Conflicts between Relevance-Sensitive andIterated Belief Revision

Pavlos Peppas and Anastasios Michael Fotinopoulos and Stella Seremetaki1

Abstract. The original AGM paradigm focuses only on one-stepbelief revision and leaves open the problem of revising a belief statewith whole sequences of evidence. Darwiche and Pearl later ad-dressed this problem by introducing extra (intuitive) postulates asa supplement to the AGM ones. A second shortcoming of the AGMparadigm, seemingly unrelated to iterated revision, is that it is tooliberal in its treatment of the notion of relevance. Once again thisproblem was addressed with the introduction of an extra (also veryintuitive) postulate by Parikh. The main result of this paper is thatParikh postulate for relevance-sensitive belief revision is inconsis-tent with each of the Darwiche and Pearl postulates for iterated beliefrevision.

1 INTRODUCTION

The original AGM paradigm for belief revision, [1, 3, 11], focusesonly on one-step transitions leaving open the problem of how to re-vise a belief state with a whole sequence of evidence. This problemwas later addressed by Darwiche and Pearl who formulated four intu-itive new postulates (known as the DP postulates) to regulate iteratedrevisions. Possible world semantics were introduced to characterizethe new postulates, and with some adjustments (see section 4) theDP postulates were shown to be compatible with the original AGMones.2

Although Darwiche and Pearl’s work has received some criticism,it remains very influential in the literature of iterated belief revisionand has served as a basis for further developments in the area [7, 5].

A shortcoming of a different nature of the original AGM paradigmis that it neglects the important role ofrelevancein belief revision.As noted by Parikh, [8], when a belief stateψ is revised by new in-formationµ, only the part ofψ that is related toµ should be effected;the rest ofψ should remain the same. Parikh proceeded to formulatea postulate, called (P), that captures this intuition (albeit in limitedcases). Postulate (P) was later shown to be consistent with the AGMpostulates and possible-world semantics were introduced to charac-terize it, [10].

The main contribution of this paper is to show that, in the pres-ence of the AGM postulates, Parikh postulate for relevance-sensitivebelief revision is inconsistent witheachof the (seemingly unrelated)Darwiche and Pearl postulates for iterated belief revision.

This of course is quite disturbing. Both the concept of relevanceand the process of iteration are key notions in Belief Revision and

1 Dept of Business Administration, University of Patras, Patras26500, Greece, emails: pavlos@upatras.gr, anfotino@upatras.gr, sers-tel@master.math.upatras.gr

2 To be precise, the DP postulates were shown to be compatible with the re-formalization of the AGM postulates introduced by Katsuno and Mendel-zon in [6].

we can do away with neither. Moreover, the encoding of these no-tions proposed by Parikh, Darwiche, and Pearl appears quite naturaland it is not obvious how one should massage the postulates in orderto reconcile them. Further to this point, subsequent postulates intro-duced to remedy problems with the (DP) ones, [7, 5], are also shownto be incompatible with postulate (P) (see section 6). On the positiveside, these incompatibility results reveal a hitherto unknown connec-tion between relevance and iteration that deepens our understandingof the belief revision process.

The paper is structured as follows. The next section introducessome notation and terminology. Following that we briefly review(Katsuno and Mendelzon’s re-formalization of) the AGM postulates,Darwiche and Pearl’s approach for iterated revisions, and Parikh’sproposal for relevance-sensitive belief revision (sections 3, 4, and 5).Section 6 contains our main incompatibility results. Finally in sec-tion 7 we make some concluding remarks.

2 PRELIMINARIES

Throughout this paper we shall be working with a finitary proposi-tional languageL. We shall denote the (finite) set of all propositionalvariables ofL by A. For a set of sentencesΓ of L, we denote byCn(Γ) the set of all logical consequences ofΓ, i.e.Cn(Γ) = {ϕ ∈L: Γ ` ϕ}. A theory K of L is any set of sentences ofL closedunder`, i.e.K = Cn(K). We shall denote the set of all theories ofL by TL. A theoryK of L is complete iff for all sentencesϕ ∈ L,ϕ ∈ K or ¬ϕ ∈ K. As it is customary in Belief Revision, hereinwe shall identify consistent complete theories with possible worlds.We shall denote the set of all consistent complete theories ofL byML. If for a sentenceϕ, Cn(ϕ) is complete, we shall also callϕcomplete. For a set of sentencesΓ of L, [Γ] denotes the set of allconsistent complete theories ofL that containΓ. Often we shall usethe notation[ϕ] for a sentenceϕ ∈ L, as an abbreviation of[{ϕ}].When two sentencesϕ andχ are logically equivalent we shall oftenwrite ϕ ≡ χ as an abbreviation of ϕ ↔ χ. Finally, the symbols> and⊥ will be used to denote an arbitrary (but fixed) tautology andcontradiction ofL respectively.

3 THE KM POSTULATES

In the AGM paradigm belief sets are represented as logical theories,new evidence as sentences ofL, and the process of belief revision ismodeled as a function mapping a theoryK and a sentenceµ to a newtheoryK ∗ µ. Moreover, eight postulates for∗ are proposed, knownas the AGM postulates, that aim to capture the notion ofrationalityin belief revision.

Katsuno and Mendelzon in [6] slightly reshaped the AGM con-stituents to make the formalization more amendable to implementa-

tion. In particular, the object languageL is set to be a finitary propo-sitional one,3 and belief sets are defined asfinite sets of sentences ofL. Since a belief set contains only finitely many elements, one canin fact represent it as a single sentenceψ; namely the conjunction ofall its elements. This is the representation eventually adopted in [6]and the one we shall use herein. To emphasize the finiteness of thenew representation we shall callψ abelief baseand reserve the termbelief setfor the closure ofψ, i.e. the theoryCn(ψ).

With the above reformulation, a revision function becomes a func-tion ∗ mapping a sentenceψ and a sentenceµ to a new sentenceψ ∗ µ; i.e. ∗ : L × L 7→ L. Moreover in the new formalization theAGM postulates are equivalent to the following six, known as theKM postulates:

(KM1) ψ ∗ µ ` µ.(KM2) If ψ ∧ µ is satisfiable thenψ ∗ µ ≡ ψ ∧ µ.(KM3) If µ is satisfiable thenψ ∗ µ is also satisfiable.(KM4) If ψ1 ≡ ψ2 andµ1 ≡ µ2 thenψ1 ∗ µ1 ≡ ψ2 ∗ µ2.(KM5) (ψ ∗ µ) ∧ ϕ ` ψ ∗ (µ ∧ ϕ).(KM6) If (ψ∗µ)∧ϕ is satisfiable thenψ∗(µ∧ϕ) ` (ψ∗µ)∧ϕ.

4 ITERATED BELIEF REVISION

One thing to notice about the KM postulates is that they all refer tosingle-step revisions; no constraints are placed on how the revisionpolicy at the initial belief baseψ may relate to the revision policies atit descendants (i.e. at the belief bases resulting fromψ via a sequenceof revisions).

Darwiche and Pearl’s solution to this problem came in the form offour additional postulates, known as the DP postulates, listed below,[2]:

(DP1) If ϕ ` µ then(ψ ∗ µ) ∗ ϕ = ψ ∗ ϕ.(DP2) If ϕ ` ¬µ then(ψ ∗ µ) ∗ ϕ = ψ ∗ ϕ.(DP3) If ψ ∗ ϕ ` µ then(ψ ∗ µ) ∗ ϕ ` µ.(DP4) If ψ ∗ ϕ 6` ¬µ then(ψ ∗ µ) ∗ ϕ 6` ¬µ.

The DP postulates are very intuitive and their intended interpreta-tion, is loosely speaking as follows (see [2, 7] for details). Postulate(DP1) says that if the subsequent evidenceϕ is logically strongerthan the initial evidenceµ thenϕ overrides whatever changesµ mayhave made. (DP2) says that if two contradictory pieces of evidencearrive sequentially one after the other, it is the later that will prevail.(DP3) says that if revisingψ by ϕ causesµ to be accepted in the newbelief base, then revising first byµ and then byϕ can not possiblyblock the acceptance ofµ. Finally, (DP4) captures the intuition that“no evidence can contribute to its own demise” [2]; if the revision ofψ by ϕ does not cause the acceptance of¬µ, then surely this shouldstill be the case ifψ is first revised byµ before revised byψ.

An initial problem with the DP postulates (more precisely with(DP2)) was that they were inconsistent with the KM postulates. How-ever this inconsistency was not deeply rooted and was subsequentlyresolved. There are in fact (at least) two ways of removing it. Thefirst, proposed by Darwiche and Pearl themselves, [2], involves sub-stituting belief bases withbelief states, and adjusting the KM pos-tulates accordingly. The second, proposed by Nayak, Pagnucco, andPeppas, [7], keeps belief bases as the primary objects of change, butmodifies the underlying assumptions about the nature of∗.3 In the original AGM paradigm, the object languageL is not necessarily

finitary nor propositional. The details ofL are left open and only a smallnumber of structural constraints are assumed ofL and its associated entail-ment relation (see [3, 11]).

In particular, notice that, with the exception of (KM4), the KMpostulates apply only to asingleinitial belief baseψ; no reference toother belief bases are made. Even postulate (KM4) can be weakenedto comply with this policy:

(KM4)’ If µ1 ≡ µ2 thenψ ∗ µ1 ≡ ψ ∗ µ2.

With the new version of (KM4), the KM postulates allow us todefine a revision function as aunary function∗ : L 7→ L, mappingthe new evidenceµ to a new belief base∗ψ(µ), given the initial beliefbaseψ asbackground. This is the first modification made by Nayaket al. The second is to make revision functionsdynamic. That is,revision functions may change as new evidence arrives. With thisrelaxation it is possible for example to have one revision function∗ψ associated initially withψ, and a totally different one after therevision ofψ by a sequence of evidence that have made the full circleand have convertedψ back to itself.4 Notice that the weakening of(KM4) to (KM4)’ is consistent with such dynamic behavior.

As shown by Nayak et al., these two modifications suffice to recon-cile the DP postulates with the KM ones, and it is these modificationswe shall adopt for the rest of the paper.5 Hence for the rest of the pa-per, unless specifically mentioned otherwise, we shall use the term“KM postulates” to refer to (KM1)-(KM6) with (KM4) replaced by(KM4)’, and we shall assume that revision function are unary anddynamic.

We close this section with a remark on notation. Although we as-sume that revision functions areunary(relative to some backgroundbelief baseψ), for ease of presentation we shall keep the original no-tation and denote the revision ofψ by µ asψ ∗ µ rather than∗ψ(µ).

5 RELEVANCE-SENSITIVE BELIEF REVISION

Leaving temporarily aside the issue of iterated belief revision, weshall now turn back to one-step revisions to review the role ofrele-vancein this process.

As already mentioned in the introduction, Parikh in [8] pointed outthat the AGM/KM postulates fail to capture the intuition that duringthe revision of a belief baseψ by µ, only the part ofψ that is relatedto µ should be effected, while everything else should stay the same.

Of course determining the part ofψ that is relevant to some newevidenceµ is not a simple matter. There is however at least one spe-cial case where the role of relevance can be adequately formalized;namely, when it is possible to break downψ into two (syntactically)independent parts such that only the fist of the two parts is syntacti-cally related to the new evidenceµ.

More precisely, for a sentenceϕ of L, we shall denote byAϕ

the smallest set of propositional variables, through which a sentencethat is logically equivalent toϕ can be expressed. For example, ifϕ is the sentence(p ∨ q ∨ z) ∧ (p ∨ q ∨ ¬z), thenAϕ = {p, q},sinceϕ is logically equivalent top ∨ q, and no sentence with fewerpropositional variables is logically equivalent toϕ. We shall denoteby Lϕ the propositional sublanguage built fromAϕ via the usualboolean connectives. ByLϕ we shall denote the sublanguage builtfrom thecomplementof Aϕ, i.e. fromA− Aϕ. Parikh proposed thefollowing postulate to capture the role of relevance in belief revision(at least for the special case mentioned above):6

4 In such cases, although the sequence of evidence does not effect the beliefsof the agent, it does however change the way the agent reacts to new input.

5 It should be noted though that our results still hold even if Darwiche andPearl’s proposal of switching to belief states was adopted.

6 The formulation of (P) in [8] is slightly different from the one presentedbelow since Parikh was working with theories rather than belief bases. Thetwo version are of course equivalent.

(P) If ψ ≡ χ ∧ ϕ whereχ, ϕ are sentences of disjoint sublan-guagesLχ, Lϕ respectively, andLµ ⊆ Lχ, thenψ ∗ µ ≡(χ ◦ µ) ∧ ϕ, where◦ is a revision function of the sublan-guageLχ.

According to postulate (P), whenever it is possible to break downthe initial belief baseψ into two independent partsχ and ϕ, andmoreover it so happens that the new evidenceµ can be expressedentirely in the language of the first part, then during the revision ofψ by µ, it is only the first part that is effected; the unrelated partϕcrosses over to the new belief base verbatim.

Notice that of the nature of the “local” revision operator◦ andits relationship with the “global” revision operator∗ is not clearlystated in (P). Peppas et al., [10], therefore proposed a re-formulationof axiom (P) in terms of two new conditions (R1) and (R2) that donot refer to a “local” revision operator◦. Only the first of these twoconditions will be needed herein:

(R1) If ψ ≡ χ∧ϕ, Lχ∩Lϕ = ∅, andLµ ⊆ Lχ, thenCn(ψ)∩Lχ

= Cn(ψ ∗ µ) ∩ Lχ.

At first (R1) looks almost identical to (P) but it is in fact strictlyweaker than it (see [10] for details). It is essentially condition (R1)that we will be using to derive our incompatibility results.

6 INCOMPATIBILITY RESULTS

As already announced in the introduction, we shall now prove thatin the presence of the KM postulates, (R1) – and therefore (P) – isinconsistent with each of the postulates (DP1)-(DP4). The proof re-lies on the semantics characterization of these postulates so we shallbriefly review it before presenting our results.

We start with Grove’s seminal representation result [4] and its sub-sequent reformulation by Katsuno and Mendelzon [6].

Let ψ be a belief base and≤ψ a total preorder inML. We denotethe strict part of≤ψ by <ψ. We shall say that≤ψ is faithful iff theminimal elements of≤ψ are all theψ-worlds:7

(SM1) If r ∈ [ψ] thenr ≤ψ r′ for all r′ ∈ML.(SM2) If r ∈ [ψ] andr′ 6∈ [ψ] thenr <ψ r′.

Given a belief baseψ and a faithful preorder≤ψ associated withit, one can define a revision function∗ : L 7→ L as follows:

(S*) ψ ∗ µ = γ(min([µ],≤ψ).

In the above definitionmin([µ],≤ψ) is the set of minimalµ-world with respect to≤ψ, while γ is a function that maps a set ofpossible worldsS to a sentenceγ(S) such that[γ(S)] = S.

The preorder≤ψ essentially represents comparative plausibility:the closer a world is to the initial worlds[ψ], the more plausible it is.Then according to (S*), the revision ofψ by µ is defined as the beliefbase corresponding to the most plausibleµ-worlds.

In [4, 6] it was shown that the function induced from (S*) satisfiesthe KM postulates and conversely, every revision function∗ that sat-isfies the KM postulates can be constructed from a faithful preorderby means of (S*).8

7 In [6] a third constraint was required for faithfulness, namely that logicallyequivalent sentences are assigned the same preorders. This is no longernecessary given the new version of (KM4).

8 We note that the weakening of (KM4) to (KM4)’ does not effect these re-sults since it is accommodated by a corresponding weakening of the notionof faithfulness.

This correspondence between revision functions and faithful pre-orders can be preserved even if extra postulates for belief revision areintroduced, as long as appropriate constraints are also imposed onthe preorders. In particular, Darwiche and Pearl proved that the fol-lowing four constraints (SI1)-(SI4) on faithful preorders correspondrespectively to the four postulates (DP1)-(DP4).

(SI1) If r, r′ ∈ [µ] thenr ≤ψ r′ iff r ≤ψ∗µ r′.(SI2) If r, r′ ∈ [¬µ] thenr ≤ψ r′ iff r ≤ψ∗µ r′.(SI3) If r ∈ [µ] andr′ ∈ [¬µ] thenr <ψ r′ entailsr <ψ∗µ r′.(SI4) If r ∈ [µ] andr′ ∈ [¬µ] thenr ≤ψ r′ entailsr ≤ψ∗µ r′.

Notice that all of the above constraints make associations betweenthe preorder≤ψ related to the initial belief baseψ and the preorder≤ψ∗µ related to the belief base that results from the revision ofψ byµ.

The semantic constraint(s) corresponding to postulate (P) havealso been fully investigated in [10].9 Herein however we shall focusonly on condition (R1); in fact we shall be even more restrictive andconsider only the semantic counterpart of (R1) in the special case ofconsistent andcompletebelief bases:

(PS) If Diff(ψ, r) ⊂ Diff(ψ, r′) thenr <ψ r′.

In the above condition, for any two worldsw, w′, Diff(w, w′) rep-resents the set of propositional variables that have different truth val-ues in the two worlds; in symbols,Diff(w, w′) = {p ∈ A : w ` pandw′ 6` p} ∪ {p ∈ A : w′ ` p andw 6` p}. Whenever a sentenceψ is consistent and complete, we useDiff(ψ, w′) as an abbreviationof Diff(Cn(ψ), w′).

Intuitively, (PS) says that the plausibility of a worldr dependson the propositional variables in which it differs from the initial(complete) belief baseψ: the more the propositional variables inDiff(ψ, r) the less plausibler is. In [10] it was shown that, for thespecial case of consistent and complete belief bases, (PS) is the se-mantic counterpart of (R1); i.e. given a consistent and complete be-lief baseψ and a faithful preorder≤ψ, the revision function∗ pro-duced from≤ψ via (S*) satisfies (R1) iff≤ψ satisfies (PS).

Although it is possible to obtain a fully-fledged semantic charac-terization of postulate (P) by generalizing (PS) accordingly (see [10])the above restricted version suffices to establish the promised results:

Theorem 1 In the presence of the KM postulates, postulate (P) isinconsistent with each of the postulates (DP1)-(DP4).

Proof. Since (P) entails (R1) it suffices to show that (R1) is incon-sistent with each of (DP1)-(DP4).

Assume that the object languageL is built from the propositionalvariablep, q, andz. Moreover letψ be the complete sentencep∧q∧zand let≤ψ be the following preorder inML:

pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz <ψ pqz

In the above definition of≤ψ, and in order to increase readabil-ity, we have used sequences of literals to represent possible worlds(namely the literals satisfied by a world), and we have representedthe negation of a propositional variablev by v.

Notice that≤ψ satisfies (PS). In what follows we shall constructsentencesµ1, µ2, µ3, andµ4, such that no preorder satisfying (PS)and related toψ ∗µ1 (respectively toψ ∗µ2, ψ ∗µ3, ψ ∗µ4) can also

9 In the same paper axiom (P) was shown to be consistent with all theAGM/KM postulates.

satisfy (SI1) (respectively (SI2), (SI3), (SI4)). Given the correspon-dence between (R1) and (PS) on one hand, and the correspondencebetween (DP1)-(DP4) and (SI1)-(SI4) on the other, this will sufficeto prove the theorem.

Inconsistency of (PS) and (SI1):Let µ1 be the sentenceq∨z. According to the definition of≤ψ, thereis only one minimalµ1-world, namelypqz, and therefore by (S*),ψ∗µ1 ≡ p∧q∧z. Consider now the possible worldsw = pqz andw′

= pqz. Clearly,Diff(ψ ∗µ1, w′) = {q} ⊂ {q, z} = Diff(ψ ∗µ1, w).

Consequently, no matter what the new preorder≤ψ∗µ1 is, as longas it satisfies (PS) it holds thatw′ <ψ∗µ1 w. On the other hand,sincew, w′ ∈ [µ1] andw ≤ψ w′, (SI1) entails thatw ≤ψ∗µ1 w′.Contradiction.

Inconsistency of (PS) and (SI2):Letµ2 be the sentencep∧q∧z. Once again there is only one minimalµ2-world, namelypqz, and thereforeψ ∗ µ2 ≡ p ∧ q ∧ z. Let wand w′ be the possible worldspqz and pqz respectively. It is nothard to verify thatDiff(ψ ∗ µ2, w

′) ⊂ Diff(ψ ∗ µ2, w) and therefore(PS) entailsw′ <ψ∗µ1 w. On the other hand, given thatw, w′ ∈[¬µ2] andw ≤ψ w′, (SI2) entails thatw ≤ψ∗µ1 w′, leading us to acontradiction.

Inconsistency of (PS) and (SI3):Let µ3 be the sentence(p ∧ q ∧ z) ∨ (p ∧ q ∧ z). Given the abovedefinition of≤ψ it is not hard to verify thatψ ∗ µ3 ≡ p ∧ q ∧ z.Once again, definew andw′ to be the possible worldspqz andpqzrespectively. ClearlyDiff(ψ∗µ3, w

′) ⊂ Diff(ψ∗µ3, w) and therefore(PS) entailsw′ <ψ∗µ3 w. On the other hand notice thatw ∈ [µ3],w′ ∈ [¬µ3] andw <ψ w′. Hence (SI3) entails thatw <ψ∗µ3 w′.Contradiction.

Inconsistency of (PS) and (SI4):Let µ4 = µ3 = (p∧ q ∧ z)∨ (p∧ q ∧ z) and assume thatw, w′ are aspreviously defined. Then clearly, like before, (PS) entailsw′ <ψ∗µ4

w. On the other hand, sincew ∈ [µ4], w′ ∈ [¬µ4] andw ≤ψ w′,(SI4) entails thatw ≤ψ∗µ4 w′. A contradiction once again.2

The above inconsistencies extend to subsequent developments ofthe Darwiche and Pearl approach as well. Herein we consider twosuch extensions in the form of two new postulates introduced to rec-tify anomalies with the original DP approach.

The first postulate, called theConjunction Postulate, was intro-duced by Nayak, Pagnucco, and Peppas in [7]:

(CNJ) If µ ∧ ϕ 6` ⊥, thenψ ∗ µ ∗ ϕ ≡ ψ ∗ (µ ∧ ϕ).

As shown in [7], in the presence of the KM postulates, (CNJ) en-tails (DP1), (DP3), and (DP4). Hence the following result is a directconsequence of Theorem 1:

Corollary 1 In the presence of the KM postulates, postulate (P) isinconsistent with postulate (CNJ).

The last postulate we shall consider herein is theIndependencePostulateproposed by Jin and Thielscher in [5]:

(Ind) If ¬µ 6∈ ψ ∗ ϕ thenµ ∈ ψ ∗ µ ∗ ϕ.

Jin and Thielscher proved that, although weaker than (CNJ), (Ind)still entails (DP3) and (DP4). Hence, from Theorem 1, it follows:

Corollary 2 In the presence of the KM postulates, postulate (P) isinconsistent with postulate (Ind).

7 CONCLUSION

In this paper we have proved the inconsistency of Parikh’s postulate(P) for relevance-sensitive belief revision, with each of the four pos-tulates (DP1)-(DP4) proposed by Darwiche and Pearl for iterated be-lief revision. This result suggests that a major refurbishment may bedue in our formal models for belief revision. Both relevance and iter-ation are central to the process of belief revision and neither of themcan be sacrificed. Moreover, the formalizations of these notions bypostulates (P) and (DP1)-(DP2) respectively seem quite intuitive andit is not clear what amendments should be made to reconcile them.

On a more positive note, the inconsistencies proved herein reveal ahitherto unknown connection between relevance and iteration, whichwill eventually lead to a deeper understanding of the intricacies ofbelief revision.

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