge base,� some formula, andV = Var(K [ f�g). Recall that, for any setEQ of equivalences,PEQ = fp 2 P j p � p0 2 EQg andPEQ = P n PEQ .

1. ConsiderB = (K; f�g; ;) in LP : Suppose thatM � V is a model ofK _+�. By Proposition 4.6, thereis a consistent belief change extensionEi0 = Cn(K 0 [EQi0 [ f�g) \ LP of B (for somei0 2 I)such thatdKei0 ^ � is true underM . DefineM1 =M \PEQi0 andM2 =M nM1. By construction

of formuladKei0 , it follows thatM1 [M 2 is a model ofK, whereM2 = (V \PEQi0 ) nM . Hence,

a simple renaming yields thatM 01 [M 02 is a model ofK 0. It follows thatM [M 01 [M 02 is a model ofK 0 ^ ^p�p02EQi0(p � p0) ^ �:By settingJ = fpeq j p � p0 2 EQi0g; we get thatM [ J [M 01 [M 02 is a model ofM[B℄ = K 0 ^ p2V �peq ! (p � p0)� ^ �;which in turn implies thatM [J is a model of9V 0M[B℄. On the other hand, sinceEi0 is a consistentbelief change extension ofB, Theorem 4.2 entails thatJ is a model ofText[B℄. Consequently,M [Jis a model ofText[B℄ ^ 9V 0M[B℄, and thereforeTm [B℄ = 9Veq�Text[B℄ ^ 9V 0M[B℄�is true underM .

Conversely, assume thatM � V is a model ofTm [B℄. Then, there is an interpretationJ � Veq suchthatM [ J is a model ofText[B℄ ^ 9V 0M[B℄: (13)

In particular, we have thatJ is a model ofText[B℄, since the free variables ofText[B℄ are fromVeq .Hence, according to Theorem 4.2, we get thatEi0 = Cn(K 0 [EQi0 [ f�g) \ LP is a consistentbelief change extension ofB, for EQ i0 = fp � p0 j peq 2 Jg: SinceK _+� � Ei0 , for showing thatM is a model ofK _+�, it suffices to show thatM is a model ofEi0 . This can be seen as follows.

Given thatM [ J is a model of (13), we have thatM [ J is a fortiori a model of9V 0M[B℄. Hence,there is some interpretationN 0 � V 0 such thatM [J [N 0 is a model ofM[B℄. From this, we obtainthatM [N 0 is a model ofK 0 ^ ^(p�p0)2EQi0(p � p0) ^ �;

20 INFSYS RR 1843-03-03

which in turn implies thatM [N 0 is a model ofCn(K 0 [EQi0 [ f�g). In particular,M must be amodel of all those elements fromCn(K 0 [EQi0 [ f�g) which contain no atoms fromV 0. In otherwords,M is a model ofEi0 = Cn(K 0 [EQi0 [ f�g) \ LP :

2. ConsiderB = (K; ;; f:�g) in LP , and assume thatM � V is a model ofK _��. From Proposi-tion 4.6, we obtain that there is some consistent belief change extensionEi0 = Cn(K 0 [EQi0)\LPofB and some�k0i0 2 �i0 such thatbK k0i0 is true underM , for�i0 = f�ki0 j �ki0 : PEQi0 ! f>;?gg:Analogous to Part 1, defineM1 = M \ PEQi0 andM2 = M nM1. SincebK k0i0 is true underM ,there is someJ � V \PEQi0 such thatK is true underM1 [ J , and thereforeM 01 [ J 0 is a model ofK 0. Hence,M1 [M 01 [ J 0 is a model ofK 0 ^ ^p�p02EQ i0(p � p0): (14)

Since no atoms fromV \PEQi0 occur in (14),M [M 01[J 0 is also a model of (14). Applying similar

arguments as in Part 1, it follows thatM is a model ofTm [B℄.The proof of the converse direction proceeds analogously toPart 1.

Proof. Since the membership relations are already argued in the main body of the paper, it remains to showthat the problemsEXT, CHOICE, andSKEPTICAL, as well as its variants for revision and contraction, arehard for the respective classes.

1. REXT isNP-hard because a formula� is satisfiable iff the belief change scenarioBR = (f�g; f>g; ;)has a consistent belief change extension. Similarly, CEXT is NP-hard because� is satisfiable iffBC = (f�g; ;; f:?g) possesses a consistent belief change extension. Either of these propertiesimplies thatEXT isNP-hard as well.

2. We show that RCHOICE and CCHOICE are�P2 -hard; similar to the above,�P2 -hardness ofCHOICE isthen an immediate consequence.

We first deal with CCHOICE. According to Proposition 5.1, checking whether a closed QBF � ofform 9P8Q�, where� is a propositional formula in disjunctive normal form andP [Q is a partitionof Var(�), is �P2 -hard. In order to show�P2 -hardness of CCHOICE, we construct a polynomial-time transformation mapping each closed QBF� of the above form into a pair(BC ; ��), whereBC = (K; ;; f:�g) is a belief change scenario and�� is a formula, such that� is valid iff there is aconsistent choice contractionK _� � containing��.The construction ofK, �, and�� is as follows. ForP = fp1; : : : ; png, letR = fr1; : : : ; rng be a setof new atoms not occurring inVar(�). DefineK = fpi ^ ri j i = 1; : : : ; ng; and� = n_i=1(pi ^ ri);and let�� be the result of replacing in� each literal:pj with pj 2 P by rj 2 R (1 � j � n). Weshow that� = 9P8Q� is valid iff there is a consistent belief change extensionCn(K 0 [ EQ i0)\LPof BC = (K; ;; f:�g) containing��.

INFSYS RR 1843-03-03 21

To begin with, observe that, for any consistent belief change extensionCn(K 0 [ EQ)\LP of BC , itholds that

(�) eitherpj � p0j 2 EQ or rj � r0j 2 EQ , but not both, for each1 � j � n.

Assume that9P8Q� is valid. Then, there is an interpretationS0 � P such that for eachU � Q,S0 [ U is a model of�. By construction of��, it follows thatS0 [ T0 [ U is a model of��, forT0 = frj 2 R j pj 2 P n S0g and eachU � Q. Now defineEQi0 = fpj � p0j j pj 2 S0g [ frj � r0j j pj 2 P n S0g:Clearly,Cn(K 0 [ EQ i0) \ LP is a consistent belief change extension ofBC . We claim that�� 2Cn(K 0 [ EQ i0) \ LP .

Since�� 2 LP , it suffices to show thatK 0 [ EQ i0 ` ��. LetM 0 [ N be a model ofK 0 [ EQ i0 ,whereM 0 � P 0 [ R0 andN � P [Q [ R. DefineN1 = N \ (P [ R) andN2 = N \Q. Hence,N = N1[N2. Now, by definition ofK andEQi0 , it must hold thatN1 = S0[T0. ButS0[T0[U isa model of��, for anyU � Q; in particular, sinceN2 � Q, S0[T0[N2 is a model of��. Therefore,N = N1 [N2 is a model of��. Since�� contains no primed atoms, it follows thatM 0 [N is also amodel of��. This proves the relationK 0 [ EQ i0 ` ��. Hence, we showed that9P8Q� is valid onlyif there is consistent belief change extension ofBC containing��.Conversely, assume that�� 2 Cn(K 0 [ EQ i0) \ LP for some consistent belief change extensionCn(K 0 [ EQ i0) \ LP of BC = (K; ;; f:�g). We show that9P8Q� is valid.

Observe that, according to Condition (�), we have eitherpj � p0j 2 EQ i0 or rj � r0j 2 EQ i0 , but notboth. DefineS0 = fpj 2 P j pj � p0j 2 EQ i0g; andT0 = frj 2 R j rj � r0j 2 EQ i0g;and letW0 = S0 [ T0. Clearly,pj 2 S0 iff rj =2 T0, for each1 � j � n. Furthermore,W0 [W 00 is amodel ofEQ i0 . But, sinceVar(EQ i0) =W0[W 00, andW 00 � P 0[R0, we have thatW0[P 0[R0 isa model ofEQ i0 , as well. Moreover,W0 [P 0 [R0 is a model ofK 0 [EQ i0 . In fact, for anyU � Q,W0 [ P 0 [ R0 [ U is a model ofK 0 [ EQ i0 . Since, by hypothesis,K 0 [ EQ i0 ` ��, it follows thatW0 [ P 0 [ R0 [ U is a model of��, for anyU � Q. But �� does not contain any primed atoms, soW0 [ U = S0 [ T0 [ U must also be a model of��, for anyU � Q. Hence, by definition of��, andsincepj 2 S0 iff rj =2 T0, we obtain thatS0 [ U is a model�, for eachU � Q. We just proved thatthere is someS0 � P such that for eachU � Q, � is true underS0 [ U . This means that9P8Q� isvalid.

Now we turn our attention to RCHOICE. Consider�, K, �, and�� as above. We claim that� =9P8Q� is valid iff there is a consistent choice revisionK _+ (:�) containing��. From this,�P2 -hardness of RCHOICE is an immediate consequence.

To prove the claim, we must show that� is valid iff there is a consistent belief change exten-sionCn(K 0 [ EQ i0 [ f:�g) \ LP of BR = (K; f:�g; ;) containing��. Recall that we demon-strated above that� is valid iff there is a consistent belief change extensionCn(K 0 [ EQ i0) \ LPof BC = (K; ;; f:�g) containing��. Furthermore, for anyEQ � fv � v0 j v 2 P [ Rg, it

22 INFSYS RR 1843-03-03

holds thatCn(K 0 [ EQ) \ LP is a consistent belief change extension ofBC = (K; ;; f:�g) iffCn(K 0 [ EQ [ f:�g) \ LP is a consistent belief change extension ofBR = (K; f:�g; ;) (cf. alsoTheorem 4.1 of[8]). Hence, it suffices to show that the following condition holds:

(��) �� 2 Cn(K 0 [ EQ) \ LP iff �� 2 Cn(K 0 [ EQ [ f:�g) \ LP , for any consistent beliefchange extensionCn(K 0 [ EQ) \ LP of BC .

Consider some consistent belief change extensionCn(K 0 [ EQ) \ LP of BC . If �� is contained inCn(K 0 [ EQ)\LP ; then�� 2 Cn(K 0 [ EQ [ f:�g)\LP ; by monotonicity ofCn(�). So suppose�� 2 Cn(K 0 [ EQ [ f�g) \ LP : We must show thatK 0 [ EQ ` �� holds.

Let M 0 [ N be some model ofK 0 [ EQ , whereM 0 � P 0 [ R0 andN � P [ R [ Q. SinceCn(K 0 [ EQ) \ LP is a consistent belief change extension ofBC , EQ satisfies Condition (�), i.e.,for P = fp1; : : : ; png andR = fr1; : : : ; rng, we have thatpj � p0j 2 EQ or rj � r0j 2 EQ , but notboth, for each1 � j � n. DefineN = N n (fpj j rj � r0j 2 EQg [ frj j pj � p0j 2 EQg):Obviously,M 0 [ N is a model ofK 0 [ EQ [ f:�g = fp0i ^ r0i j i = 1; : : : ; ng [ EQ [ ni=1(:pi _ :ri):Hence, since�� 2 Cn(K 0 [ EQ [ f:�g); M 0[ N is a model of��. Moreover, since�� is a formulain disjunctive normal form, and no atom fromP or R occurs negated in��, it follows that, for anyS � P and anyT � R, M 0 [ N [ S [ T is a model of��. In particular,M 0 [ N is a model of��.This provesK 0 [ EQ ` ��.

3. Again, we only show�P2 -hardness of RSKEPTICAL and CSKEPTICAL. To this end, we exploit someresults from[13] and[7; 8].

Let P = fp1; : : : ; png andQ = fq1; : : : ; qmg be two distinct sets of variables, and consider a closedQBF� of form 8P9Q�, where� is a propositional formula such thatP [Q = Var(�). Furthermore,let R = fr1; : : : ; rng be a set of variables distinct fromVar(�), and letv be a further variable notoccurring inVar(�) or R. For K and� as defined in the proof of CCHOICE above, define thefollowing knowledge baseKS and formulas� and :KS = K [Q [ fvg;� = :� ^ (v ! �) ^ ((q1 _ : : : _ qm) ! v); and = ni=1(pi _ ri):As shown in[13], � is valid iff v 2 KS _+s(� ^ ), where _+s is the Satoh revision operator. SinceKS _+s(� ^ ) is equivalent toKS _+(� ^ ) (cf. [7; 8]), we get that� is valid iff v 2 KS _+(� ^ ). (15)

INFSYS RR 1843-03-03 23

Hence, RSKEPTICAL is�P2 -hard.

As for CSKEPTICAL, �P2 -hardness follows from (15) by showing thatv 2 KS _+(� ^ ) iff v 2KS _�(:� _ : ). To see this, analogous to the argumentation given above forRCHOICE, it sufficesto show thatv 2 Cn(K 0S [ EQ [ f(� ^ )g) \ LP iff v 2 Cn(K 0S [ EQ) \ LP ;for any maximal setEQ � fw � w0 j w 2 Var (KS) [ Var(� ^ )g such thatK 0S [ EQ [f(� ^ )g 6` ?: But this property is obviously satisfied, because any suchEQ containsv � v0, andK 0S [ fv � v0g ` v.

Proof.DP -membership can be seen as follows. According to Proposition 4.1, given a belief change scenarioB = (K;R;C) and some setEQ � fp � p0 j p 2 Var(B)g of equivalences, deciding whetherEQdetermines a consistent belief change extension ofB is equivalent to

(i) deciding whetherK 0 [ EQ [R [ C is consistent, and

(ii) deciding whetherK 0 [ EQ [ (p � p0) [ R [ C is inconsistent, for eachp 2 Var(B) such that(p � p0) =2 EQ .

Clearly, Task (i) is inNP and Task (ii) is in co-NP. Hence, the combined problem is inDP .For showingDP -hardness, we consider the following well-knownDP -complete problem (cf.[26]):

SAT-UNSAT : Given two propositional formulas� and , decide whether� is satisfiable and is unsatisfi-able.

We construct a polynomial transformation mapping each pair(�; ) of propositional formulas into abelief change scenarioB and some setEQ � fp � p0 j p 2 Var(B)g of equivalences such that

(�) if � is satisfiable and is unsatisfiable, thenEQ determines a consistent belief change scenario ofB,and vice versa.

The construction ofB andEQ is as follows. Let� and be propositional formulas. Without loss ofgenerality, we can assume thatVar(�) andVar( ) are disjoint. Furthermore, letp andq be distinct atomsnot occurring inVar (�) [Var( ). Then, defineB = (fp ! �; q ! g; fp; qg; ;); andEQ = fp � p0g [ fv � v0 j v 2 Var(�) [Var( )g:

It is easy to see that this construction obeys Condition (�). �References

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