Distance Semantics for Relevance-Sensitive Belief Revision

Pavlos Peppas Samir Chopra Norman Foo Dept of Business Administration Dept of Computer and Knowledge Systems Group University of Patras Information Science School of Computer Science 265 00 Patras, Greece Brooklyn College of the and Engineering [email protected] City University of New York University of New South Wales Brooklyn, NY 11210, USA Sydney, NSW 2052, Australia [email protected] [email protected]

Abstract is not fully captured by the AGM paradigm. To remedy this shortcoming, Parikh introduced an additional axiom, named Possible-world semantics are provided for Parikh’s (P), as a supplement to the AGM postulates. Loosely speak- relevance-sensitive model for belief revision. Having ing, axiom (P) says that when new information ϕ is received, Grove’s system-of-spheres construction as a base, we only part of the initial belief set K will be affected; namely consider additional constraints on measuring distance the part that shares common propositional variables with the between possible worlds, and we prove that, in the pres- minimal language of ϕ. Parikh’s approach is also known as ence of the AGM postulates, these constraints character- the language splitting model. ize precisely Parikh’s axiom (P). These additional con- straints essentially generalize a criterion of similarity Although axiom (P) is just a first step towards captur- that predates axiom (P) and was originally introduced ing the role of relevance in belief revision 1, Parikh’s work in the context of Reasoning about Action. A by-product has received considerable attention since the publication of of our study is the identification of two possible read- (Parikh 1999) (see for example (Chopra & Parikh 1999), ings of Parikh’s axiom (P), which we call the strong and (Chopra & Parikh 2000), (Chopra, Georgatos, & Parikh the weak versions of the axiom. An interesting feature 2001)). Yet, despite all the research on axiom (P), no se- of the strong version is that, unlike classical AGM be- lief revision, it makes associations between the revision mantics for it have yet been formulated. This is the gap that policies of different theories. the present article aims to fill. We examine new constraints on systems-of-spheres and, building on Grove’s result, we prove that in the presence of the AGM postulates, axiom (P) is sound and complete with respect to these new semantic Introduction constraints. Much of the work in the field of Belief Revision is based on What is particularly pleasing about our result is that the the classical work of Alchourron, Gardenfors and Makin- new constraints on systems of spheres are in fact not new at son, (Alchourron, Gardernfors, & Makinson 1985), that has all; they essentially generalize a very natural condition that given rise to a formal framework for studying this process, predates axiom (P) and has been motivated independently by commonly referred to as the AGM paradigm. Within the Winslett in the context of Reasoning about Action (Winslett AGM paradigm there are two constituents that are of partic- 1988). This connection between Belief Revision with Rea- ular interest for this paper. The first is the set of rationality soning about Action further confirms intuitions about the re- postulates for belief revision, known as the AGM postulates lationship between the two areas (see (Peppas & Wobcke (Alchourron, Gardernfors, & Makinson 1985). The second 1992), (Peppas 1994), and (Peppas, Foo, & Nayak 2000)). is a special kind of preorder on possible worlds, called a sys- tem of spheres, based on which Grove defined a constructive In the course of formulating semantics for axiom (P) we model for belief revision; Grove has shown in (Grove 1988) observed that there are in fact two possible readings of this that the AGM postulates are sound and complete with re- axiom, which we call the strong and the weak versions of spect to his system-of-spheres semantics. (P). We present both these versions herein and we show that the strong version of (P) brings with it a new feature in the Studying the AGM paradigm, Parikh (Parikh 1999) ob- picture of classical AGM revision: it makes associations be- served that is it rather liberal in its treatment of the notion of tween the revision policies of different theories. relevance. More precisely, Parikh argues that during belief revision a rational agent does not change her entire belief The outline of the paper is as follows. We first present corpus, but only the portion of it that is relevant to the new some background material on the AGM paradigm and the information. This intuition of local change, Parikh claims, 1Work on relevance in general in Artificial Intelligence has been Copyright c 2004, American Association for Artificial Intelli- ongoing for a while. A comprehensive collection of papers dealing gence (www.aaai.org). All rights reserved. with relevance could be found in (AIJ 1997).

KR 2004 319 language splitting model (first three sections). The crucial cussion, we often project operations defined earlier for the axiom (P) is then examined in greater detail to fully flesh entire language L, to one of its sublanguages L . When this out its possible readings (4th section). We proceed with the happens, all notation will be subscripted by the sublanguage formulation of semantics for axiom (P). For ease of expo- L . For example, for a set of sentences Γ ⊂L, the term sition and clarity, we start by focusing on the special case CnL (Γ) denotes the logical closure of Γ in L . Similarly, of “opinionated” agents, that is, agents whose belief set is a [Γ]L denotes the set of all maximally consistent supersets of consistent complete theory (5th section). Then we consider Γ in L . When no subscript is present, it is understood that the general case of incomplete theories (6th section). The the operation is relevant to the original language L. last section contains some concluding remarks. The AGM Paradigm Formal Preliminaries Much research in belief revision is based on the work of Al- Throughout this paper we work with a finite set of propo- chourron, Gardenfors and Makinson (Alchourron, Gardern- sitional variables P = {p1,...pm}. We define L to be the fors, & Makinson 1985), who have developed a research propositional language generated from P (using the stan- framework for this process, known as the AGM paradigm. dard boolean connectives ∧, ∨, →, ¬ and the special sym- In this section we shall briefly review two of the main mod- bols , ⊥) and governed by classical propositional logic . els for belief revision within the AGM paradigm; the first is A sentence ϕ ∈Lis contingent iff  ϕ and ¬ϕ. For a based on a set of rationality postulates, and the second on a set of sentences Γ of L, we denote by Cn(Γ) the set of all preorder on worlds known as a system of spheres. logical consequences of Γ, i.e., Cn(Γ) = {ϕ ∈L: Γ  ϕ}. Cn(ϕ ,ϕ ,...,ϕ ) ϕ We shall often write 1 2 n , for sentences 1, The AGM Postulates ϕ2,...,ϕn, as an abbreviation of Cn({ϕ1,ϕ2,...,ϕn}). A theory T of L is any set of sentences of L closed under In the AGM paradigm, belief revision is modeled as a func- , i.e., T = Cn(T ). In this paper we focus only on consistent tion ∗ mapping a theory T and a sentence ϕ to the theory theories. Hence from now on, whenever the term “theory” T ∗ϕ. In this paper we assume that the theory T and the sen- appears unqualified, it is understood that it refers to a con- tence ϕ are individually consistent. Alchourron, Gardenfors, sistent theory. We denote the set of all consistent theories and Makinson have proposed the following set of postulates of L by KL. A theory T of L is complete iff for all sen- for belief revision: tences ϕ ∈L, ϕ ∈ T or ¬ϕ ∈ T . We denote the set of all consistent complete theories of L by ML. In the context of (K*1) T ∗ ϕ is a theory of L. systems of spheres, consistent complete theories essentially (K*2) ϕ ∈ T ∗ ϕ. play the role of possible worlds. Following this convention, in the rest of the article we use the terms “possible world” (K*3) T ∗ ϕ ⊆ T + ϕ. (or simply “world”) and “consistent complete theory” inter- (K*4) If ¬ϕ ∈ T then T + ϕ ⊆ T ∗ ϕ. changeably. For a set of sentences Γ of L, [Γ] denotes the set of all consistent complete theories of L that contain Γ. (K*5) T ∗ ϕ = L iff ¬ϕ. Often we use the notation [ϕ] for a sentence ϕ ∈L,asan (K*6) If  ϕ ↔ ψ then T ∗ ϕ = T ∗ ψ. [{ϕ}] T abbreviation of . For a theory and a set of sentences T ∗ (ϕ ∧ ψ) ⊆ (T ∗ ϕ)+ψ Γ of L, we denote by T +Γthe closure under  of T ∪ Γ, (K*7) . i.e., T +Γ=Cn(T ∪ Γ). For a sentence ϕ ∈Lwe often (K*8) If ¬ψ ∈ T ∗ ϕ then (T ∗ ϕ)+ψ ⊆ T ∗ (ϕ ∧ ψ). write T + ϕ as an abbreviation of T + {ϕ}. In the course of this paper, we often consider sublan- Systems of Spheres guages of L. Let P be a subset of the set of proposi- P tional variables P .ByL we denote the sublanguage of Apart from axiomatic approaches to belief revision, a num- L defined over P . In the limiting case where P is empty, ber of explicit constructions for this process have been pro- P we take L to be the language generated by , ⊥ and the posed. One popular construction is that proposed by Grove boolean connectives. For a sublanguage L of L defined over (Grove 1988) based on a total preorder on possible worlds. P P L a subset of ,by we denote the sublanguage defined T L S over the propositional variables in the complement of P i.e., Definition 1 (Grove 1988) Let be a theory of , and T ML L L(P −P ) χ L L a collection of sets of possible worlds i.e., ST ⊆ 2 . ST = . For a sentence of ,by χ we denote the [T ] minimal sublanguage of L within which χ can be expressed is a system of spheres centered on iff the following con- ditions are satisfied:3 (i.e., Lχ contains a sentence that is logically equivalent to χ, L and moreover no proper sublanguage of χ contains such (Parikh 1999) for details. 2 a sentence). Finally we note that in the forthcoming dis- 3We include condition (S4) for reasons of comprehensiveness, even though in the finite propositional case, this condition is redun- 2 It is not hard to verify that for every χ, Lχ is unique – see dant.

320 KR 2004 (S1) ST is totally ordered with respect to set inclusion; In order to block revision functions like ∗ t Parikh intro- that is, if V,U ∈ ST then V ⊆ U or U ⊆ V . duced in (Parikh 1999) a new axiom, named (P), as a supple- ment to the AGM postulates. The main intuition that axiom (S2) The smallest sphere in ST is [T ]; that is, [T ] ∈ ST , (P) aims to capture is that an agent’s beliefs can be subdi- and if V ∈ ST then [T ] ⊆ V . vided into disjoint compartments, referring to different sub- M ∈ S M ject matters, and that when revising, the agent modifies only (S3) L T (and therefore L is the largest sphere the compartment(s) affected by the new information: in ST ). T Cn(χ, ψ) χ, ψ (S4) For every ϕ ∈L, if there is any sphere in ST inter- (P) If = where are sentences of dis- L L ϕ ∈ secting [ϕ] then there is also a smallest sphere in joint sublanguages 1, 2 respectively, and L1 T ∗ ϕ (CnL (χ)◦ϕ)+ψ ◦ ST intersecting [ϕ]. , then = 1 , where is a revision operator of the sublanguage L 1. For a system of spheres ST and a sentence ϕ ∈L, the 4 smallest sphere in ST intersecting [ϕ] is denoted CT (ϕ). It was shown in (Parikh 1999) that (P) is consistent with With any system of spheres ST , Grove associates a function the AGM postulates, (K*1) - (K*6) (known as the basic ML AGM postulates). The results presented later in this paper fT : L → 2 defined as follows: entail that (P) is in fact consistent with all eight AGM pos- tulates (K*1) - (K*8). fT (ϕ)=[ϕ] ∩ CT (ϕ)

Grove uses the system of spheres S and its associated func- Two Readings of Axiom (P) tion fT , to define constructively the process of revising T , Before proceeding with the formulation of semantics for ax- by means of the following condition: iom (P), it is worth taking a closer look at it.
(S*) T ∗ ϕ = fT (ϕ) Consider two sentences χ, ψ ∈L, such that Lχ ∩Lψ = ∅, and let T be the theory T = Cn({χ, ψ}). Moreover, let Grove proved that the class of functions generated from sys- ϕ be any sentence in Lχ. According to axiom (P), anything tems of spheres by means of (S*) is precisely the family outside Lχ will not be affected by the revision of T by ϕ. of revision functions satisfying the eight AGM postulates This however is only one side of axiom (P). The other side (K*1) - (K*8). One of the main aims of this paper is to concerns the part of the theory T that is related to ϕ, which characterize the subclass of systems of spheres which cor- according to axiom (P) will change to CnLχ (χ)◦ϕ, where ◦ respond (via (S*)) to revision functions that, in addition to is a revision function defined over the sublanguage L χ.Itis (K*1) - (K*8), also satisfy axiom (P) (see below). this second side of axiom (P) that needs closer examination. Axiom (P) is open to two different interpretations. Ac- Relevance-Sensitive Belief Revision cording to the first reading, which we call the weak version of axiom (P), the revision function ◦ that modifies the rele- When revising a theory T by a sentence ϕ it seems plau- vant part of T – call it the local revision function – may vary sible to assume that only the beliefs that are relevant to ϕ from theory to theory, even when the relevant part Cn(χ) should be affected, while the rest of the belief corpus is un- stays the same. To give a concrete example, let a, b, c be changed. For example, an agent that is revising her beliefs propositional variables, let T be the theory T = Cn(a∧b, c), about planetary motion is unlikely to revise her beliefs about and let T be the theory T = Cn(a ∧ b, ¬c). Denote by L1 Malaysian politics. This simple intuition is not fully cap- the sublanguage defined over {a, b} and by L 2 the sublan- tured in the AGM paradigm. To see this consider the trivial guage defined over {c}. Moreover, let ϕ be the sentence ϕ revision function ∗t defined below: = ¬a ∨¬b. The part of T and T that is relevant to ϕ (in the  sense of the language-splitting model) is the same for both T + ϕ ϕ T theories, namely Cn(a ∧ b). Nevertheless, according to the T ∗ ϕ = if is consistent with t Cn(ϕ) otherwise weak version of axiom (P), the local revision operators ◦ and ◦ that modify the two identical relevant parts of T and T respectively, may very well differ. For example, it could be It is not hard to verify that ∗t satisfies all the AGM pos- the case that CnL1 (a ∧ b)◦(¬a ∨¬b) = CnL1 (¬a ∧ b), and tulates, and yet it has the rather counter-intuitive effect of Cn (a∧b)◦(¬a∨¬b) Cn (a∧¬b) T L1 = L1 , from which it fol- throwing away all non-tautological beliefs in whenever lows that T ∗ ϕ = Cn(¬a, b, c), and T ∗ ϕ = Cn(a, ¬b, ¬c). the new information ϕ is inconsistent with T , regardless of ϕ In other words, the weak version of axiom (P) allows the lo- whether these beliefs are related to or not. cal revision function to be context-sensitive. In the scenario c T 4In the limiting case where ϕ is inconsistent, Grove defines described above, the presence of in leads to a local revi- ◦ Cn (a∧b) Cn (¬a∧b) CT (ϕ) to be the set ML. However, since in this paper we only sion function for L1 that produces L1 consider revision by consistent sentences, these limiting cases are as the result of revising by ¬a ∨¬b; on the other hand, the irrelevant. presence of ¬c in T , induces a local revision function ◦

KR 2004 321 for CnL1 (a ∧ b) that produces CnL1 (a ∧¬b) for the same For (R2), let us denote by T the theory Cn(χ). Firstly no- input. Therefore, while c (or ¬c) remains unaffected during tice that T ∗ ϕ is equal to the closure in L of CnLχ (χ)◦ϕ the (global) revision by ¬a ∨¬b (since it is not relevant to i.e., T ∗ ϕ = Cn(CnLχ (χ)◦ϕ). Indeed, T can be written the new information), its presence influences the way that as T = Cn(χ, ψ ∨¬ψ) and therefore, by (the strong ver- the relevant part of the theory is modified. sion of) axiom (P), T ∗ ϕ = (CnLχ (χ)◦ϕ)+(ψ ∨¬ψ) = Cn(CnL (χ)◦ϕ). Consequently, T ∗ ϕ = (Cn(χ) ∗ ϕ)+ψ, To prevent such an influence we need to resort to the χ and moreover LCn(χ)∗ϕ ∩Lψ = ∅. Therefore, (T ∗ ϕ) ∩Lχ strong version of axiom (P) which makes the local revision = (Cn(χ) ∗ ϕ) ∩Lχ as desired. function ◦ context-independent. According to the strong in- terpretation of (P), for any two theories T = Cn(χ, ψ) and ( ⇐ ) T = Cn(χ, ψ ), such that Lχ ∩Lψ = Lχ ∩Lψ = ∅, there ∗ T exists a single local revision function ◦ such that T ∗ ϕ = Assume that satisfies (R1) and (R2). Let be a theory L T Cn(χ, ψ) χ, ψ ∈L L ∩L (CnL (χ)◦ϕ)+ψ and T ∗ϕ = (CnL (χ)◦ϕ)+ψ , for any of such that = , where and χ ψ χ χ ∅ ϕ ∈Lχ. = . As a first step in proving (P), we shall show that for any ϕ ∈Lχ, the theory T ∗ ϕ is also split between Lχ and It should be noted that although axiom (P) is open to both Lψ. Assume on the contrary that this is not the case for a the weak and the strong interpretations, the discussion and particular ϕ ∈Lχ. Then, as shown by Lemma-A in (Parikh some results in (Parikh 1999) suggest that the strong ver- 1999), there is a world r such that both r ∩Lχ and r ∩ Lχ sion of axiom (P) is intended. Following Parikh, we shall are individually consistent with T ∗ ϕ, and yet r ∈ [T ∗ ϕ]. also adopt the strong version of axiom (P) in this paper. To Let α be the conjunction of literals in Lχ that hold at r, make this assumption explicit and to avoid any ambiguity, and similarly, let β be the conjunction of literals in Lχ that we make use of the following two conditions which together hold at r. Since α is consistent with T ∗ ϕ, from (K*7) and are shown to be equivalent to the strong version of axiom (K*8) it follows that T ∗ (ϕ ∧ α) = (T ∗ ϕ)+α. Therefore, (P): r ∈ [T ∗(ϕ∧α)]. Consider now any world r in [T ∗(ϕ∧α)]. Clearly, r  α, and given that r = r, it follows that r  T Cn(χ, ψ) L ∩L ∅ ϕ ∈L (R1) If = , χ ψ = , and χ, then ¬β. Since all worlds in [T ∗ (ϕ ∧ α)] satisfy ¬β it follows (T ∗ ϕ) ∩ Lχ = T ∩ Lχ. that ¬β ∈ T ∗(ϕ∧α). Then, by (R1) we derive that ¬β ∈ T , ¬β ∈ T ∗ ϕ (R2) If T = Cn(χ, ψ), Lχ ∩Lψ = ∅, and ϕ ∈Lχ, then which again by (R1) entails that . This however r ∩ L T ∗ ϕ (T ∗ ϕ) ∩Lχ = (Cn(χ) ∗ ϕ) ∩Lχ. contradicts the fact that χ is consistent with . Hence we have shown that, given (R1), the theory T ∗ ϕ is L L ϕ ∈L Condition (R1) is straightforward: when revising a theory split between χ and ψ for all χ. T by a sentence ϕ, the part of T that is not related to ϕ is Continuing with the proof of condition (P), we define ◦ to not affected by the revision. Condition (R2) is what imposes be the following operator of Lχ: for any theory T of Lχ and the strong version of axiom (P). To see this, consider a re- any sentence γ ∈Lχ, T ◦γ = (Cn(T ) ∗ γ) ∩Lχ.Itisnot vision function ∗ (which defines a revision policy for all the ◦ hard to verify that is indeed an AGM revision operator, i.e., theories of L), and let T = Cn(χ, ψ) and T = Cn(χ, ψ ) be it satisfies the postulates (K*1) - (K*8). Consider now any L ∩L L ∩L ∅ two theories such that χ ψ = χ ψ = . Consider sentence ϕ ∈Lχ. We conclude the proof of this theorem by now any sentence ϕ ∈Lχ. The relevant part to ϕ of T and showing that T ∗ ϕ = (CnLχ (χ)◦ϕ)+ψ. T is in both cases the same. Then, according to (R2), the way that this relevant part is modified in both T and T is Indeed, from (R2) it follows that (T ∗ϕ)∩Lχ = (Cn(χ)∗ also the same; namely, as dictated by the revision function ϕ)∩Lχ, and therefore by the construction of ◦, (T ∗ϕ)∩Lχ Cn (χ)◦ϕ L ∩L ∅ ∗ itself when applied to Cn(χ) (once again, notice that ∗ is = Lχ . Moreover, since χ ψ = , we derive (T ∗ ϕ) ∩L ((Cn (χ)◦ϕ)+ψ) ∩L defined for all theories, including T , T , and Cn(χ)). that χ = Lχ χ. On the other hand, from (R1) we have that (T ∗ ϕ) ∩ L χ = T ∩ Lχ The following result shows that (R1) and (R2) are indeed Cn(χ, ψ) ∩ L ((Cn (χ)◦ϕ)+ψ) ∩ L equivalent with the strong version of axiom (P). = χ = Lχ χ (the last equation follows from the fact that (CnLχ (χ)◦ϕ) is in Lχ which in turn is disjoint from Lψ). Putting together the Theorem 1 Let ∗ be a revision function satisfying the AGM above observations we notice that the two theories, T ∗ ϕ postulates (K*1) - (K*8). Then ∗ satisfies (P) iff ∗ satisfies and (CnL (χ)◦ϕ)+ψ are identical when projected on Lχ (R1) and (R2). χ as well as when projected on to its complement Lχ.Given that, as shown earlier, T ∗ ϕ is split between Lχ and Lψ,we Proof. 5 derive the desired identity; i.e. T ∗ ϕ = (CnLχ (χ)◦ϕ)+ψ ( ⇒ ) Assume that ∗ satisfies (P). Let T be a theory of L such The strong version of axiom (P) brings about a new fea- that T = Cn(χ, ψ), where χ, ψ ∈Land Lχ ∩Lψ = ∅. ture in the picture of classical AGM revision: it makes asso- Consider now any sentence ϕ ∈Lχ. By (P) it follows T ∗ ϕ (Cn (χ)◦ϕ)+ψ ◦ 5 that = Lχ , where is a revision op- Notice the use of (R2) in proving the strong version of axiom erator of Lχ. From the above, (R1) follows immediately. (P).

322 KR 2004 ciations between the revision policies of different theories. According to condition (PS), the less a world r differs None of the AGM postulates have this property – they all from the initial belief set T in propositional variables, the refer to a single theory T – making any combination of closer it is to the center of ST . Notice that condition (PS) revision policies on different theories permissible (as long places no constraints on the relative order of worlds that are of course as each policy individually satisfies the AGM ax- Diff-incomparable. In other words, for two worlds r and r ioms). This is no longer the case when (R2) (or the strong such that neither Diff(T,r) ⊂ Diff(T,r) nor Diff(T,r) ⊂ version of (P)) is brought into the picture. This condition in- Diff(T,r), their relative order in ST is not constrained by troduces dependencies between the revisions carried out on (PS). different (overlapping) theories. It turns out that, in the special case of consistent complete belief sets, condition (PS) is the counterpart of (R1) in the The Special Case of Complete Theories realm of systems of spheres. Before however presenting the formal result, let us consider intuitively why this might be Let us now turn to our main objective in this article, which is so. to formulate system-of-spheres semantics for axiom (P). The Let ST be a system of spheres centered on [T ] that sat- semantics will be developed progressively in two steps. In isfies (PS). Moreover let ϕ be any consistent sentence that the first step, undertaken in this section, we limit ourselves contradicts T (i.e., ¬ϕ ∈ T ). The set of ϕ-worlds occupy to the first side of axiom (P), i.e., condition (R1). Moreover a territory in ST that is disjoint from the center [T ]. At the we consider only complete theories as belief sets. Then, in outskirts of this ϕ-territory there are worlds that look very the second step (next section), we generalize our results to different from T . However, as we move closer to the center arbitrary theories and we also bring (R2) into the picture. of ST the ϕ-worlds that we meet agree with T in progres- sively more and more propositional variables. By the time The reason for this two-phase approach is mainly to in- ϕ crease readability and enhance the motivation of the con- we reach the boundary of the -territory with the center of ST , all the ϕ-worlds there agree with T in every proposi- cepts that will be introduced later. Conditions (R1) and (R2) L are quite independent of one another so it makes sense to tional variable outside ϕ. Hence, the intersection of these worlds (which by (S*) is the revision of T by ϕ) also agrees study them separately. Moreover, the characterization of T L (R1) in terms of systems of spheres is much more intuitive entirely with outside ϕ; thus (R1). when confined to complete theories; once this characteriza- The above intuitive explanation of the relationship be- tion is well understood for the special case, its generaliza- tween (PS) and (R1) is formally established with following tion, although not trivial, is easier to follow. result: Let T be a consistent complete theory, and let ST be a system of spheres centered on [T ]. The intended reading Theorem 2 Let ∗ be a revision function satisfying (K*1) - T L S of ST is that it represents comparative similarity between (K*8), a consistent complete theory of , and T the sys- possible worlds i.e., the further away a world is from the tem of spheres centered on [T ], corresponding to ∗ by means 6 ∗ T S center of ST , the less similar it is to [T ]. None of the condi- of (S*). Then satisfies (R1) at iff T satisfies (PS). tions (S1) - (S4) however indicate how similarity between worlds should be measured. In (Peppas, Foo, & Nayak Proof. 2000) a specific criterion of similarity is considered, orig- ⇒ inally introduced in the context of Reasoning about Action ( ) with Winslett’s Possible Models Approach (PMA) (Winslett Assume that ∗ satisfies (R1) at T . Moreover assume that, 1988). This criterion, called PMA’s criterion of similarity, contrary to the theorem, ST violates (PS). Let V be the measures “distance” between worlds based on propositional smallest sphere in ST that violates (PS). That is, V is r, r variables. In particular, let be any two possible worlds ST r the smallest sphere in that contains a world for of L.ByDiff(r, r ) we denote the set of propositional vari- which there exists another world r such that Diff(T,r) ⊂ ables that have different truth values in the two worlds i.e., (T,r) r Diff , and moreover the smallest sphere containing , Diff(r, r ) = {pi ∈ P : pi ∈ r and pi ∈ r }∪ {pj ∈ P : U V V ⊆ U call it , is not a subset of (and therefore ). Let pj ∈ r and pj ∈ r }. A system of spheres ST is a PMA sys- ϕ be the conjunction of all the literals7 in r than are not in tem of spheres iff it satisfies the following condition (Peppas, T . Notice that from Diff(T,r) ⊂ Diff(T,r), it follows that Foo, & Nayak 2000) (throughout this paper, the symbols r ϕ ∈ r. Moreover, it is easily verified that V is the small- r and always represent consistent complete theories): est sphere in ST that intersects [ϕ]. Indeed, assume on the contrary that a sphere smaller than V , call it V , contains a (PS) If Diff(T,r) ⊂ Diff(T,r) then there is a sphere world z satisfying ϕ, i.e., ϕ ∈ z. Then clearly Diff(T,r) V ∈ ST that contains r but not r . ⊆ Diff(T,z), and since r = z, it follows that Diff(T,r ) ⊂ Diff(T,z). Hence V violates (PS), which contradicts our 6Perhaps “comparative plausibility” would have been a bet- initial assumption that V is the smallest sphere in ST that ter term in the present context. However we shall tolerate this slight abuse of terminology mainly to comply with (Peppas, Foo, 7A literal is a propositional variable or the negation of a propo- & Nayak 2000). sition variable.

KR 2004 323 violates (PS). Therefore V is indeed the smallest sphere in ϕ2, ..., ϕn). Parikh has shown in (Parikh 1999) that for ev- ST intersecting [ϕ]. Consequently, r ∈ [T ∗ ϕ]. Consider ery theory T there is a unique finest T -splitting, i.e. one 8 now a literal l ∈ r , such that l ∈ T and l ∈Lϕ. Note which refines every other T -splitting. We shall denote the that since Diff(T,r) ⊂ Diff(T,r), such a literal l indeed finest T -splitting of T by F(T ). exists. Clearly then, ¬l ∈ T , and ¬l ∈ T ∗ ϕ. This however ¬l ∈L Using the notion of a finest T -splitting, we define the dif- contradicts (R1) since ϕ, and therefore it should have T L remained unaffected from the revision by ϕ. ference between a (possibly incomplete) theory of and a world r as follows: ( ⇐ ) T L S χ, ψ Definition 2 Let be a consistent theory of (possibly in- Assume that T satisfies (PS), and let be sentences in complete) and r a possible world. The difference Diff(T,r) L, such that T = Cn(χ, ψ) and Lχ ∩Lψ = ∅. Consider now between T and r is the following subset of P : any sentence ϕ ∈Lχ, and let r be any world in [T ∗ ϕ] i.e.,  ϕ ∈ r and r belongs to the smallest sphere CT (ϕ) in ST T,r {P ∈F(T ): ϕ ∈LP T  ϕ Diff( )= for some , and that intersects [ϕ]. Firstly we show that Diff(T,r ) ⊆Lχ. r ¬ϕ} Assume on the contrary that there is a literal l ∈ T ∩ Lχ, such that l ∈ r . Let r be the consistent complete theory It is not hard to verify that in the special case of a consis- that agrees with r in all literals except l. Clearly then, since T tent complete theory , the above definition of Diff collapses ϕ ∈ r and l ∈Lϕ, we derive that ϕ ∈ r . Moreover, by the to the one given in the previous section. construction of r, Diff(T,r) ⊂ Diff(T,r). Consequently, by (PS), there exists a sphere V that contains r and does not Notice that if T is incomplete, then for any world w com- T w ∈ [T ] (T,w) ∅ contain r . Therefore V ⊂ CT (ϕ). This leads to a contra- patible with (i.e. ), Diff = . Moreover, r (T,r) ⊆ (w, r) diction since V contains a ϕ-world (namely, r ), and at the for any world , Diff w∈[T ] Diff ; the precise same time it is smaller than the smallest sphere intersecting relationship between Diff(T,r) and Diff(w, r) for w ∈ [T ], [ϕ]. Hence Diff(T,r ) ⊆Lχ. This shows that all worlds r is given by the following result: in [T ∗ ϕ] agree with T on everything outside L χ. Therefore T ∩ Lχ = (T ∗ ϕ) ∩ Lχ as desired. Theorem 3 Let T be a consistent theory of L and r a pos- sible world. Then, Diff(T,r) = {P ∈F(T ):for all It is worth noting that in (Peppas, Foo, & Nayak 2000), w ∈ [T ], P ∩ Diff(w, r) = ∅}. there exists a characterization of condition (PS) in terms of epistemic entrenchments (Gardenfors & Makinson 1988). Proof. Consequently, since by Theorem 2 (R1) is equivalent to (PS), the results in (Peppas, Foo, & Nayak 2000) can be (LHS) ⊆ (RHS) used to provide a characterization of (R1) in terms of epis- temic entrenchments. Assume that p ∈ Diff(T,r). Then for some P ∈F(T ), P As already mentioned in the introduction, what is quite p ∈ P and for some ϕ ∈L , T  ϕ and r ¬ϕ. Let appealing about Theorem 2 is that it characterizes (R1), P l1,...ln be the literals in L that hold in r. Clearly then, not in terms of some technical non-intuitive condition, but l1 ∧···∧ln ¬ϕ, and therefore ϕ ¬l1 ∨···∨¬ln. From rather by a natural constraint on similarity between possi- this we derive that for all w ∈ [T ], since w  ϕ, there is at ble worlds, that in fact predates (R1) and was motivated in- least one 1 ≤ k ≤ n such that w ¬lk. Consequently, for dependently in a different context (Winslett 1988). More- all w ∈ [T ], P ∩ Diff(w, r) = ∅. over, as we will show in the next section, the essence of this characterization of (R1) in terms of constraints on similar- ⊇ ity, carries over into the general case of incomplete belief (LHS) (RHS) sets (albeit with some modifications). Assume that for some P ∈F(T ), P ∩ Diff(w, r) = ∅ P for all w ∈ [T ]. Let l1,...ln be the literals in L that The General Case hold in r. Then for all w ∈ [T ] there is a 1 ≤ j ≤ n such that ¬lj ∈ w. Therefore, T ¬l1 ∨···∨¬ln. Since To elevate Theorem 2 to the general case, we first need to (¬l ∨···∨¬l ) ∈LP P ⊆ (T,r) extend the definition of Diff to cover comparisons between 1 n , it follows that Diff as a world r and an arbitrary, possibly incomplete, theory T . desired. The generalization of Diff that we shall use herein takes into account the notion of a T -splitting introduced by Parikh in Condition (R1) and Systems of Spheres his language-splitting model (Parikh 1999). Having generalized Diff, let us re-examine condition (PS) Let T be a theory of L and P1,P2,...,Pn a partition of for a system of spheres ST related to a belief set T that is not the set P of all propositional variables in L. We say that {P ,P ,...,P } T 1 2 n is a -splitting iff there exist sentences 8A partition Z refines another partition Z , if for every element P1 P2 Pn ϕ1 ∈L , ϕ2 ∈L , ..., ϕn ∈L , such that T = Cn(ϕ1, of Z there is a superset of it in Z .

324 KR 2004 necessarily complete. It turns out that in this case (PS) no Multiple T -duals (external and internal ones as we will longer corresponds to (R1); in particular, (PS) is too strong. see later) add more structure to a system of spheres, and ren- der condition (PS) too strong for the general case. The possi- To see this, consider the following counter-example. As- T L a b bility of placing external -duals in different spheres, opens sume that is built from the propositional variables , , up new ways of ordering worlds that still induce relevance- c, d, and let T be the theory T = Cn(a ↔ b, c ↔ d). {{a, b} {c, d}} T S sensitive revision functions without however submitting en- Clearly, , is a -split. Next, let T be the tirely to the demands of (PS). following system of spheres (represented as a total preorder on worlds):9 Let us elaborate on this point. Consider a system of spheres ST centered on the theory T , and let r, r be any two abcd abcd (T,r) ⊂ (T,r) abcd abcd worlds such that Diff Diff . Theorem 2 tells us abcd abcd abcd ≤ ≤ ≤ abcd ≤ abcd that in the special case of complete theories, to ensure local abcd abcd abcd change (alias, condition (R1)) the world r should be placed abcd abcd abcd abcd (strictly) closer to the center [T ] of ST than r . In the gen- eral case however, and with the aid of external T -duals, one It is not hard to verify that the revision function ∗ induced can perhaps afford to be a bit more liberal about the location r by the above system of spheres ST , satisfies (R1). At the of ; perhaps all that is needed is that at least one external T r r r [T ] same time however, ST violates (PS). In particular consider -dual of (and not necessarily itself) be closer to r the worlds r = {a, b, ¬c, d} and r = {¬a, b, c, ¬d}}. Al- than . It turns out that, in fact, this is pretty much the case, r r r though Diff(T,r) = {c, d}⊂Diff(T,r ) = {a, b, c, d}, the expect that the world “covering” for (in relation to ) T r two worlds r and r are equidistant from the center of ST . is not just any external -dual of but a very specific one: it is the external T -dual of r that agrees with r on all liter- Despite its failure to generalize, (PS) should not be disre- als in Diff(T,r). We shall call this external T -dual of r, the garded altogether. It can still serve as a guide in formulating r -cover for r at T , and we shall denote it by ϑT (r, r ). the appropriate counterpart(s) of (R1) for the general case; as we prove later in this section, the two general conditions Definition 4 Let T be a theory of L, let r, r be two possible (Q1) and (Q2) that correspond to (R1) are both in the spirit worlds such that Diff(T,r) ⊂ Diff(T,r), and let r be an of (PS) (and not surprisingly, they collapse to (PS) in the external T -dual of r. The world r is the r-cover for r at special case of complete belief sets). T iff r ∩ Diff(T,r) = r ∩ Diff(T,r). We shall denote the r -cover for r at T by ϑT (r, r ). To formulate the conditions (Q1) and (Q2), we first need to introduce some further concepts related to the notion of distance between a world and an incomplete theory. A simple example will help to clarify the above definition. Suppose that the language L is built from the propositional Consider a theory T and let r be a world not compatible variables a, b, c, d, e, and let T be the theory T = Cn(a ↔ b, with T i.e., r ∈ [T ]. Clearly Diff(T,r) = ∅. Is there another b ↔ c, d ↔ e). Let r be the world r = Cn(a, ¬b, c, d, e) world r that differs from T on exactly the same proposi- and r the world r = Cn(¬a, b, ¬c, d, ¬e). The finest T - tional variables, i.e., Diff(T,r)=Diff(T,r)? If T is com- splitting is {{a, b, c}, {d, e}}. Then according Definition 2, plete, the answer is obviously “no”: for any set of proposi- Diff(T,r) = {a, b, c} and Diff(T,r) = {a, b, c, d, e}. Hence tional variables P , there can only be one world r such that Diff(T,r) ⊂ Diff(T,r). The world r has many external Diff(T,r) = P . If however T is incomplete (i.e., [T ] con- T -duals like Cn(¬a, ¬b, c, d, e), Cn(a, b, ¬c, d, e), etc. Yet tains more than one world), this is no longer the case. For out of all these external T -duals, only one is a r -cover for r example, suppose that T = Cn(a ↔ b, c ↔ d) – where a, b, at T , namely the world ϑT (r, r ) = Cn(¬a, b, ¬c, d, e). c, d, are the propositional variables of the language – and let r, r be the possible worlds r = Cn({¬a, b, c, d}), and r = As mentioned earlier, the notion of “covering” will be Cn({a, ¬b, c, d}). It is not hard to see that, although r and used to weaken condition (PS). In particular, consider the r are different, Diff(T,r) = Diff(T,r) = {a, b}. The two condition (Q1) below: worlds r and r, have also another thing in common: they (T,r) ⊂ (T,r) agree on the propositional variables outside Diff(T,r).We (Q1) If Diff Diff then there is a sphere V ∈ S ϑ (r, r) r call such worlds external T -duals (for the definition below, T that contains T but not . recall that P is the set of all propositional variables in the language L): Condition (Q1) formalizes the intuition mentioned earlier about weakening (PS) with the aid of external T -duals. It Definition 3 Let r, r be possible worlds, and let T be a is not hard to show that (PS) entails (Q1), and that (Q1) theory of L. The worlds r and r are external T -duals collapses to (PS) when the initial belief set T is complete. iff Diff(T,r) = Diff(T,r) and r ∩ (P − Diff(T,r)) = Moreover, (Q1) is strictly weaker than (PS). To see this, con- r ∩ (P − Diff(T,r)). sider the first example in this section; the system of spheres ST satisfies (Q1) but violates (PS). 9Notice that in order to increase readability, in this example we are representing worlds as sequences of literals rather than theories; Yet despite all its nice properties, condition (Q1) in itself moreover, the negation of a propositional variable p is denoted p. does not suffice to guarantee local change; it seems that from

KR 2004 325 something too strong for (R1) (condition (PS)), we have now quently, in that case, (Q1) reduces to (PS), while (Q2) de- moved to something too weak. Consider in particular the generates to a vacuous condition. following counter-example: the language L is build over three propositional variables a, b, c, the initial belief set T The promised correspondence between (R1) and the two conditions (Q1) - (Q2) is given by the theorem below: is T = Cn(a ↔ b), and the system of sphere ST centered on [T ] is the one represented below: Theorem 4 Let ∗ be a revision function satisfying (K*1) abc - (K*8), T a consistent theory of L, and ST a system of abc abc spheres centered on [T ], that corresponds to ∗ by means of ≤ abc ≤ abc ≤ abc abc (S*). Then ∗ satisfies (R1) at T iff ST satisfies (Q1) - (Q2). abc Proof. In this example all the worlds outside [T ] (i.e. in ML − ( ⇒ ) [T ]) differ from [T ] on precisely the same propositional vari- ables, namely on {a, b}. Consequently ST trivially satis- Assume that ∗ satisfies (R1) at T . Starting with (Q1), fies (Q1) since its antecedent Diff(T,r) ⊂ Diff(T,r ) never let r, r be two consistent complete theories such that holds for r, r ∈ [T ]. Yet despite the compliance with (Q1), Diff(T,r) ⊂ Diff(T,r).IfDiff(T,r) = ∅, then r is con- the revision function * induced from S T violates (R1) at T sistent with T , and therefore r ∈ [T ]. Then, given that (simply consider the revision of T by a ∧¬b).10 r ∈ [T ], (Q1) follows trivially from (S2). Assume there- fore that Diff(T,r) = ∅. Let L be the sublanguage of L To secure the correspondence with (R1), condition (Q1) defined over Diff(T,r), and let ϕ be the conjunction of all needs to be complimented with a second condition, called literals in L that hold at r i.e., ϕ = l1 ∧ ··· ∧ lk, where (Q2). This second condition uses the notion of an internal for each 1 ≤ i ≤ k, li is a literal in L , and li ∈ r . T -dual defined below: By (S*), [T ∗ ϕ] = fT (ϕ) = [ϕ] ∩ CT (ϕ), where CT (ϕ) is the smallest sphere in ST intersecting [ϕ]. Clearly, since Definition 5 Let r, r be possible worlds, and let T be a L r r T ϕ is consistent, fT (ϕ) = ∅. Consider now any world r theory of . The worlds and are internal -duals iff f (ϕ) ϕ Diff(T,r) = Diff(T,r), and r ∩Diff(T,r) = r ∩Diff(T,r). in T . From (R1) and the construction of , it follows that Diff(T,r) ⊆ Diff(T,r). Moreover, again by the con- T struction of ϕ, Diff(T,r) ⊆ Diff(T,r ). Hence Diff(T,r) = To give a concrete example of internal -duals and high- (T,r) r ϕ r ∩ (T,r) light their difference from external ones, suppose that L is Diff . Moreover, since satisfies , Diff = r ∩ Diff(T,r). Finally notice that, because of (R1), built over the propositional letters a, b, c, d, and let the ini- T T Cn(a ↔ b, c ↔ d) r r ∈ CT (ϕ). We have therefore shown that all worlds in tial theory be = . The worlds f (ϕ) S r = Cn(¬a, b, c, d) and r = Cn(¬a, b, ¬c, ¬d) differ from T T are closer to the center of T than , that they all (T,r) differ from T on exactly the same propositional variables as on exactly the same propositional variables, i.e., Diff r (T,r) r = Diff(T,r) = {a, b}.Yetr and r are not external T -duals , and that within Diff they agree with . What is still (T,r) left to show in order to prove (Q1) is that there is at least since outside Diff the two worlds are not identical. On f (ϕ) r the other hand, r and r are identical inside Diff(T,r). This one world in T that agrees with on the propositional T variables outside Diff(T,r). Let ψ be the conjunction of lit- makes them internal -duals. erals in L that hold at r. Since r differs from T only in Clearly, for any theory T and any two worlds r, r ,ifr L ¬ψ ∈ T the propositional variables in , it follows that . and r are both internal and external T -duals, then they are Consequently, by (R1), ¬ψ ∈ T ∗ ϕ. This again entails identical. that there is at least one world fT (ϕ), that satisfies ψ and therefore agrees with r on all propositional variables outside We can now proceed with the presentation of condition (T,r) (Q2), which together with (Q1), brings about the correspon- Diff as desired. This concludes the proof of (Q1). T dence with (R1). As usual, in the following condition is an For (Q2), assume on the contrary that it is not true at S T , arbitrary consistent theory of L (possibly incomplete), S T is V S and let be the smallest sphere in T that violates (Q2). a system of spheres centered on [T ], and r, r are possible Then, there exist two worlds r, r that are internal T -duals worlds. such that r ∈ V and r ∈ V . From r ∈ V we firstly derive that Diff(T,r) = ∅, and therefore Diff(T,r) = ∅, which r r T (Q2) If and are internal -duals, then they belong to again entails that [T ] ⊂ V . Next, let L be the sublanguage S V ∈ the same spheres in T ; i.e., for any sphere of L defined over Diff(T,r). Define ϕ to be the conjunction S r ∈ V r ∈ V T , iff . of all literals in L that hold at r (and therefore also hold at r ). Clearly CT (ϕ) ⊆ V . Moreover, from (R1) and the con- Notice that in the special case that T is complete, no world ϕ r f (ϕ) r T struction of it follows that any world in T differs has internal or external -duals (other than itself). Conse- from T on exactly the same propositional variables as r i.e., 10 Diff(T,r ) = Diff(T,r). In addition, since r satisfies ϕ, r It is worth noting that in this example, there is only one system of spheres ST whose revision function * satisfies (R1) at T . This and r are in fact internal T -duals, which also makes r and is the system containing only the spheres [T ] and ML. r internal T -duals. Consider now the sentence ψ defined

326 KR 2004 as the conjunction of literals in L that hold at r. Since counterpart of (R2). As usual, P is a subset of P , T and r ∈ CT (ϕ) (recall that CT (ϕ) ⊆ V ), and moreover all T are theories of L, and ST , ST are systems of spheres worlds in fT (ϕ) satisfy ϕ, it follows that no world in fT (ϕ) centered on [T ] and [T ] respectively: satisfies ψ, or equivalently, all worlds in fT (ϕ) satisfy ¬ψ. Hence, ¬ψ ∈ T ∗ ϕ. From (R1) this entails that ¬ψ ∈ T , (Q3) If {P , (P − P )} is both a T - splitting and a T - P P which again entails that Diff(T,r ) ∩ L = ∅, leading us to a splitting, and moreover, T ∩L = T ∩L , then P P contradiction. Hence (Q2) is true. ST /L = ST /L . ( ⇐ ) The following result shows that (Q3) is the system-of- Assume that ST satisfies (Q1) and (Q2). Let T be such spheres counterpart of (R2): that T = Cn(χ, ψ), for some sentences χ, ψ ∈Lsuch that Lχ ∩Lψ = ∅. Moreover, let ϕ be any sentence in Lχ.If Theorem 5 Let * be a revision function satisfying (K*1) - [T ] ∩ [ϕ] = ∅ , then (R1) trivially holds. Assume therefore (K*8), and {ST }T ∈KL a family of systems of spheres (one that [T ]∩[ϕ] = ∅. Firstly we show that T ∩Lχ ⊆ (T ∗ϕ)∩Lχ. for each theory T in KL), corresponding to ∗ by means of ∗ {S } Let γ be any sentence in T ∩ Lχ. Assume, contrary to the (S*). Then satisfies (R2) iff T T ∈KL satisfies (Q3). theorem, that for some r in fT (ϕ), ¬γ ∈ r. Let w be any world in [T ]. Construct r as follows: r agrees with r in Lχ Proof. and it agrees with w outside Lχ i.e., r ∩Lχ = r ∩Lχ, and ( ⇒ ) r ∩Lχ = w ∩Lχ. Clearly, Diff(T,r ) ⊂ Diff(T,r). Then by (Q1), there is a sphere V smaller than CT (ϕ) that contains Assume that ∗ satisfies (R2), and let T , T be two theories ϑT (r ,r) (i.e. the r-cover for r at T ). It is not hard to verify of L. Moreover, assume that for some P ⊆ P , the sets P , that ϕ ∈ ϑT (r ,r), which however contradicts the fact that (P − P ) are a splitting of P relative to both T and T , and P P ϑT (r ,r) ∈ V ⊂ CT (ϕ). Hence T ∩ Lχ ⊆ (T ∗ ϕ) ∩ Lχ as T ∩L = T ∩L .IfP = ∅ or P = P , then (Q3) is trivially desired. true. Assume therefore that ∅ = P ⊂ P . Then, for some χ ψ ψ T Cn(χ, ψ) T Cn(χ, ψ) L γ L sentences , , and , = , = , χ = For the converse, let be any sentence in χ such that P (P −P ) γ ∈ T . Then there is a world w ∈ [T ] such that ¬γ ∈ L , and ψ, ψ ∈L = Lχ. Consequently, by (R2), for ϕ ∈L (T ∗ϕ)∩L (T ∗ϕ)∩L (Cn(χ)∗ϕ)∩L w. Let r be any world in fT (ϕ). Construct the world r as any χ, χ = χ = χ. r r Lχ w follows: agrees with in and it agrees with outside S S Next consider the systems of spheres T and T cen- Lχ (i.e., r ∩Lχ = r ∩Lχ, and r ∩ Lχ = w ∩ Lχ). Since, tered on [T ] and [T ] respectively, and assume that, contrary T ∩ L χ ⊆ P P as we have shown in the first part of the proof, to Theorem 5, ST /L = ST /L . Without loss of gener- (T ∗ ϕ) ∩ Lχ, it follows that Diff(T,r) ⊆Lχ. Then, by P ality, we can assume that ST /L contains an element that the construction of r , it follows that r and r are internal P T r ∈ C (ϕ) r is not in ST /L . Let V be the smallest sphere in ST such -duals. Consequently, by (Q2), T , and since P P satisfies ϕ, it follows that r ∈ fT (ϕ). Finally notice that, by that V/L ∈ ST /L . Moreover, let V be the small- P P construction, ¬γ ∈ r . Consequently γ ∈ T ∗ ϕ as desired. est sphere in ST such that V/L ⊂ V /L . Clearly, Combining the above we derive that T ∩ Lχ = (T ∗ ϕ) ∩ Lχ. [T ] ⊂ V ⊂ML and [T ] ⊂ V ⊂ML. Consider now r ∈ V r ∩LP ∈ V/LP a world such that  . Next, con- P sider a world r ∈ V such that r ∩L ∈ ( {U ∈ ST : P Condition (R2) and Systems of Spheres U ⊂ V })/L . It is not hard to verify that such a world P  r indeed exists, and moreover, r ∩L ∈ ( {U ∈ ST : We now turn to the second side of axiom (P), encoded by U ⊂ V })/LP l , ···l LP condition (R2). As noted previously, a ramification of (R2) . Let 1 m be the literals in that P is that it introduces dependencies between revision policies hold in r, and let l1, ···lm be the literals in L that hold associated with different theories. Not surprisingly, the con- in r . Finally, let ϕ be the sentence ϕ = (l1 ∧···∧lm) ∨ dition corresponding to (R2) in the realm of systems of (l1 ∧···∧lm). Then the smallest sphere intersecting [ϕ] in spheres, is one that makes associations between systems of ST and in ST ,isV and V respectively. From this we de- P P P P spheres with different centers. rive that r /L ∈ fT (ϕ)/L and r /L ∈ fT (ϕ)/L . Consequently, (T ∗ϕ)∩Lχ =(T ∗ϕ)∩Lχ, which however Definition 6 Let V be a set of worlds in ML, and let L be contradicts (R2). a sublanguage of L.ByV/L we denote the restriction of ( ⇐ ) V to L ; that is, V/L = {r ∩L: r ∈ V }. Moreover, for a system of spheres ST ,byST /L we denote the restriction of {S } T Assume that T T ∈KL satisfies (Q3) and let be a the- ST to L ; that is, ST /L = {V/L : V ∈ ST }. ory of L such that for some χ, ψ ∈L, T = Cn(χ, ψ) and Lχ ∩Lψ = ∅. Let T be the theory T = Cn(χ). Notice that for any sublanguage L of L, ST /L is also a Clearly, T ∩Lχ = T ∩Lχ, and therefore by (Q3), ST /Lχ system of spheres. The condition (Q3) below is the semantic = ST /Lχ. Next, consider any sentence ϕ ∈Lχ, and let

KR 2004 327 CT (ϕ) and CT (ϕ) be the smallest spheres in ST and ST References respectively intersecting [ϕ]. We show that CT (ϕ)/Lχ = AIJ. 1997. Special issue on relevance. Artificial Intelli- CT (ϕ)/Lχ. Assume on the contrary that CT (ϕ)/Lχ = gence 97. CT (ϕ)/Lχ. Since ST /Lχ = ST /Lχ, without loss of gen- erality we can then assume that CT (ϕ)/Lχ ⊂ CT (ϕ)/Lχ. Alchourron, C.; Gardernfors, P.; and Makinson, D. 1985. Moreover, again from ST /Lχ = ST /Lχ, we have that On the logic of theory change: Partial meet functions CT (ϕ)/Lχ ∈ ST /Lχ. Let V be the sphere in ST whose for contraction and revision. Journal of Symbolic Logic restriction to Lχ is equal to CT (ϕ) i.e., V/Lχ = CT (ϕ)/Lχ. 50:510–530. Clearly, V ⊂ CT (ϕ). On the other hand however, since Borgida, A. 1985. Language features and flexible handling ϕ ∈Lχ and CT (ϕ) contains as least one ϕ-world, it follows of exceptions in information systems. ACM Transactions that V also contains a ϕ-world. This of course contradicts on Database Systems 10:563–603. C (ϕ) S the fact that T is the smallest sphere in T intersect- Chopra, S., and Parikh, R. 1999. An inconsistency toler- ing [ϕ], and proves that CT (ϕ)/Lχ = CT (ϕ)/Lχ. Conse- (T ∗ ϕ) ∩L (T ∗ ϕ) ∩L ant model for belief representation and belief revision. In quently, χ = χ as desired. Proceedings of the 16th International Joint Conference on Putting together the results reported in Theorems 1, 4, and Artificial Intelligence, 192–197. Morgan-Kaufmann. 5, we obtain immediately the following theorem that pro- Chopra, S., and Parikh, R. 2000. Relevance sensitive be- vides possible-world semantics for (the strong version of) lief structures. Annals of Mathematics and Artificial Intel- axiom (P): ligence 28(1-4):259–285. Chopra, S.; Georgatos, K.; and Parikh, R. 2001. Rel- Theorem 6 Let ∗ be a revision function satisfying (K*1) - evance sensitive non-monotonic inference for belief se- {S } (K*8) and T T ∈KL a family of systems of spheres (one quences. Journal of Applied Non-Classical Logics. for each theory T in KL), corresponding to ∗ by means of ∗ {S } Dalal, M. 1988. Investigations into a theory of knowledge (S*). Then satisfies (P) iff T T ∈KL satisfies (Q1) - (Q3). base revisions: Preliminary report. In Proceedings of the 7th AAAI Conference, 475–479. Conclusion Gardenfors, P., and Makinson, D. 1988. Revisions of knowledge systems using epistemic entrenchment. In The main contribution of this paper is Theorem 6 that pro- Proceedings of Theoretical Aspects of Reasoning about vides system-of-spheres semantics for Parikh’s axiom (P). Knowledge, 83–95. Morgan-Kaufmann. What is quite appealing about this result is that the semantic Grove, A. 1988. Two modellings for theory change. Jour- conditions (Q1) - (Q3) that characterize axiom (P) are quite nal of Philosophical Logic 17:157–170. natural constraints on similarity between possible worlds. In Lehmann, D.; Magidor, M.; and Schlechta, K. 2001. Dis- fact, conditions (Q1) - (Q2) essentially generalize a measure tance semantics for belief revision. Journal of Symbolic of similarity that predates axiom (P), and was motivated in- Logic 66(1):295–317. dependently in the context of Reasoning about Action by Winslett. This intuitive nature of the semantics is more ev- Parikh, R. 1999. Beliefs, belief revision, and splitting lan- ident in the special case of consistent complete belief sets. guages. In Lawrence Moss, J. G., and de Rijke, M., eds., An interesting by-product of our study is the identification Logic, Language, and Computation Volume 2 - CSLI Lec- of the two possible readings of axiom (P), both of which are ture Notes No. 96. CSLI Publications. 266–268. plausible depending on the context. Peppas, P., and Wobcke, W. 1992. On the use of epistemic entrenchment in reasoning about action. In Proceedings It should be noted that apart from Winslett, other authors of the 10th European Conference on Artificial Intelligence, have also made specific proposals for measuring distance 403–407. John Wiles and Sons. between possible worlds (see for example, (Borgida 1985), (Dalal 1988), and (Satoh 1988)). 11 It would be a worthwhile Peppas, P.; Foo, N.; and Nayak, A. 2000. Measuring simi- exercise to investigate whether any of these measures of dis- larity in belief revision. Journal of Logic and Computation tance also yield some kind of “local change effect” for their 10(4). associated revision functions. Peppas, P. 1994. Belief Revision and Reasoning about Ac- tion. Ph.D. Dissertation, Basser Department of Computer Science, University of Sydney. Acknowledgements Satoh, K. 1988. Nonmonotonic reasoning by minimal be- lief revision. In Proceedings of the International Confer- We would like to thank the anonymous referees for many ence on Fifth Generation Computer Systems, 455–462. useful comments on this article. The first author is also grateful to the School of Computer Science and Engineer- Winslett, M. 1988. Reasoning about action using a pos- ing at the University of New South Wales for funding his sible models approach. In Proceedings of the 7th AAAI visits to Sydney and thus enabling this joint work. Conference, 89–93.

11For a more general study on the notion of distance in belief revision, refer to (Lehmann, Magidor, & Schlechta 2001).

328 KR 2004