Prime Forms and Minimal Change in Propositional Belief Bases

Transcription

1 Annals of Mathematics and Artificial Intelligence manuscript No. (will be inserted by the editor) Prime Forms and Minimal Change in Propositional Belief Bases J. Marchi G. Bittencourt L. Perrussel Received: date / Accepted: date Abstract This paper proposes to use prime implicants and prime implicates normal forms to represent belief sets. This representation is used, on the one hand, to define syntactical versions of belief change operators that also satisfy the rationality postulates but present better complexity properties than those proposed in the literature and, on the other hand, to propose a new minimal distance that adopts as a minimal belief unit a fact, defined as a prime implicate of the belief set, instead of the usually adopted Hamming distance, i.e., the number of propositional symbols on which the models differ. Some experiments are also presented that show that this new minimal distance allows to define belief change operators that usually preserve more information of the original belief set. Keywords belief change minimal change prime implicates prime implicants knowledge compilation Mathematics Subject Classification (2000) Primary 03B42 Secondary 68T30 J. Marchi Departamento de Ciência da Computação Universidade Federal de Lavras Lavras MG - Brazil Tel.: jerusa@das.ufsc.br G. Bittencourt In Memoriam. L. Perrussel IRIT - Institut de Recherche en Informatique de Toulouse Université de Toulouse Université Toulouse 1 Capitole 2 rue du Doyen Gabriel Marty F Toulouse Cedex 9 - France Tel.: laurent.perrussel@univ-tlse1.fr

2 2 J. Marchi et al. 1 Introduction The main goal of the Belief Change endeavor is to propose methods to incorporate new information in a belief set [20,23] in such a way that the resulting set of beliefs is consistent and that the initial belief set was changed in a minimal way [1,14,18, 19, 27]. Formally, the notion of minimal change is usually represented by a closeness criterion based on a distance between the models of the belief set and the models of the new information [9,22,27]. This criterion is widely used in the Belief Revision [1,11,15] and Belief Update areas [17,27,52]. Belief revision and belief update are similar activities: changing an initial belief set in order to incorporate a new piece of information that may contradict these beliefs. In the context of belief revision, beliefs are supposed to represent a state of a world that is assumed to be more or less stable and the new piece of information adds new beliefs or corrects old beliefs. On the other hand, belief update supposes that this state is evolving and the new piece of information describes how the world has changed. In the belief revision area, Dalal has proposed a method based on a notion of distance which considers as the minimal belief unit a propositional symbol [11]. Dalal claims that this minimal unit represents the minimal unit of change. Dalal has shown that his revision method satisfies the AGM rationality postulates (Alchourròn, Gärdenfors, Makinson) [1, 20]. These postulates characterize how a rational agent should revise its belief set. In the belief update area, Forbus [17] and Winslett [52] have proposed similar operators. Their update operators satisfy the KM postulates (Katsuno, Mendelzon) [27]. These postulates are similar to AGM postulates and describe how an agent should update its belief set in a rational way. All these three operators focus on a semantic definition of belief sets. Several other methods for revising and updating belief sets have been proposed, e.g., syntax-based methods [39,40] or probability-based methods [37]. This work presents two main contributions. The first one is to produce syntactic belief change operators of the Dalal, Winslett and Forbus model-based change operators, using the same distance definitions. We focus on these operators for two reasons: they are among the most popular operators and they are among the few belief update operators that satisfy the KM postulates. To achieve our intent, we propose to represent belief sets and incoming information as sets of Prime Implicants [26, 30]. We demonstrate that when belief sets are represented in this specific syntax we obtain, in the case of belief revision, operators that are semantically equivalent to the classical ones. In the case of update operators, we show that it is not possible to reformulate these methods in terms of prime implicants. However, we show that if we consider that models are defined by implicants instead of by models, we can obtain a new definition of update. We revisit the KM postulates in order to characterize update operations from an axiomatic point of view based on this new definition and then we show that the new update operators satisfy the reviewed KM postulates. This formulation presents the following advantages: although we consider belief change in a syntactic context, by using prime implicants, we avoid the syntax dependency problem, which is a key issue in the syntax-based methods [39, 40]; and, as long as belief sets and incoming information are represented using this specific syntax, the change task complexity is improved, because in general, there are less prime implicants than models. The second contribution of this work is to propose a new definition of minimal change by introducing a new distance between belief sets. Dalal s distance considers

3 Prime Forms and Minimal Change 3 a propositional symbol as the primitive belief unit and assumes that flipping the truth value of any one of them is equivalent to any other. This definition does not take into account the specific structure of the belief set, in which this change can have different impact according to the relative importance of the propositional symbol with respect to this structure. We propose to use the relation between the prime normal forms [26,30] to take into account the relative impact of each literal in the context of a given belief set. We consider that the Prime Implicates represent minimal facts that are entailed by a belief set and the Prime Implicants represent the possible situations in which the belief set is true. The new notion considers as a minimal belief unit a fact, instead of the truth value of an isolated propositional symbol. We show that this new distance leads to more informed change operators than those based on the Dalal distance. The structure of this paper is as follows. Section 2 introduces the logical framework. Section 3 briefly presents the belief revision and update problems and the well known AGM and KM postulates and also discusses the importance of preferences and distances in the belief change area. The section concludes with an overview of Dalal and Winslett distances. Section 4 introduces a syntactical way to express belief changes, setting preferences over terms in the prime implicants context. We demonstrate that the syntactical operation is equivalent to the semantical one for the revision method. Section 5 presents a new distance that is based on an explicit representation of the relation between literals and clauses. This new distance exploits the specific syntax of prime normal forms and allows the introduction of a new minimal change notion that combines Dalal minimal belief unit and the frequency of occurrence of literals in the clauses. Section 6 presents experimental results of these new approaches. In section 7 some related works are discussed. Finally, section 8 presents some concluding remarks. 2 Preliminaries In the following, we consider a finite set of propositional symbols P = {p 1,..., p n}; let LIT = {L 1,..., L 2n } be the set of the associated literals where L i = p j or L i = p j. Beliefs, i.e., formulas are expressed with respect to propositional language L(P). A clause C is a disjunction [16] of literals: C = L 1 L kc and a dual clause, or term, is a conjunction of literals: D = L 1 L kd. We note L the opposite value of a literal L: if L = p then L = p, otherwise if L = p then L = p. Belief bases and new pieces of information are defined as sets of beliefs; hereafter usually denoting as ψ and µ. A belief base is said to be complete if it contains only one term D and iff for all p P either literal p or p belongs to D. We note D D the subtraction operation over terms, that results on literals of D that have no correspondence with literals of D, i.e, if D = p 1 p 2 and D = p 1 p 2, D D = { p 2 }. Let D be a term where all literals have been flipped, e.g. if D = p 1 p 2 then D = p 1 p 2. Let D be the set of all possible terms defined over P. In the sequel, conjunctions and disjunctions of literals, clauses and terms are seen as sets. An interpretation is a function from P to = {true, false}. Let W be the set of all possible interpretations also called possible worlds. A world w is a model of a formula ϕ (w = ϕ) iff ϕ is true in the interpretation w, according to the usual meaning of =. For any formula ϕ, [[ϕ]] denotes the set of the models of ϕ.

4 4 J. Marchi et al. Let be a total pre-order over a set W and < the associated strict pre-order: w < w iff w w and w w (w, w W). Let min(w, ) be the set of minimal elements of W such that min(w, ) = {w W w W s.t. w < w}. The relation N represents the ordinal pre-order over the set of integers N. 2.1 Prime Implicants and Prime Implicates Given a propositional language L(P) and an ordinary formula ψ L(P), there are algorithms to convert ψ into a conjunctive normal form (CNF) and into a disjunctive normal form (DNF) (e.g., [50]). The CNF is defined as a conjunction of clauses, CNF ψ = C 1 C m, and the DNF as a disjunction of terms, DNF ψ = D 1 D w, such that ψ CNF ψ DNF ψ. Let us formally define the notions of implicate and implicant: Definition 1 (Implicate and Implicant) Let C be a clause, D be a term and ψ L(P) a formula. C is an implicate of ψ iff ψ = C and D is an implicant of ψ iff D = ψ. Let D ψ be the set of all possible implicants of ψ. Definition 2 (Prime Implicate and Prime Implicant) Let clause C be an implicate and term D be an implicant of a formula ψ. C is a prime implicate iff for all implicates C of ψ s.t. C = C we have C = C and D is a prime implicant iff for all implicants D of ψ s.t. D = D, we have D = D. These semantic definitions can be expressed syntactically [45] by: C is a prime implicate iff it is an implicate and for all literals L C, ψ = (C {L}), and D is a prime implicant iff it is an implicant and for all literals L D, (D {L}) = ψ. Alternatively, prime implicates and prime implicants can be defined [30, 45] as special cases of CNF and DNF forms. These forms consist of the smallest sets of clauses/terms closed for inference, without any subsumed clauses/terms. Although the definition of prime implicate/implicant includes tautological or contradictory terms, because a tautology is an implicate of any formula and a contradiction is an implicant of any formula, in the sequel we only consider clauses/terms that do not have any pair of tautological or contradictory literals. We define PI ψ as a conjunction of all non tautological prime implicates of ψ and IP ψ as a disjunction of all non contradictory prime implicants of ψ such that ψ PI ψ IP ψ. To simplify the notation, in the sequel we omit non tautological/non contradictory when we mention prime implicates/implicants. In propositional logic, conjunctive and disjunctive normal forms, as well as prime implicant and prime implicate are dual notions Prime Form Properties Prime forms offer key properties which give fruitful strategy for knowledge representation and knowledge compilation activities. For knowledge representation strategy, the most important property presented by prime forms is uniqueness (up to the order of the clauses and terms and of the literals that occur within them). The representations in prime implicants and prime implicates are unique in the sense that, given a set X of propositional symbols, every proposition builds up with symbols of X has exactly one representation

5 Prime Forms and Minimal Change 5 in prime implicants and prime implicates. The structure of any pair in prime implicants and prime implicates only depends of the relations between the propositional symbols and not on their identity with respect to the set X. Therefore, each pair represents a whole family of congruent propositions, i.e., propositions that are equivalent with respect to the group of permutations and complementarity. Besides uniqueness up to congruence, prime forms representation presents the following interesting characteristics: It is minimal, in the sense that it is closed under inference and subsumption. Although it is not minimal with respect to syntactical size, it is, in general, smaller than the complete disjunctive normal form (models). It can be calculate deterministically, without the use of any heuristic. Given the two prime forms representations of a given formula ψ, the prime representations of its negation can be obtained directly(where dual is a function switching conjunction symbols with disjunction symbols and vice-versa): PI ψ dual(ip ψ ), IP ψ dual(pi ψ ) That is the prime forms of a concept includes the prime form of its negation. Prime implicants and prime implicates of a proposition present a holographic relation (presented in section 5.1), where each literal in a clause is associated with the set of dual clauses to which it belongs and conversely. If we consider PI ψ as a set of valid rules or facts and IP ψ as a set of possible situations, then this relation connects situations to relevant facts, and facts to relevant situations. The holographic relation between prime implicants and prime implicates can also be used to explicit the propositional symbols used in a formula as well as finding equivalent propositional symbols, i.e., symbols that can be exchanged without modifying the syntactical expression of the prime forms which represents a concept. These characteristics can be used in learning and reasoning activities in order to improve the cognitive capabilities of an intelligent agent [5,6,4], as well as to develop better belief change [8,35,36] and belief merging methods [44]. Knowledge compilation consists of transforming a propositional theory into a target language; this compilation stage is usually performed off-line. The resulting theory is then used on-line for efficiently answering to a large number of queries [10, 34] Prime forms are also useful for compiling propositional theories; even if in the worst case the number of prime implicants could be exponential with respect to the number of propositional symbols [49]. However, for the great majority of knowledge bases that are not the worst case [48], the resultant prime form can be queried in polynomial time for consistency, validity, clause entailment, implicants, equivalence, sentential entailment and model enumeration [13,12]. Specifically in this work, prime forms allow us to construct belief change operators that respect the syntax irrelevance principle and, most importantly, they allow to introduce a new minimal change notion that preserves information, as presented as the second contribution of this article (section 5).

6 6 J. Marchi et al. 3 Belief Change Belief change consists of modelling the dynamics of beliefs. When an agent changes its belief base, it can either add a new statement or delete an existing belief [20]. In the first case, the new belief belongs to the resulting belief base, while in the second case the retracted belief does not belong to the resulting belief base. It means that the change operation has to succeed (w.r.t. the intended meaning) [20, 23, 27]. This principle of success is one of the pre-requisites for change functions that has been widely studied (e.g [14,20,24]). These pre-requisites are called rationality postulates for belief change functions and describe how these functions should behave. The well-known AGM postulates refer to belief revision functions while the KM postulates refer to update functions. The difference between revision and update functions [52] is linked to the status of the incoming information [24]: if it reflects some changes of the current world, the belief base is updated; otherwise if the input reexamines and corrects the described universe, the belief base is revised. 3.1 Belief Revision and AGM Postulates The AGM postulates describe prerequisites for the belief contraction and revision functions from an axiomatic point of view. These postulates describe how changes should occur based on minimal change 1 and syntax independence principles. Let ψ and µ be two propositional formulas such that ψ represents the current belief base and µ the incoming piece of information. ψ µ denotes the revised belief base. According to [27] which considers a finite belief base where beliefs are expressed in propositional logic, the AGM postulates for belief revision are defined as follows: (R1) ψ µ implies µ. (Success) (R2) If ψ µ is satisfiable then ψ µ ψ µ. (Preservation) (R3) If µ is satisfiable then ψ µ is also satisfiable. (Vacuity) (R4) If ψ 1 ψ 2 and µ 1 µ 2 then ψ 1 µ 1 ψ 2 µ 2. (Extensionality) (R5) (ψ µ) ϕ implies ψ (µ ϕ). (Subexpansion) (R6) If (ψ µ) ϕ is satisfiable then ψ (µ ϕ) implies (ψ µ) ϕ. (Superexpansion) According to [41], the first postulate reflects the primacy of the incoming piece of information. Preservation means that if the new information is consistent with the belief base, the revision process reduces to an expansion. According to the vacuity postulate, the only situation in which the belief base can become inconsistent occurs when new information is inconsistent. Extensionality postulate expresses the irrelevance of syntax principle. Subexpansion and Superexpansion postulates focus on conjunctions, expressing the minimal change principle. 3.2 Belief Update and KM Postulates The KM postulates describe how update functions should behave, from an axiomatic point of view. Let ψ and µ be two propositional formulas; ψ µ denotes 1 See [14] for a discussion on the notion of minimal change.

7 Prime Forms and Minimal Change 7 the updated belief base. To guide the construction of update operators, Katsuno and Mendelzon [27] have proposed the following postulates: (U1) ψ µ implies µ. (Success) (U2) If ψ implies µ then ψ µ is equivalent to ψ. (Preservation) (U3) If both ψ and µ are satisfiable then ψ µ is also satisfiable. (Consistency) (U4) If ψ 1 ψ 2 and µ 1 µ 2 then ψ 1 µ 1 ψ 2 µ 2. (Extensionality) (U5) (ψ µ) ϕ implies ψ (µ ϕ). (Subexpansion) (U6) If ψ µ 1 implies µ 2 and ψ µ 2 implies µ 1 then ψ µ 1 ψ µ 2. (Symmetric entailment) (U7) If ψ is complete then (ψ µ 1 ) (ψ µ 2 ) implies ψ (µ 1 µ 2 ). (Disjunctive equivalence) (U8) (ψ 1 ψ 2 ) µ (ψ 1 µ) (ψ 2 µ). (Disjunction rule) Postulates (U1) (U5) are similar to the postulates for revision, except for postulate (U2) that is quite different from the corresponding revision postulate (R2): if a new sentence µ is not contradictory with the belief base then updating by µ does not influence the belief base. Moreover, if a belief base is inconsistent, the resulting base is also inconsistent. Symmetric entailment postulate (U6) says that if we have µ 2 as a consequence of the belief base updated by µ 1 and we also have µ 1 as a consequence of the belief base updated by µ 2, then the two update operations give the same results. Disjunctive equivalence postulate (U7) is only applied to complete belief bases. Postulate (U8), called disjunctive rule, expresses the monotonicity of belief update. 3.3 Belief Change and Preferences Because of the success postulate, whenever an agent adds a new belief to its belief base, it should avoid that the resulting belief state become logically inconsistent. In that context, the agent may have to discard some of its original beliefs in order to guarantee the consistency of the resulting belief state. The question is how an agent decides which formulas should be discarded. A common idea is that it should take this decision in a rational way: if an agent has the choice among the beliefs that should be discarded, it will choose the ones it considers as the less important. Thus, rationality entails preferences: an agents discards its less preferred beliefs. There are numerous ways to represent preferences: P. Gärdenfors has proposed to use epistemic entrenchment [20], A. Grove has proposed the systems of spheres [22]; in [38] B. Nebel claims that basic beliefs should be distinguished from inferred beliefs and this fact should appear in preferences. All these approaches do not presuppose how preferences are defined: they can be given to the agent or it can use an algorithm to construct these preferences. In that former case, preferences over beliefs are usually defined with the help of a notion of distance [8,11]. This distance is defined between worlds [11,24,27] or between sets of worlds (and consequently between formulas) [8, 51]. A distance between two worlds is a measure of how close these two worlds are. Let d be

8 8 J. Marchi et al. a function which calculates the distance between two worlds. In the following we suppose that for any w, w W, d(w, w ) = d(w, w) and d(w, w ) is null 2 iff w = w. The intuitive meaning of a distance is the following. Suppose a world w and a belief base ψ such that w = ψ; suppose that a change occurs, a new contradictory information µ should be included in the base, and there are two worlds w, w s.t. {w, w } = µ. If the world w is closer to the initial world w than w, i.e. d(w, w ) < d(w, w ), then the agent should prefer w rather than w and choose w as a model of the new belief base. Thus distances between worlds give an explicit method to change a belief base [22,27,29]. Besides rationality postulates that characterize belief change functions in a semantic way, belief change can also be defined through explicit belief change functions [15]. In the following, we briefly review belief change in terms of explicit revision and update functions based on preferences Pre-orders and Belief Revision In [27, 29], Katsuno and Mendelzon characterize belief revision in terms of minimal distances and pre-orders. Let ψ be a pre-order over W associated with a propositional formula ψ such that: (P1) if w, w [[ψ]] then w ψ w ; (P2) if w [[ψ]] and w [[ψ]] then w < ψ w ; (P3) if ψ ϕ then ψ = ϕ. Whenever (P1) (P3) hold, pre-order ψ is said to be a faithful pre-order. In [27, 29], Katsuno and Mendelzon establish the following theorem: Theorem 1 (Revision operator) Let ψ and µ be two propositional formulas. Revision operator satisfies (R1) (R6) if and only if there exists a total pre-order ψ s.t. (P1), (P2) and (P3) hold and [[ψ µ]] = min([[µ]], ψ ) Pre-orders and Belief Update Given a belief base ψ and new information µ, an update operator is a function that transforms ψ in the updated belief base ψ µ [27]. In semantical terms, an update operation is defined as follows: Given a formula µ that contradicts ψ, the updated belief base ψ µ is obtained by selecting for each model w of ψ, the set of models of µ that are closest to w. Let w be a pre-order over W associated with a world w so that [27]: (P4) for all w W if w w then w < w w Formally: Theorem 2 (Update operator) Let ψ and µ be two propositional formulas. Update operator satisfies (U1) (U8) if and only if (P4) holds for all pre-orders w and the models of ψ µ are given by: [[ψ µ]] = [ w [[ψ]] min([[µ]], w) 2 More precisely, d(w, w ) = 0 if the distance is represented by numerical values or d(w, w ) = if the distance is represented by a set.

9 Prime Forms and Minimal Change 9 Since belief change can be expressed with the help of preferences, what remains is to define explicit distances and show which preferences they entail. 3.4 Dalal Distance A popular criterion used in the belief change area to build distances between worlds has been proposed by Dalal [11], based on the intuition that an isolated propositional symbol represents the minimal unit of knowledge. According to this criterion, the distance between worlds which is a Hamming distance is given by the set of propositional symbols that have different truth values in each world. The Dalal distance has been used both in belief revision and belief update. Definition 3 (Distance between worlds) Let w and w be two worlds. The distance between w and w is the set of symbols whose truth values differs [11,24,27]: d(w, w ) = {p w(p) = 1 and w (p) = 0} {p w(p) = 0 and w (p) = 1} In the following, p w stands for w(p) = 1 and p w stands for w(p) = 0. Example 1 Given a set of propositional symbols P = {p, q, r} and two worlds w and w such that w = {p, q, r} and w = {p, q, r}; we get the following distance d(w, w ) = {q} {r} = {q, r} The Dalal distance entails the following preferences. Let us consider three worlds w, w and w. The world w is said to be preferred to the world w w.r.t. w if the distance between w and w is shorter than the distance between w and w. In other words w and w share more truth values than w and w : Definition 4 (Dalal preferences) Let w W. Let D w be a total pre-order over W associated with w defined as follows: w D w w d(w, w) N d(w, w) Preferences over W with respect to a propositional formula ψ are defined as follows: w D ψ w w [[ψ]], w [[ψ]] d(w, w ) N d(w, w ) Pre-orders D w and D ψ are total since they are based on numerical values: the number of contradicting symbols. Dalal operator, denoted by D, follows the previously presented revision operator definition (Theorem 1), in which pre-order ψ is represented by pre-order D ψ. The update operator proposed by Forbus [17] which is the counterpart of Dalal s semantical revision operator, uses the same total pre-order, as presented in definition 4. This operator, denoted by F, follows the update operator definition previously presented (Theorem 2).

10 10 J. Marchi et al. 3.5 Winslett Distance The possible models approach (PMA) [52] satisfies the update operator definition presented in Theorem 2; it is similar to Forbus: the difference concerns the closeness criterion. Forbus considers the closeness criterion from a numerical point of view (the cardinalities) while PMA evaluates the closeness from a set inclusion point of view. Formally: Definition 5 (PMA preferences) Let w, w and w W be three models. The closeness relation between w and w, with respect to w is given by the partial preorder relation W w : w W w w d(w, w ) d(w, w ) In the following we need to consider PMA-references over W indexed by a propositional formula ψ [47]; these preferences are defined as follows: w W ψ w ( w [[ψ]] w [[ψ]]) d(w, w ) d(w, w ) Let W be the update operator associated with the PMA preferences. In [25] Herzig and Rifi have shown that among numerous update operators only PMA [52] and Forbus operators respect all KM postulates. Example 2 Consider belief base ψ = ( p 3 p 2 ) ( p 3 p 1 p 4 ) ( p 2 p 4 ) and new information µ = ( p 4 p 3 ) (p 1 p 2 ). The models of ψ denoted as w 1, w 2,... w 7 are: [[ψ]] = {w 1 = {p 1, p 2, p 3, p 4 }, w 2 = { p 1, p 2, p 3, p 4 }, w 3 = {p 1, p 2, p 3, p 4 }, w 4 = { p 1, p 2, p 3, p 4 }, w 5 = {p 1, p 2, p 3, p 4 }, w 6 = { p 1, p 2, p 3, p 4 }, w 7 = { p 1, p 2, p 3, p 4 }} The models of µ, denoted as u 1, u 2,... u 7 are: [[µ]] = {u 1 = {p 1, p 2, p 3, p 4 }, u 2 = {p 1, p 2, p 3, p 4 }, u 3 = { p 1, p 2, p 3, p 4 }, u 4 = { p 1, p 2, p 3, p 4 }, u 5 = {p 1, p 2, p 3, p 4 }, u 6 = {p 1, p 2, p 3, p 4 }, u 7 = {p 1, p 2, p 3, p 4 }} The following tables present the distances d(u i, w j ) calculated w.r.t. the models of ψ and µ. w 1 w 2 w 3 w 4 u 1 {p 2, p 3, p 4 } {p 1, p 2, p 3, p 4 } {p 2, p 3 } {p 1, p 2, p 3 } u 2 {p 3, p 4 } {p 1, p 3, p 4 } {p 3 } {p 1, p 3 } u 3 {p 1, p 2, p 3, p 4 } {p 2, p 3, p 4 } {p 1, p 2, p 3 } {p 2, p 3 } u 4 {p 1, p 3, p 4 } {p 3, p 4 } {p 1, p 3 } {p 3 } u 5 {p 2, p 3 } {p 1, p 2, p 3 } {p 2, p 3, p 4 } {p 1, p 2, p 3, p 4 } u 6 {p 2 } {p 1, p 2 } {p 2, p 4 } {p 1, p 2, p 4 } u 7 {p 2, p 4 } {p 1, p 2, p 4 } {p 2 } {p 1, p 2 }

11 Prime Forms and Minimal Change 11 w 5 w 6 w 7 u 1 {p 2, p 4 } {p 1, p 2, p 4 } {p 1, p 3, p 4 } u 2 {p 4 } {p 1, p 4 } {p 1, p 2, p 3, p 4 } u 3 {p 1, p 2, p 4 } {p 2, p 4 } {p 3, p 4 } u 4 {p 1, p 4 } {p 4 } {p 2, p 3, p 4 } u 5 {p 2 } {p 1, p 2 } {p 1, p 3 } u 6 {p 2, p 3 } {p 1, p 2, p 3 } {p 1 } u 7 {p 2, p 3, p 4 } {p 1, p 2, p 3, p 4 } {p 1, p 4 } According to Dalal preferences based on the cardinality of the sets, we get the following preferred worlds: min([[µ]], D ψ) = {u 2, u 4, u 5, u 6, u 7 } And thus: [[ψ D µ]] = {{p 1, p 2, p 3, p 4 }, { p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }} Now, let us consider belief update; first the minimal worlds defined by Forbus preferences are: min([[µ]], D w 1 ) = {u 6 } min([[µ]], D w 2 ) = {u 4, u 6 } min([[µ]], D w 3 ) = {u 2, u 7 } min([[µ]], D w 4 ) = {u 4 } min([[µ]], D w 5 ) = {u 2, u 5 } min([[µ]], D w 6 ) = {u 4 } min([[µ]], D w 7 ) = {u 6 } Therefore, [[ψ F µ]] = {u 2, u 4, u 5, u 6, u 7 }. The minimal worlds according to the PMA distance and preferences are: min([[µ]], W w 1 ) = {u 2, u 6 } min([[µ]], W w 2 ) = {u 4, u 6 } min([[µ]], W w 3 ) = {u 2, u 7 } min([[µ]], W w 4 ) = {u 4, u 7 } min([[µ]], W w 5 ) = {u 2, u 5 } min([[µ]], W w 6 ) = {u 4, u 5 } min([[µ]], W w 7 ) = {u 3, u 6 } Hence we get that [[ψ W µ]] = {u 2, u 3, u 4, u 5, u 6, u 7 }; that is: [[ψ W µ]] = {{p 1, p 2, p 3, p 4 }, { p 1, p 2, p 3, p 4 }, { p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }, {p 1, p 2, p 3, p 4 }} The different results obtained applying Forbus and PMA in this example leads us to present the concept of strength.

12 12 J. Marchi et al. 3.6 Strength In order to compare belief update operators, Herzig and Rifi [25] have proposed the concept of strength. This concept only applies to update operators because only in the update context the real state of affairs evolves. New incoming information represents some change in the real world and becomes the most important piece of information. By definition, to update a belief base means to accept as true the new information and to maintain in the belief set the beliefs that are closest to the incoming information. The certainty about the real world grows as few models are returned by the update operation. Therefore, the strength is defined as: Definition 6 (strength) An operator 1 is said to be stronger than a second operator 2 if the set of models of ψ 1 µ is included in the set of models of ψ 2 µ. 1 < 2 iff ψ, µ [[ψ 1 µ]] [[ψ 2 µ]] Using this notion, we conclude that the update operator proposed by Forbus is stronger than PMA: F < W 4 Expressing Belief Change with Prime Implicants Using the specific syntax of prime implicants, we redefine the belief change operators in a syntactic way. The main idea is that the syntactical operators calculate the revised belief base using a distance between prime implicants, instead of between models. The first step of the proposed approach is to compile the belief base ψ and the new information µ as prime implicants, noted IP ψ and IP µ. Several algorithms to calculate the prime implicant and prime implicate sets have been proposed (e.g., [7, 26, 30, 45]). The second step consists of the construction of a set of terms with the help of the calculated prime implicants. Each of these terms combines literals from prime implicants of ψ and µ in a consistent way. The final stage consists of evaluating the terms with respect to a criterion, e.g. the usual Hamming distance. Based on this evaluation a subset of terms is extracted and this subset characterizes the resulting belief base. In the following, we describe the main elements which lead us to rephrase Theorems 1 and 2 in the context of prime implicants. 4.1 Expressing Belief Revision In the following we suppose that we can express preferences over a set of terms. Let be a pre-order representing these preferences. At this stage, we do not require to link preferences and distances. We define what are minimal terms in a set of terms Γ; let min(γ, ) be the subset of minimal terms w.r.t. a preference relation such that min(γ, ) = {D Γ D Γ s.t. D < D}. Among all the possible terms we characterize a set of relevant terms which will help us to calculate the revised belief set. These terms are constructed with the help of prime implicants IP ψ and IP µ as follows: for every D ψ and D µ, a new term is obtained by adding to D ψ all the literals of D µ which are not conflicting with the literals to D ψ. Let Γ be a function which computes this set of relevant terms:

13 Prime Forms and Minimal Change 13 Definition 7 (Γ) Let IP ψ and IP µ be two sets of prime implicants. Let Γ : L(P) L(P) 2 D be a function that combines terms as follows: Γ(ψ, µ) = {D µ (D ψ D µ) D ψ IP ψ and D µ IP µ} The set Γ(ψ, µ) contains all prime implicants of IP µ where each of them has been extended with the biggest consistent part of every prime implicant of IP ψ. Example 3 Consider the following sets of prime implicants: IP ψ = ( p 3 p 2 ) ( p 2 p 4 ) ( p 3 p 1 p 4 ) and IP µ = ( p 4 p 3 ) (p 1 p 2 ), that represent the belief base and new information introduced in Example 2. The following table presents the set of terms Γ(ψ, µ): D ψ D µ (D ψ D µ) D i Γ(ψ, µ) p 3 p 2 p 4 p 3 { p 2 } p 2 p 3 p 4 (D 1 ) p 3 p 2 p 1 p 2 { p 3 } p 1 p 2 p 3 (D 2 ) p 2 p 4 p 4 p 3 { p 2 } p 2 p 3 p 4 (D 3 ) p 2 p 4 p 1 p 2 {p 4 } p 1 p 2 p 4 (D 4 ) p 3 p 1 p 4 p 4 p 3 { p 1 } p 1 p 3 p 4 (D 5 ) p 3 p 1 p 4 p 1 p 2 { p 3, p 4 } p 1 p 2 p 3 p 4 (D 6 ) Our aim is to state theorems that describe belief change operations in terms of preferences over terms. These theorems will be similar to the Theorems 1 and 2. First we rewrite the constraints that characterize the notion of faithful pre-order. Let ψ and µ be two formulas. Let ψ be a preference relation defined over the whole set of possible terms D: (P1-T) if D u, D v D ψ then D u ψ D v; (P2-T) if D u D ψ and D v D ψ then D u < ψ D v; (P3-T) if IP ψ = IP ϕ then ψ = ϕ. Whenever (P1-T) (P3-T), relation ψ is said to be a faithful relation. The three conditions redefine conditions (P1), (P2) and (P3) by considering terms rather than worlds. Terms characterize sets of models and the following condition (PI-T) enables to relate preferences over sets of models. That is, we constrain preferences in such a way that, removing some models from a set of models (represented by a term D u), can not lead to a new set (represented by D v) that is more preferred. In other words, this constraint enforces minimal change by avoiding preferences that favor too specific terms. Formally, let D u D and be a pre-order; condition (PI-T) holds iff D v D ((D u D v) (D u D v and D v D u)) (PI-T) The proposed revision operator IP can now be defined. This operator is less general than operator since it only focuses on formulas that can be obtained by combining prime implicants of ψ and µ. It means that the result given by IP is a subset of set Γ(ψ, µ) while the result given by is a subset of W. It also means that preferences only need to be defined over Γ(ψ, µ) while operator entails that preferences have to be defined over the whole set of valuations. This restriction can easily be understood since in this paper we mainly focus on preferences that are

14 14 J. Marchi et al. computed in an automatic way based on a specific notion of distance. The following theorem rephrases Theorem 1; the main difference concerns the preferences: as mentioned they are only defined over Γ(ψ, µ) and the additional constraint (PI-T) has to hold. Theorem 3 (Revision operator) Let IP ψ and IP µ be two sets of prime implicants. Revision operator IP satisfies (R1) (R6) if and only if (i) there exists a relation ψ defined over the subset of terms Γ(ψ, µ) which is a total pre-order, (ii) constraints (P1- T) (P3-T) hold, (iii) for all D Γ(ψ, µ) condition (PI-T) holds w.r.t. Γ(ψ, µ) and ψ and (iv) DNF ψ IP µ = min(γ(ψ, µ), ψ ) The proof is straightforward since it is based on the proof of the Theorem 1 [28]. The proof is detailed in the appendix, Section A.1. In the following, we show that if the preferences over terms are linked to the preferences overs worlds, then Theorems 1 and 3 are similar. That is the revised belief sets are equivalent whether we use worlds or terms. Constraint (KP-1) states that preferences over terms and worlds have to be closely connected; i.e. if a term D u is preferred to a term D v then we have the same preferences between the worlds in which these two terms are satisfied. Let ψ be a pre-order over D and W associated with ψ such that: D u ψ D v u 0 [[D u]], v [[D v]] u 0 ψ v (KP-1) A second constraint (KP-2) requests that all worlds which belong to the set of the models of a term are equally preferred. Condition (KP-2) is the extension of condition (PI-T) to worlds. u, v [[D]](u ψ v and v ψ u) (KP-2) As long as the two constraints hold, the results given by and IP are logically equivalent. Formally: Theorem 4 Let ψ and µ be two formulas, let ψ be a faithful pre-order over W and Γ(ψ, µ). Revision operators and IP produce identical belief sets, that is [[ψ IP µ]] = [[ψ µ]] if and only if for all D u, D v Γ(ψ, µ) constraint (KP-1) and (KP-2) hold. The proof is detailed in the appendix, Section A.2. The next step is to extend this result to belief update. 4.2 Expressing Belief Update In the previous section, we have shown that we can re-express belief revision methods by using prime implicants. The update operator, as previously shown, operates world by world. Since we consider in our context prime implicants as minimal units of belief rather than worlds, we have to rewrite the update operator so that preferences are indexed by prime implicants. More formally, we first rewrite constraint (P4) in our context:

15 Prime Forms and Minimal Change 15 (P4-T) For all D u, D v Γ(ψ, µ), if D u D v then D u < Du D v As shown by constraint (P4-T), preferences are now indexed by terms instead of worlds. It follows that postulates (U1) (U8) that characterize the notion of update by applying change to each world of the initial belief set have to be slightly reformulated in order to accommodate our own notion of change. That is, postulates have to reflect that update should be viewed as applying change to each prime implicant of the initial belief set. It follows that our update notion differs from Katsuno-Mendelzon definition. In our context, the minimal object of change is the minimal set of literals that entails the belief set while Katsuno-Mendelzon consider as minimal object of change a world of the belief set. It means that we consider a prime implicant as a minimal unit of interpretation of a belief set rather than a world; in fact we consider as minimal unit the answer to the question what is the minimal set of literals that is required to entail a belief set?. Let ψ and µ be two propositional formulas; ψ IP µ denotes the updated belief base. The following postulates characterize operator IP : (U1-T) ψ IP µ implies µ. (U2-T) If ψ implies µ then ψ IP µ is equivalent to ψ. (U3-T) If both ψ and µ are satisfiable then ψ IP µ is also satisfiable. (U4-T) If ψ 1 ψ 2 and µ 1 µ 2 then ψ 1 IP µ 1 ψ 2 IP µ 2. (U5-T) (ψ IP µ) ϕ implies ψ IP (µ ϕ). (U6-T) If ψ IP µ 1 implies µ 2 and ψ IP µ 2 implies µ 1 then ψ IP µ 1 ψ IP µ 2. (U7-T) If IP ψ = {D ψ } then (ψ IP µ 1 ) (ψ IP µ 2 ) implies ψ IP (µ 1 µ 2 ). (U8-T) (ψ 1 ψ 2 ) IP µ (ψ 1 IP µ) (ψ 2 IP µ). All postulates are identical to postulates (U1) (U8) except postulate (U7-T). This postulate rephrases condition ψ has to be complete as ψ has to be represented by only one prime implicant. That is, in both cases ψ is represented by one unit of interpretation: either a world or a prime implicant. Now we rephrase Theorem 2. Again, as for operator IP, preferences are only defined over a specific set of terms given by Γ. As mentioned, our unit of interpretation is a prime implicant and thus the definition of the output given by IP will be obtained by selecting minimal elements w.r.t. to preference relations indexed by terms rather than worlds as expressed by Theorem 2. The remaining question is what is the relevant set of terms that has to be considered for extracting minimal elements? Since the selection is relative to a prime implicant D ψ, the relevant set of terms will only be defined w.r.t. D ψ, that is set Γ(D ψ, µ). Hence, as for revision, a key difference between and IP is that the result given by IP is a subset of Γ(D ψ, µ) while the result given by is a subset of W. Following [27], we now show that whenever constraints (P4-T) and (PI-T) hold, the eight update postulates are satisfied: Theorem 5 (Update operator) Let IP ψ and IP µ be two sets of prime implicants. Update operator IP satisfies (U1-T) (U8-T) if and only if for all D ψ IP ψ (i) relation Dψ defined over the subset of terms Γ(D ψ, µ) is a pre-order, (ii) constraint (P4-T) holds, (iii) condition (PI-T) holds w.r.t. Γ(D ψ, µ) and Dψ and DNF ψ IP µ = [ D ψ IP ψ min(γ(d ψ, µ), Dψ )

16 16 J. Marchi et al. The proof is almost a direct translation of the proof of Theorem 2 given in [27]. In the appendix, Section A.3, we detail the main points of the proof. Even if the two update operators and IP are different, they obey postulates which are almost identical. Thus, if we connect preferences over terms and preferences over worlds, we will be able to establish a link between update operators and IP. First, we show that there is a relation in terms of strength and next we show that, under specific conditions, operators and IP produce equivalent results. As for the revision operator, we relate preferences by formulating a constraint denoted (KPW). This constraint states that, for each D ψ IP ψ, for all D µ IP µ, and for all D u and D w Γ(ψ, µ), at least one model in which D u holds is preferred to all models in which D w holds. Formally: ( D u, D w Γ(ψ, µ), D ψ ) D u Dψ D w u 0 [[D u]], w [[D w]] u 0 Dψ w (KPW) Suppose that constraint (KPW) holds. It entails that our update operator produces less models than the initial ones since preferences are indexed by prime implicants; i.e. each prime implicant represent a set of worlds. That is, all the worlds that can be obtained by considering the propositional symbols that do not appear in the prime implicant. It follows that in terms of strength, we get that operator IP is stronger than the initial operator. Formally: Theorem 6 Let ψ and µ be two formulas. Assume preferences Dψ such that (P4-T) holds for every Dψ and for all D Γ(ψ, µ), condition (PI-T) holds w.r.t. Γ(ψ, µ) and every Dψ ; assume preferences w such that (P4) holds for every w and constraint (KPW) hold. The set of models of [[ψ IP µ]] is included in [[ψ µ]] and thus IP < The proof is detailed in the appendix, Section A.4. The remaining question is to set specific conditions, so that both update operators produce equivalent results. First if ψ is inconsistent or µ is inconsistent then the resulting belief sets are inconsistent and thus equivalent. Second, postulates (U2) and (U2-T) entail that if ψ µ is consistent, then the resulting belief set is consistent and equivalent. Finally, resulting belief sets are also equivalent if ψ is complete. Proposition 1 Let ψ and µ be two propositional formulas. Assume update operator IP s.t. (U1-T) (U8-T) hold and update operator s.t. (U1) (U8) hold. [[DNF ψ IP µ]] = [[ψ µ]] holds if ψ is inconsistent or µ is inconsistent or ψ µ is consistent or ψ is complete and constraint (KPW) holds. The proof is presented in the appendix, Section A.5.

17 Prime Forms and Minimal Change Distance between Terms In this section, we show how Dalal and Winslett preference relations can be rephrased in order to handle terms. This will lead us to show that the distance between terms is similar to the distance between worlds and thus the Dalal-based preferences over terms satisfy constraints (KP-1), (KP-2) and (KPW). First, we rewrite the notion of distance so that it handles terms. Indeed, distances only need to be defined for relevant terms that belong to the set given by Γ(ψ, µ), where ψ and µ are two propositional formulas that represent the belief base and the new information. Every term that belongs to Γ(ψ, µ) can be rewritten as D µ (D ψ D µ) s.t. D ψ IP ψ and D µ IP µ. Hence, the set D ψ D µ represents the contradicting literals associated with the term D µ (D ψ D µ). Formally, we introduce a function κ that returns the set of propositional symbols associated with this set of contradicting literals. Definition 8 (κ) Let D, D u, D v be three terms such that D = D v (D u D v): κ(d) = {p P p (D u D v) or p (D u D v)} Example 4 Consider the set Γ(ψ, µ) presented in Example 3. The following table presents the sets κ associated to each term belonging to Γ(ψ, µ): D ψ D µ D Γ(ψ, µ) κ(d) p 3 p 2 p 4 p 3 p 2 p 3 p 4 {p 3 } p 3 p 2 p 1 p 2 p 1 p 2 p 3 {p 2 } p 2 p 4 p 4 p 3 p 2 p 3 p 4 {p 4 } p 2 p 4 p 1 p 2 p 1 p 2 p 4 {p 2 } p 3 p 1 p 4 p 4 p 3 p 1 p 3 p 4 {p 3, p 4 } p 3 p 1 p 4 p 1 p 2 p 1 p 2 p 3 p 4 {p 1 } The following proposition characterizes the links between worlds and terms. Proposition 2 Let IP ψ and IP µ be two sets of prime implicants; let D and D be two terms belonging to Γ(ψ, µ) such that D = D µ (D ψ D µ) and D = D µ (D ψ D µ). u 0 [[D µ]], v 0 [[D ψ ]], w [[D µ]]d(u 0, v 0 ) d(w, v 0 ) κ(d) κ(d ) The proof is presented in appendix, Section A.6. The main consequence of this proposition is that Dalal distance satisfies constraints (KP-1), (KP-2) and (KPW) and PMA distance satisfies constraint (KPW), i.e. the constraints that link preferences over terms and preferences over worlds. 4.4 Dalal and PMA based Preferences over Terms In this section, we extend relations W and D in order to take into account terms and we show that they satisfy the notion of faithful pre-order. Intuitively, we would like to establish a direct link between distances and preferences such as: D W D v D κ(d) κ(d )

18 18 J. Marchi et al. for PMA-based preferences. Unfortunately, this definition of preferences entails that constraint (P4-T) does not hold if ψ µ is consistent. Suppose ψ = p q and µ = ; we get that Γ(ψ, µ) = {p, q} and consequently the naive translation leads to p W p q and q W p p. Thus, we have to define preferences in a two-step process. For any term D, we first enforce (P4-T) by setting preferences relative to D and next we consider preferences indexed by other terms with the help of function κ. Notice that preferences are only defined for terms that belong to the set Γ(ψ, µ). Formally: Definition 9 ( W and D ) Let IP ψ and IP µ be two sets of prime implicants; let D and D be two terms belonging to Γ(ψ, µ) such that D = D µ (D ψ D µ) and D = D µ (D ψ D µ). Pre-order W D ψ is defined as follows: Pre-order D D ψ is defined similarly: D W D ψ D ( if Dψ D κ(d) κ(d ) otherwise D D D ψ D ( if Dψ D κ(d) N κ(d ) otherwise Pre-order D ψ characterizes preferences indexed by a set of prime implicants. Preferences indexed by statement ψ are defined as follows. Let D and D be two terms belonging to Γ(ψ, µ) : D D ψ D iff κ(d) N κ(d ) According to this definition, we get the following preferences for terms presented in Example 4. Example 5 Consider set Γ(ψ, µ) presented in Example 3 and κ values given Example 4. First, we detail PMA-based preferences; there are three preference relations indexed by the prime implicants of ψ. As shown by κ values, the only preferences which can be defined are the reflexive ones: p 2 p 3 p 4 W p 3 p 2 p 2 p 3 p 4 p 1 p 2 p 3 W p 3 p 2 p 1 p 2 p 3 p 2 p 3 p 4 W p 2 p 4 p 2 p 3 p 4 p 1 p 2 p 4 W p 2 p 4 p 1 p 2 p 4 p 1 p 3 p 4 W p 3 p 1 p 4 p 1 p 3 p 4 p 1 p 2 p 3 p 4 W p 3 p 1 p 4 p 1 p 2 p 3 p 4 Dalal-based preferences entail more interesting preferences (reflexive preferences are omitted). Notice that since κ( p 1 p 3 p 4 ) = 2, term p 1 p 2 p 3 p 4 is strictly preferred to p 1 p 2 p 3 p 4 since κ(p 1 p 2 p 3 p 4 ) = 1. p 2 p 3 p 4 = D p 3 p 2 p 1 p 2 p 3 p 2 p 3 p 4 = D p 2 p 4 p 1 p 2 p 4 p 1 p 2 p 3 p 4 D p 3 p 1 p 4 p 1 p 3 p 4

19 Prime Forms and Minimal Change 19 Finally, preferences related to statement ψ are the following: all terms are equally preferred since for all of them κ(d) = 1 except for term p 1 p 3 p 4 since κ( p 1 p 3 p 4 ) = 2: p 2 p 3 p 4 = D ψ p 1 p 2 p 3 = D ψ p 2 p 3 p 4 = D ψ p 1 p 2 p 4 = D ψ p 1 p 2 p 3 p 4 D ψ p 1 p 3 p 4 An immediate consequence of Proposition 2 is that constraints which relate preferences over terms and worlds hold. First, we consider constraint (KPW) Proposition 3 Let D ψ IP ψ. Let Dψ be a relation indexed by D ψ such that Dψ { D D ψ, W D ψ }; let D and D be two terms such that D = D µ (D ψ D µ) and D = D µ (D ψ D µ): D Dψ D u 0 [[D]], v 0 [[D ψ ]], w [[D ]] u 0 v0 w and thus (KPW) holds: ( D u, D w Γ(ψ, µ), D ψ ) D u Dψ D w u 0 [[D u]], w [[D w]] u 0 Dψ w The proof is immediate. Proposition 2 also entails that Dalal-based pre-order satisfies constraint (KP-1) and constraint (KP-2): Proposition 4 D D ψ D u 0 [[D]], v [[D ]] u 0 D ψ v Proposition 5 For any D Γ(ψ, µ): u, v [[D]](u D ψ v and v D ψ u) The proof of Proposition 4 is straightforward; the proof of Proposition 5 is detailed in the appendix, Section A Revision Operator D IP In order to formally state the behavior of our Dalal-based belief revision operator D IP, we first show that Dalal-based preferences satisfy constraint (P1-T) (P3-T) and (PI-T). That is: Proposition 6 D ψ is a faithful pre-order. and Proposition 7 Let ψ and µ be two formulas and Γ(ψ, µ) be a set of terms defined w.r.t. definition 7. Condition (PI-T) holds: D, D Γ(ψ, µ) ((D D ) (D D ψ D and D D ψ D)) It follows that revision operator D IP based on Dalal distance satisfies postulates (R1) (R6).

20 20 J. Marchi et al. Corollary 1 Let ψ and µ be two propositional formulas. Postulates (R1) (R6) hold for ψ D IP µ. The proofs of the two propositions and the corollary are detailed in the appendix, respectively Sections A.8, A.9 and A.10. Since D ψ satisfies constraints KP-1 and KP-2, we get that [[ψ D IP µ]] = [[ψ D µ]] and thus our operator is equivalent to Dalal s proposal. The following example illustrates that result. Example 6 Let us pursue our example. The terms chosen by our Dalal operator are minimal w.r.t. D ψ ; that is p 2 p 3 p 4, p 1 p 2 p 3, p 2 p 3 p 4, p 1 p 2 p 4 and p 1 p 2 p 3 p 4 ; we get the following DNF: DNF ψ D IP µ = ( p 2 p 3 p 4 ) (p 1 p 2 p 3 ) ( p 2 p 3 p 4 ) (p 1 p 2 p 4 ) (p 1 p 2 p 3 p 4 ) If we eliminate redundant terms and simplify (using subsumption), we obtain the following prime implicants: IP ψ D IP µ = (p 1 p 2 p 3 ) ( p 2 p 3 p 4 ) (p 1 p 2 p 4 ) Notice that this set of prime implicants exactly corresponds to the worlds given by [[ψ D µ]] described in Example Update operators W IP and F IP In order to show that PMA-based preferences define an update operator W IP that satisfies postulates (U1-T) through (U8-T), we first show that PMA-based preferences satisfy constraint (P4-T) and (PI-T). Proposition 8 Let D u, D v Γ(ψ, µ) s.t. D u D v. It holds that D u < W D u D v. The proof is detailed in the appendix, Section A.11. Proposition 9 Let ψ and µ be two formulas and Γ(ψ, µ) be a set of terms defined w.r.t. definition 7. Condition (PI-T) holds for all D ψ IP ψ : D ψ IP ψ D u, D v Γ(ψ, µ) ((D u D v) (D u W D ψ D v and D v W D ψ D u)) The proof of this last proposition is similar to the proof of Proposition 7 which has been detailed in the appendix, Section A.9. It follows that update operator W IP based on PMA preferences satisfies postulates (U1-T) (U8-T). Corollary 2 Let ψ and µ be two propositional formulas. Postulates (U1-T) (U8-T) hold for ψ W IP µ. The corollary is a direct consequence of Propositions 8 and 9 and Theorems 5. Now, let us focus on Dalal-based preferences in order to define update operator F IP and to show that our Forbus-like update operator satisfies postulate (U1-T) (U8-T). Proposition 10 Let D u, D v Γ s.t. D u D v. It holds that D u < D D u D v.