Decrement Operators in Belief Change
DDecrement Operators in Belief Change
Kai Sauerwald and Christoph Beierle
FernUniversität in Hagen, 58084 Hagen, Germany {kai.sauerwald,christoph.beierle}@fernuni-hagen.de
Abstract.
While research on iterated revision is predominant in thefield of iterated belief change, the class of iterated contraction operatorsreceived more attention in recent years. In this article, we examine anon-prioritized generalisation of iterated contraction. In particular, theclass of weak decrement operators is introduced, which are operatorsthat by multiple steps achieve the same as a contraction. Inspired byDarwiche and Pearl’s work on iterated revision the subclass of decrementoperators is defined. For both, decrement and weak decrement operators,postulates are presented and for each of them a representation theoremin the framework of total preorders is given. Furthermore, we presenttwo sub-types of decrement operators.
Keywords: belief revision, belief contraction, non-prioritized change,gradual change, forgetting, decrement operator
Changing beliefs in a rational way in the light of new information is one ofthe core abilities of an agent - and thus one of the main concerns of artificialintelligence. The established AGM theory [1] deals with desirable propertiesof rational belief change. The AGM approach provides properties for differenttypes of belief changes. If new beliefs are incorporated into an agent’s beliefswhile maintaining consistency, this is called a revision. Expansion adds a beliefunquestioned to an agent’s beliefs, and contraction removes a belief from anagent’s beliefs. Building upon the characterisations of these kinds of changes andthe underlying principle of minimal change, the theory fanned out in differentdirections and sub-fields.The field of iterated belief revision examines the properties of belief revisionoperators which, due to their nature, can be applied iteratively. In this sub-field,one of the most influential articles is the seminal paper [7] by Darwiche andPearl (DP), establishing the insight that belief sets are not a sufficient repre-sentation for iterated belief revision. An agent has to encode more informationabout her belief change strategy into her epistemic state - where the revisionstrategy deeply corresponds with conditional beliefs. This requires additionalpostulates that guarantee intended behaviour in forthcoming changes. The com-mon way of encoding, also established by Darwiche and Pearl [7], is an extension a r X i v : . [ c s . A I] J u l Sauerwald and Beierle of Katsuno and Mendelzon’s characterisation of AGM revision in terms of plau-sibility orderings [12], where it is assumed that the epistemic states contain anorder over worlds (or interpretations).Similar work has been done in recent years for iterated contraction. Chopra,Ghose, Meyer and Wong [6] contributed postulates for contraction on epistemicstates. Caridroit, Konieczny and Marquis [4] provided postulates for contractionin propositional logic and a characterisation with plausibility orders in the styleof Katsuno and Mendelzon. By this characterisation, the main characteristic ofa contraction with α is that the worlds of the previous state remain plausibleand that the most plausible counter-models of α become plausible.However, in the sub-field of non-prioritised belief change, or more specifically,in the field of gradual belief change much work remains to be done on contrac-tion. An important generalisation of iterated revision operators are the class ofimprovement operators by Konieczny and Pino Pérez [14], which achieve thestate of an revision by multiple steps in a gradual way. These kind of changeswhere intensively studied by Konieczny, Pino Pérez, Booth, Fermé and Gres-pan [3, 13]. A counterpart of improvement operators for the case of contractionis missing. This article fills this gap. We investigate the contraction analogon toimprovement operators, which we call decrement operators. The leading idea isto examine a class of operators which lead, after enough consecutive applications,to the same states as an (iterative) contraction would do.The research presented in this paper is also motivated by the quest for aformalisation of forgetting operators within the field of knowledge representationand reasoning (KRR). In a recent survey article by Eiter and Kern-Isberner [8]the connection between contraction and forgetting of a belief is dealt with froma KRR point of view. Steps towards a general framework for kinds of forgettingin common-sense based belief management, revealing links to well-known KRRmethods, are taken in [2]. However, for the fading out of rarely used beliefsthat takes places in humans gradually over time, or for the change of routines,e.g. in established workflows, often requiring many iterations and the intentionalforgetting of the previous routines, counterparts in the formal methods of KRRare missing. With our work on decrement operators, we provide some basicbuilding blocks that may prove useful for developing a formalisation of thesepsychologically inspired forgetting operations.In summary, the main contributions of this paper are : – Postulates for operators which allow one to perform contractions gradually. – Representation theorems for these classes in the framework or epistemicstates and total preorders. – Define two special types of decrement operators.The rest of the paper is organised as follows. Section 2 briefly presents the re-quired background on belief change. Section 3 introduces the main idea and thepostulates along with a representation theorem for weak decrement operators.In Section 4 the weak decrement operators are restricted by DP-like iteration This version of the paper contains the full proofs.ecrement Operators in Belief Change 3 postulates, leading to the class of decrement operators; we give also a represen-tation theorem for the class of decrement operators. In Section 5 two specialtypes of decrement operators are specified. We close the paper with a discussionand point out future work in Section 6.
Let Σ be a propositional signature. The propositional language L Σ is the smallestset, such that a ∈ L Σ for every a ∈ L Σ and ¬ α ∈ L Σ , α ∧ β, α ∨ β ∈ L Σ if α, β ∈ L Σ . We omit often Σ and write L instead of L Σ . We write formulas in L with lower Greek letters α, β, γ, . . . , and propositional variables with lower caseletters a, b, c, . . . ∈ Σ . The set of propositional interpretations Ω , also called set ofworlds, is identified with the set of corresponding complete conjunctions over Σ .Propositional entailment is denoted by | = , with (cid:74) α (cid:75) we denote the set of modelsof α , and Cn ( α ) = { β | α | = β } is the deductive closure of α . This is lifted to a set X by defining Cn ( X ) = { β | X | = β } . For two sets of formulas X, Y we say X isequivalent to Y with respect to the formula α , written X = α Y , if Cn ( X ∪{ α } ) = Cn ( Y ∪{ α } ) . For two sets of interpretations Ω , Ω ⊆ Ω we say Ω is equivalentto Ω with respect to the formula α , written Ω = α Ω , if Ω and Ω containthe same set of models of α , i.e. { ω ∈ Ω | ω | = α } = { ω ∈ Ω | ω | = α } . Fora set of worlds Ω (cid:48) ⊆ Ω and a total preorder ≤ (reflexive and transitive relation)over Ω , we denote with min( Ω (cid:48) , ≤ ) = { ω | ω ∈ Ω (cid:48) and ∀ ω (cid:48) ∈ Ω (cid:48) ω ≤ ω (cid:48) } theset of all worlds in the lowest layer of ≤ that are elements in Ω (cid:48) . For a totalpreorder ≤ , we denote with < its strict variant, i.e. x < y iff x ≤ y and y (cid:54)≤ x ;with (cid:28) the direct successor variant, i.e. x (cid:28) y iff x < y and there is no z suchthat x < z < y ; and we write x (cid:39) y iff x ≤ y and y ≤ x . Every agent is equipped with an epistemic state , sometimes also called beliefstate, that maintains all necessary information for her belief apparatus. With E we denote the set of all epistemic states. Without defining what a epistemicstate is, we assume that for every epistemic state Ψ ∈ E we can obtain the setof plausible sentences Bel ( Ψ ) ⊆ L of Ψ , which is deductively closed. We write Ψ | = α iff α ∈ Bel ( Ψ ) and we define (cid:74) Ψ (cid:75) = { ω | ω | = α for each α ∈ Bel ( Ψ ) } . Abelief change operator over L is a (left-associative) function ◦ : E × L → E . Wedenote with Ψ ◦ n α the n-times application of α by ◦ to Ψ [14].Darwiche and Pearl [7] propose that an epistemic state ψ should be equippedwith an ordering ≤ Ψ of the worlds (interpretations), where the compatibility withBel ( Ψ ) is ensured by the so-called faithfulness. Based on the work of Katsuno and Cn ( X ∪ { α } ) matches belief expansion with α on belief sets. However, in the contexthere, the context of iterative changes, we understand this purely technically. Theproblem of expansion in this context is more complex [9]. Sauerwald and Beierle Medelezon [12], a mapping Ψ (cid:55)→ ≤ Ψ is called faithful assignment if the followingis satisfied [7]: if ω ∈ (cid:74) Ψ (cid:75) and ω ∈ (cid:74) Ψ (cid:75) , then ω (cid:39) Ψ ω if ω ∈ (cid:74) Ψ (cid:75) and ω / ∈ (cid:74) Ψ (cid:75) , then ω < Ψ ω Konieczny and Pino Pérez give a stronger variant of faithful assignments foriterated belief change [14], which ensures that the mapping Ψ (cid:55)→≤ Ψ is compatiblewith the belief change operator with respect to syntax independence. Definition 1 (Strong Faithful Assignment [14]).
Let ◦ be a belief changeoperator. A function Ψ (cid:55)→≤ Ψ that maps each epistemic state to a total preorderon interpretations is said to be a strong faithful assigment with respect to ◦ if:if ω ∈ (cid:74) Ψ (cid:75) and ω ∈ (cid:74) Ψ (cid:75) , then ω (cid:39) Ψ ω (SFA1) if ω ∈ (cid:74) Ψ (cid:75) and ω / ∈ (cid:74) Ψ (cid:75) , then ω < Ψ ω (SFA2) if α ≡ β , . . . , α n ≡ β n , then ≤ Ψ ◦ α ◦ ... ◦ α n = ≤ Ψ ◦ β ◦ ... ◦ β n (SFA3)We will make use of strong faithful assignments for the characterisation theorems. Postulates for AGM contraction in the framework of epistemic states were givenby Chopra, Ghose, Meyer and Wong [6] and by Konieczny and Pino Pérez [15].We give here the formulation by Chropra et al. [6]:Bel ( Ψ − α ) ⊆ Bel ( Ψ ) (C1) if α / ∈ Bel ( Ψ ) , then Bel ( Ψ ) ⊆ Bel ( Ψ − α ) (C2) if α (cid:54)≡ (cid:62) , then α / ∈ Bel ( Ψ − α ) (C3) Bel ( Ψ ) ⊆ Cn ( Bel ( Ψ − α ) ∪ α ) (C4) if α ≡ β , then Bel ( Ψ − α ) = Bel ( Ψ − β ) (C5) Bel ( Ψ − α ) ∩ Bel ( Ψ − β ) ⊆ Bel ( Ψ − ( α ∧ β )) (C6) if β / ∈ Bel ( Ψ − ( α ∧ β )) , then Bel ( Ψ − ( α ∧ β )) ⊆ Bel ( Ψ − β ) (C7)For an explanation of these postulates we refer to the article of Caridroit etal. [4]. A characterisation in terms of total preorders on epistemic states is givenby the following proposition. Proposition 1 (AGM Contraction for Epistemic State [15]).
A beliefchange operator − fulfils the postulates (C1) to (C7) if and only if there is afaithful assignment Ψ (cid:55)→≤ Ψ such that: (1) (cid:74) Ψ − α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) In addition to the postulates (C1) to (C7), Konieczny and Pino Pérez give DP-like postulates for intended iteration behaviour of contraction [15]. In the fol-lowing, we call these class of operators iterated contraction operators, which arecharacterized by the following proposition. ecrement Operators in Belief Change 5
Proposition 2 (Iterated Contraction [15]).
Let − be a belief change opera-tor − which satisfies (C1) to (C7) . Then − is an iterated contraction operator if and only if there exists a faithful assignment Ψ (cid:55)→≤ Ψ such that (1) holds andthe following is satisfied:if ω , ω ∈ (cid:74) α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ − α ω if ω , ω ∈ (cid:74) ¬ α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ − α ω if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω < Ψ ω ⇒ ω < Ψ − α ω if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω ≤ Ψ ω ⇒ ω ≤ Ψ − α ω The idea of (weak) improvements is to split the process of an AGM revisionfor epistemic states [7, p. 7ff] into multiple steps of an operator ˝ . For such agradual operator ˝ define Ψ ‚ α = Ψ ˝ n α , where n ∈ N is smallest integer suchthat α / ∈ Bel ( Ψ ˝ n α ) . In the initial paper about improvement operators [14],Konieczny and Pino Pérez gave postulates for ˝ , such that ‚ is an AGM revisionfor epistemic states. Due to space reasons, we refer the interested reader to theoriginal paper for the postulates [14]. The following representation theorem givesan impression on weak improvement operators. Proposition 3 (Weak Improvement Operator [14, Thm. 1]).
A beliefchange operator ˝ is a weak improvement operator if and only if there exists astrong faithful assignment Ψ (cid:55)→≤ Ψ such that: (cid:74) Ψ ‚ α (cid:75) = min( (cid:74) α (cid:75) , ≤ Ψ ) Furthermore, the class of weak improvement operators is restricted by DP-like iteration postulates to the so-called improvement operators [14], which areunique . Again, we refer to the work of Konieczny and Pino Pérez [14] for thesepostulates, and only present the characterisation in the framework of total pre-orders. Proposition 4 (Improvement Operator [14, Thm. 2]).
A weak improve-ment operator ˝ is an improvement operator if and only if there exists a strongfaithful Ψ (cid:55)→≤ Ψ assignment such thatif ω , ω ∈ (cid:74) α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ ˝ α ω (S1) if ω , ω ∈ (cid:74) ¬ α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ ˝ α ω (S2) if ω ∈ (cid:74) α (cid:75) and ω ∈ (cid:74) ¬ α (cid:75) , then ω ≤ Ψ ω ⇒ ω < Ψ ˝ α ω (S3) if ω ∈ (cid:74) α (cid:75) and ω ∈ (cid:74) ¬ α (cid:75) , then ω < Ψ ω ⇒ ω ≤ Ψ ˝ α ω (S4) if ω ∈ (cid:74) α (cid:75) and ω ∈ (cid:74) ¬ α (cid:75) , then ω (cid:28) Ψ ω ⇒ ω ≤ Ψ ˝ α ω (S5) holds and the following is satisfied: (cid:74) Ψ ‚ α (cid:75) = min( (cid:74) α (cid:75) , ≤ Ψ ) Note that the notion of improvement operators is not used consistently in the liter-ature. For instance, the improvement operators as defined in [13] are not unique. Sauerwald and Beierle
In the following section we use the basic ideas of (weak) improvement operatorsas a starting point for developing the weak decrement operators.
A property of a contraction operator − is that the success condition of contrac-tion is instantaneously achieved, i.e., if α is believed in a state ( α ∈ Bel ( Ψ ) )then after the contraction with α , it is not believed any more ( α / ∈ Bel ( Ψ − α ) ).As a generalisation, we define hesitant contractions as operators who achieve thesuccess condition of contraction after multiple consecutive applications. Definition 2.
A belief change operator ◦ is called a hesitant contraction oper-ator if the following postulate is fulfilled:if α (cid:54)≡ (cid:62) , then there exists n ∈ N such that α / ∈ Bel ( Ψ ◦ n α ) (hesitance)If ◦ is an hesitant contraction operator, then we define a corresponding operator • by Ψ • α = Ψ ◦ n α , where n = 0 if α ≡ (cid:62) , otherwise n is the smallest integersuch that α / ∈ Bel ( Ψ ◦ n α ) .The following Example 1 shows a modelling application for hesitant beliefchange operators. Example 1.
Addison bought a new mobile with much easier handling. She doesno longer have to press a sequence of buttons to access her favourite application.However, it takes multiple changes of her epistemic state before she contracts thebelief of having to press the sequence of buttons for her favourite application.We now introduce weak decrement operators, which fulfil AGM-like contrac-tion postulates, adapted for the decrement of beliefs.
Definition 3 (Weak Decrement Operator).
A belief change operator ◦ iscalled a weak decrement operator if the following postulates are fulfilled:Bel ( Ψ • α ) ⊆ Bel ( Ψ ) (D1) if α / ∈ Bel ( Ψ ) , then Bel ( Ψ ) ⊆ Bel ( Ψ • α ) (D2) ◦ is a hesitant contraction operator (D3) Bel ( Ψ ) ⊆ Cn ( Bel ( Ψ • α ) ∪ { α } ) (D4) if α ≡ β , ..., α n ≡ β n , then Bel ( Ψ ◦ α ◦ ... ◦ α n ) = Bel ( Ψ ◦ β ◦ ... ◦ β n ) (D5) Bel ( Ψ • α ) ∩ Bel ( Ψ • β ) ⊆ Bel ( Ψ • ( α ∧ β )) (D6) if β / ∈ Bel ( Ψ • ( α ∧ β )) , then Bel ( Ψ • ( α ∧ β )) ⊆ Bel ( Ψ • β ) (D7)The postulates (D1) to (D7) correspond to the postulates (C1) to (C7). By(D1) a weak decrement does not add new beliefs, and together with (D2) thebeliefs of an agent are not changed if α is not believed priorly. (D3) ensures thatafter enough consecutive application a belief α is removed. (D4) is the recoverypostulate, stating that removing α and then adding α again recovers all initial ecrement Operators in Belief Change 7 beliefs. The postulate (D5) ensures syntax independence in the case of iteration.(D6) and (D7) state that a contraction of a conjunctive belief is constrained bythe results of the contractions with each of the conjuncts alone.For the class of weak decrement operators the following representation the-orem holds: Theorem 1 (Representation Theorem: Weak Decrement Operators).
Let ◦ be a belief change operator. Then the following items are equivalent:(a) ◦ is a weak decrement operator(b) there exists a strong faithful assignment Ψ (cid:55)→≤ Ψ with respect to ◦ such that:there exists n ∈ N such that (cid:74) Ψ ◦ n α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) (decrement sucess) and n is the smallest integer such that (cid:74) Ψ ◦ n α (cid:75) (cid:54)⊆ (cid:74) α (cid:75) From Theorem 1 we easily get the following corollary:
Corollary 1. If ◦ is a weak decrement operator, then • fulfils (C1) to (C7) .Furthermore, every belief change operator that fulfils (C1) to (C7) and (D5) isa weak decrement operator. This shows that weak decrement operators are (up to (D5)) a generalisation ofAGM contraction for epistemic states in the sense of Proposition 1.
We now introduce an ordering on the formulas in order to shorten our notion inthe following postulates.
Definition 4.
Let ◦ be a hesitant contraction operator, then we define for everyepistemic state Ψ and every two formula α, β : α (cid:22) ◦ Ψ β iff Bel ( Ψ • αβ ) ⊆ Bel ( Ψ • α ) With ≺ ◦ Ψ we denote the strict variant of (cid:22) ◦ Ψ and define α Î ◦ Ψ β if α ≺ ◦ Ψ β andthere is no γ such that α ≺ ◦ Ψ γ ≺ ◦ Ψ β . Intuitively α ≺ ◦ Ψ β means that in the state Ψ the agent is more willing to giveup the belief α than the belief β .For the iteration of decrement operators we give the following postulates:if ¬ α | = β , then Bel ( Ψ ◦ α • β ) = α Bel ( Ψ • β ) (D8) if α | = β , then Bel ( Ψ ◦ α • β ) = ¬ β Bel ( Ψ • β ) (D9) if α | = γ , then Ψ ◦ α • β | = γ ⇒ Ψ • β | = γ (D10) if ¬ α | = γ , then Ψ • β | = γ ⇒ Ψ ◦ α • β | = γ (D11) if α | = β and ¬ α | = γ , then γ Î ◦ Ψ β ⇒ β (cid:22) ◦ Ψ ◦ α γ (D12) Bel ( Ψ ◦ α ) ⊆ Bel ( Ψ ) (D13) Sauerwald and Beierle (D8) states that a prior decrement with α does not influence the beliefs of andecrement with β if ¬ α | = β . (D9) states that a prior decrement with α does notinfluence the beliefs of an decrement with β if α | = β . The postulate (D10) statesthat if a belief in γ is believed after a decrement of α and the removal of β , thenonly a removal of β does not influence the belief in γ if α ¬ γ implies β . By (D11),if γ and α do not share anything, then a decrease of α does not influence thisbelief. By (D12), if in the state Ψ the agent prefers removing a consequence of ¬ α minimally more than removing a consequence of α , then after a decrementof α , she is more willing to remove the consequence of α . The postulate (D13)axiomatically enforces that a single step does not add beliefs.We call operators that fulfil these postulates decrement operators. Definition 5 (Decrement Operator). A ◦ weak decrement operator is calleda decrement operator if ◦ satisfies (D8) – (D13) . On the semantic side, we define a specific form of strong faithful assignmentwhich implements decrementing on total preorders.
Definition 6 (Decreasing Assignment).
Let ◦ be a hesitant belief changeoperator. A strong faithful assignment Ψ (cid:55)→≤ Ψ with respect to ◦ is said to be a decreasing assignment (with respect to ◦ ) if the following postulates are satisfied:if ω , ω ∈ (cid:74) α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ ◦ α ω (DR8) if ω , ω ∈ (cid:74) ¬ α (cid:75) , then ω ≤ Ψ ω ⇔ ω ≤ Ψ ◦ α ω (DR9) if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω ≤ Ψ ω ⇒ ω ≤ Ψ ◦ α ω (DR10) if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω < Ψ ω ⇒ ω < Ψ ◦ α ω (DR11) if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω (cid:28) Ψ ω ⇒ ω ≤ Ψ ◦ α ω (DR12) if ω ∈ (cid:74) ¬ α (cid:75) , ω ∈ (cid:74) α (cid:75) and ω ≤ Ψ ω for all ω , then ω ≤ Ψ ◦ α ω (DR13)The postulates (DR8) to (DR11) are the same as given by Konieczny and PinoPérez [15] for iterated contraction (cf. Proposition 2). The postulate (DR12)states that a world of ¬ α which is minimally less plausible than a world of α should be made at least as plausible as this world of α . (DR13) ensures that(together with the other postulates) that world in (cid:74) Ψ (cid:75) stays plausible after adecrement.The main result is that decrement operators are exactly those which arecompatible with a decreasing assignment. Theorem 2 (Representation Theorem: Decrement Operators).
Let ◦ bea belief change operator. Then the following items are equivalent:(a) ◦ is a decrement operator(b) there exists a decreasing assignment Ψ (cid:55)→≤ Ψ with respect to ◦ that satisfies (decrement sucess) , i.e.:there exists n ∈ N such that (cid:74) Ψ ◦ n α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) and n is the smallest integer such that (cid:74) Ψ ◦ n α (cid:75) (cid:54)⊆ (cid:74) α (cid:75) ecrement Operators in Belief Change 9 Ψ Ψ ◦ a Ψ ◦ a Layer 2 a ¬ b ¬ a ¬ b a ¬ b Layer 1 ¬ ab a ¬ b ¬ a ¬ b ¬ a ¬ b Layer 0 (cid:74) Ψ (cid:75) ab ab ¬ ab ab ¬ ab Table 1.
Example changes by two decrement operators ◦ and ◦ . The following proposition presents a nice property of decrement operators:Like AGM contraction for epistemic sates (cf. Proposition 1) a decrement oper-ators keeps plausible worlds; and only the least unplausible counter-worlds maybecome plausible.
Proposition 5.
Let ◦ be a hesitant belief change operator. If there exists a de-creasing assignment Ψ (cid:55)→≤ Ψ with respect to ◦ , then we have: (partial success) (cid:74) Ψ (cid:75) ⊆ (cid:74) Ψ ◦ α (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) Unlike improvement operators [14], there is no unique decrement operator. Thereason for this is, that if ω (cid:39) Ψ ω for ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , and it isnot required otherwise by (DR12), then the relative plausibility of ω and ω might not be changed by a decrement operator ◦ , i.e. ω (cid:39) Ψ ◦ α ω . Example 2demonstrates this.
Example 2.
Let Σ = { a, b } and Ψ be an epistemic state as given in Table 1.Then the change from Ψ to Ψ ◦ a in Table 1 is a valid change by a decrementoperator. Likewise, the change from Ψ to Ψ ◦ a from Table 1 is also a validchange for a decrement operator.We capture this observation by two types of decrement operators. In thefirst case, the decrement operator improves the plausibility of a counter-modelwhenever it is possible. Definition 7 (Type-1 Decrement Operator).
A decrement operator ◦ is a type-1 decrement operator if there exists a decreasing assignment Ψ (cid:55)→≤ Ψ with:if ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) , then ω (cid:39) Ψ ω ⇒ ω (cid:28) Ψ ◦ α ω (DR14)The second type of decrement operators keeps the order ω (cid:39) Ψ ω wheneverpossible. We capture the cases when this is possible by the following notion. If ≤⊆ Ω × Ω is a total preorder on worlds, we say ω is frontal with respect to α , if(1.) there is no ω ∈ (cid:74) α (cid:75) such that ω (cid:28) ω , and (2.) there is no ω ∈ (cid:74) ¬ α (cid:75) suchthat ω (cid:28) ω . We define the second type of decrement operators as follows. Definition 8 (Type-2 Decrement Operator).
A decrement operator ◦ is a type-2 decrement operator if there exists a decreasing assignment Ψ (cid:55)→≤ Ψ with:if ω ∈ (cid:74) ¬ α (cid:75) , ω ∈ (cid:74) α (cid:75) and ω is frontal w.r.t α , then ω (cid:39) Ψ ω ⇒ ω (cid:39) Ψ ◦ α ω (DR15) Example (continuation of Example 2) . The change from Ψ to Ψ ◦ a in Table1 can be made by a type-1 decrement operator, but not by a type-2 decrementoperator. Conversely, the change from Ψ to Ψ ◦ a from Table 1 can be madeby a type-2 decrement operator, but not by a type-1 decrement operator We provide postulates and representation theorems for gradual variants of AGMcontractions in the Darwich-Pearl framework of epistemic states. These so-calledweak decrement operators are a generalisation of AGM contraction for epistemicstates. Additionally, we give postulates for intended iterative behaviour of theseoperators, forming the class of decrement operators. For both classes of opera-tors we presented a representation theorem in the framework of total preorders.For the definition of the postulates, the new relation (cid:22) ◦ Ψ (see Definition 4) isintroduced. While (cid:22) ◦ Ψ is related to epistemic entrenchment [10], it can be shownthat (cid:22) ◦ Ψ is not an epistemic entrenchment. The exploration of the exact natureof (cid:22) ◦ Ψ remains an open task.The next natural step will be to investigate the interrelation between (weak)decrement operators and (weak) improvement operators. One approach is togeneralize the Levi identity [16] and Haper identity [11] to these operators. An-other approach could be the direct definition of a contraction operator fromimprovement operators, as suggested by Konieczny and Pino Pérez [14]. Forsuch operators, after achieving success, a next improvement may make certainmodels unplausible, while a decrement operator keeps the plausibility. While thisalready indicated a difference between the operators, the study of their specificinterrelationship is part of future work. Another goal for future work is to gen-eralize (weak) decrement operators to a more general class of gradual changeoperators [17]. Such operators are candidates for a formalisation of psycholog-ically inspired forgetting operations. An immediate target towards this goal isto take a closer look at subclasses and interrelate them with the taxonomy ofimprovement operators [13]. Acknowledgements:
We thank the reviewers for their valuable hints and com-ments that helped us to improve the paper and we thank Gabriele Kern-Isbernerfor fruitful discussions and her encouragement to follow the line of research lead-ing to this paper. This work was supported by DFG Grant BE 1700/9-1 givento Christoph Beierle as part of the priority program "Intentional Forgetting inOrganizations" (SPP 1921). Kai Sauerwald is supported by this Grant. ecrement Operators in Belief Change 11
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A Proofs
This appendix contains full proofs for the two representation theorems and forProposition 5. These proofs rely on three lemmata which are also proven here.
Lemma 1.
Let ◦ an operator satisfying (D1) to (D4) and ω ∈ Ω , then: (cid:74) Ψ • ¬ ω (cid:75) = (cid:74) Ψ (cid:75) ∪ { ω } Proof.
The proof is analogue to a proof by Caridroit et. al [5, Lem 13.]. (cid:117)(cid:116)
Theorem 1 (Representation Theorem: Weak Decrement Operators).
Let ◦ be a belief change operator. Then the following items are equivalent:(a) ◦ is a weak decrement operator(b) there exists a strong faithful assignment Ψ (cid:55)→≤ Ψ with respect to ◦ such that:there exists n ∈ N such that (cid:74) Ψ ◦ n α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) and n is the smallest integer such that (cid:74) Ψ ◦ n α (cid:75) (cid:54)⊆ (cid:74) α (cid:75) (decrement success) Proof.
We proof the theorem under the assumption that the signature has morethan 2 elements, i.e. | Σ | > . For the (a) to (b)-direction, ◦ is an hesitantcontraction operator, and the corresponding operator • is defined. We define thetotal preorder ≤ Ψ as follows: ω ≤ Ψ ω iff ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) We show that ≤ Ψ is a total preorder: Totality
Let ω , ω ∈ Ω . By definition (cid:74) ( ω ∨ ω ) (cid:75) = { ω , ω } , and therefore ¬ ( ω ∨ ω ) has at least one model and ¬ ( ω ∨ ω ) (cid:54)≡ (cid:62) . By (hesitance)there is an n (and we choose here the smallest) such that ¬ ( ω ∨ ω ) / ∈ Bel ( Ψ • ¬ ( ω ∨ ω )) . Therefore, either ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) or ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) . Reflexivity
Follows from totality.
Transitivity
Let ω , ω , ω ∈ Ω such that ω ≤ Ψ ω and ω ≤ Ψ ω . We differ-entiate by case: – If ω , ω , ω are not pairwise distinct, then transitivity is easily fulfilled(since ≤ Ψ is reflexive). – Assume that ω , ω , ω are pairwise distinct and for at least one ≤ i ≤ we have ω i ∈ (cid:74) Ψ (cid:75) . Then in each case it is easy to see that ω ∈ (cid:74) Ψ (cid:75) andthus, by (D1), for all α it follows ω ∈ (cid:74) Ψ ◦ α (cid:75) . – Assume that ω , ω , ω are pairwise distinct and ω , ω , ω / ∈ (cid:74) Ψ (cid:75) .Towards a contradiction, assume that ω (cid:54)≤ Ψ ω . By assumption of ω (cid:54)≤ Ψ ω we have ω / ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) . By Lemma 1 we have ω ∈ (cid:74) Ψ (cid:75) ∪ { ω } = (cid:74) Ψ • ¬ ω (cid:75) . From (D7) we get (cid:74) Ψ • ¬ ω (cid:75) ⊆ (cid:74) Ψ • ( ¬ ω ∧ ¬ ω ) (cid:75) and by (D5) we have (cid:74) Ψ • ( ¬ ω ∧ ¬ ω ) (cid:75) = (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) , a contradic-tion, since ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) and ω / ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) . .2 Sauerwald and Beierle We show that ≤ Ψ is a strong faithful assignment with respect to ◦ .(SFA1) Let ω , ω ∈ (cid:74) Ψ (cid:75) . Then by (D1) we have ω , ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) .Therefore by definition of ≤ Ψ we have ω (cid:39) Ψ ω .(SFA2) Let ω ∈ (cid:74) Ψ (cid:75) and ω / ∈ (cid:74) Ψ (cid:75) . Then by (D1) we have ω ∈ (cid:74) Ψ •¬ ( ω ∨ ω ) (cid:75) .Towards a contradiction assume ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) . Since ω (cid:54) = ω ,by ω ∈ (cid:74) Ψ (cid:75) we know that Ψ (cid:54)| = ¬ ( ω ∨ ω ) . Thus, by (D2) we have (cid:74) Ψ •¬ ( ω ∨ ω ) (cid:75) ⊆ (cid:74) Ψ (cid:75) .(SFA3) Follows directly from (D5).We show that (decrement sucess) is fulfilled. We differentiate by case: – Case with α ≡ (cid:62) . Then (cid:74) ¬ α (cid:75) = ∅ and by definition of • we have Ψ = Ψ • α ,especially (cid:74) Ψ (cid:75) = (cid:74) Ψ • α (cid:75) = (cid:74) Ψ (cid:75) ∪ (cid:74) ¬ α (cid:75) . – Case with α / ∈ Bel ( Ψ ) . Then by (D1) and (D2) we have (cid:74) Ψ (cid:75) = (cid:74) Ψ • α (cid:75) , resp.Bel ( Ψ ) = Bel ( Ψ • α ) . Then there is an ω ∈ (cid:74) Ψ (cid:75) such that ω (cid:54)| = α , thus, wehave min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ⊆ (cid:74) Ψ (cid:75) . We conclude (cid:74) Ψ • α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . – Case with α ∈ Bel ( Ψ ) . Then by (D1) we have (cid:74) Ψ (cid:75) ⊆ (cid:74) Ψ • α (cid:75) . We show thatevery ω ∈ (cid:74) Ψ • α (cid:75) \ (cid:74) Ψ (cid:75) is an element of the set min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) .First, by (D4) we have (cid:74) Ψ • α (cid:75) ∩ (cid:74) α (cid:75) ⊆ (cid:74) Ψ (cid:75) . Then every ω ∈ (cid:74) α (cid:75) which is anelement of (cid:74) Ψ • α (cid:75) \ (cid:74) Ψ (cid:75) leads to a violation of (D4). Thus, we observe thatevery ω ∈ (cid:74) Ψ • α (cid:75) \ (cid:74) Ψ (cid:75) is an element of (cid:74) ¬ α (cid:75) .Second, towards a contradiction suppose ω ∈ (cid:74) Ψ • α (cid:75) \ (cid:74) Ψ (cid:75) such that ω / ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . Let ω (cid:48) ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) , and therefore ω (cid:48) < Ψ ω . By def-inition we have ω (cid:48) ∈ (cid:74) Ψ •¬ ( ω (cid:48) ∨ ω ) (cid:75) and ω / ∈ (cid:74) Ψ •¬ ( ω (cid:48) ∨ ω ) (cid:75) . By (D5) we have (cid:74) Ψ •¬ ( ω (cid:48) ∨ ω ) (cid:75) = (cid:74) Ψ • ( ¬ ω (cid:48) ∧¬ ω ) (cid:75) . Then by (D7) and by Lemma 1 we conclude (cid:74) Ψ (cid:75) ∪ { ω } ⊆ (cid:74) Ψ • ( ¬ ω (cid:48) ∧ ¬ ω ) (cid:75) . This shows (cid:74) Ψ • α (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) .Suppose ω is an element of min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) such that ω / ∈ (cid:74) Ψ • α (cid:75) . Withoutloss of generality we can assume α (cid:54)≡ (cid:62) ; thus, there exists at least one ω (cid:48) ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) such that ω (cid:48) ∈ (cid:74) Ψ • α (cid:75) . By definition of ≤ Ψ we have ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω (cid:48) ) (cid:75) . Clearly ¬ α = γ ∨ ω ∨ ω (cid:48) , and thus, α ≡ ¬ γ ∧ ¬ ( ω ∨ ω (cid:48) ) .Since ω (cid:48) ∈ (cid:74) Ψ • α (cid:75) , we have Ψ • α (cid:54)| = ¬ ( ω ∨ ω (cid:48) ) . Therefore from (D7) weconclude (cid:74) Ψ • ¬ ( ω ∨ ω (cid:48) ) (cid:75) ⊆ (cid:74) Ψ • α (cid:75) and thus the contradiction ω ∈ (cid:74) Ψ • α (cid:75) .This completes the proof of (cid:74) Ψ • α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) .For the (b) to (a)-direction let ◦ be a belief change operator and ≤ Ψ a strongfaithful assignment with respect to ◦ such that (decrement sucess) is fulfilled.(D3) For Ψ and α let n Ψα be the smallest integer such that (cid:74) Ψ ◦ n Ψα α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . By (decrement sucess) the existence of n Ψα guaranteed. For α (cid:54)≡ (cid:62) , then α / ∈ Bel (cid:16) Ψ ◦ n Ψα α (cid:17) and therefore ◦ is a hesitant contractionoperator.Since ◦ satisfies (D3) the corresponding operator • is defined.(D1) Follows directly by (decrement sucess).(D2) Suppose Ψ (cid:54)| = α . Then, min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) is non-empty and by (SFA1) and(SFA2) we have min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ⊆ (cid:74) Ψ (cid:75) . ecrement Operators in Belief Change A.3 (D4) Let γ ∈ Bel ( Ψ ) and therefore (cid:74) Ψ (cid:75) ⊆ (cid:74) γ (cid:75) . Then γ ∈ Cn ( Bel ( Ψ • α ) ∪ { α } ) if and only if (cid:74) Ψ • α (cid:75) ∩ (cid:74) α (cid:75) ⊆ (cid:74) γ (cid:75) . By (decrement sucess) we conclude (cid:74) Ψ • α (cid:75) ∩ (cid:74) α (cid:75) = ( (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ )) ∩ (cid:74) α (cid:75) = (cid:74) Ψ (cid:75) \ (cid:74) ¬ α (cid:75) . Clearly, (cid:74) Ψ (cid:75) \ (cid:74) ¬ α (cid:75) ⊆ (cid:74) Ψ (cid:75) ⊆ (cid:74) γ (cid:75) .(D5) Follows by (SFA3).(D6) By (decrement sucess) we have (cid:74) Ψ ◦ ( α ∧ β ) (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) ∪ (cid:74) ¬ β (cid:75) , ≤ Ψ ) and we have (cid:74) Ψ ◦ α (cid:75) ∪ (cid:74) Ψ ◦ β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Further-more, it holds that min( (cid:74) ¬ α (cid:75) ∪ (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) and therefore, we have: (cid:74) Ψ ◦ ( α ∧ β ) (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = (cid:74) Ψ ◦ α (cid:75) ∪ (cid:74) Ψ ◦ β (cid:75) (D7) Assume Ψ • αβ (cid:54)| = β . Then by (decrement sucess) and (SFA3) we have (cid:74) Ψ • αβ (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α ∨ ¬ β (cid:75) , ≤ Ψ ) (cid:54)⊆ (cid:74) β (cid:75) . This implies that min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ α ∨¬ β (cid:75) , ≤ Ψ ) . By basic set theory we get (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α ∨ ¬ β (cid:75) , ≤ Ψ ) . By (decrement sucess) this is equivalent to (cid:74) Ψ • β (cid:75) ⊆ (cid:74) Ψ • αβ (cid:75) In summary, the operator ◦ is an weak decrement operator. (cid:117)(cid:116) Lemma 2.
Let ◦ be a belief change operator. If there exists a strong faithfulassignment Ψ (cid:55)→≤ Ψ with respect to ◦ which satisfies (DR8) , (DR9) and (DR11) ,then for every Ψ and α ∈ L we have: (cid:74) Ψ ◦ α (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) Proof.
Let ω ∈ (cid:74) Ψ ◦ α (cid:75) . If ω ∈ (cid:74) Ψ (cid:75) we are done, so it remains to show that ω ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) in the case of ω / ∈ (cid:74) Ψ (cid:75) .We first show that if ω / ∈ (cid:74) Ψ (cid:75) , then ω ∈ (cid:74) ¬ α (cid:75) . Towards a contradictionsuppose this is not the case, i.e. ω / ∈ (cid:74) Ψ (cid:75) and ω ∈ (cid:74) α (cid:75) . Then there a two cases:1. There exists ω (cid:48) ∈ (cid:74) α (cid:75) such that ω (cid:48) ∈ (cid:74) Ψ (cid:75) . We easy conclude that ω (cid:48) < Ψ ω andthus, by (DR8), we have ω (cid:48) < Ψ ◦ α ω . Due to the faithfulness of the assignment ω / ∈ (cid:74) Ψ ◦ α (cid:75) , which is a contradiction. 2. For all ω (cid:48) ∈ (cid:74) α (cid:75) we have ω (cid:48) / ∈ (cid:74) Ψ (cid:75) . Then,by using (cid:74) Ψ (cid:75) (cid:54) = ∅ , for all ω (cid:48)(cid:48) ∈ (cid:74) Ψ (cid:75) we must have ω (cid:48)(cid:48) ∈ (cid:74) ¬ Ψ (cid:75) . Thus, ω (cid:48)(cid:48) < Ψ ω andfrom (DR11) we get ω (cid:48)(cid:48) < Ψ ◦ α ω . Again, due to the faithfulness of the assignment,we have ω / ∈ (cid:74) Ψ ◦ α (cid:75) , which is a contradiction. So every ω ∈ (cid:74) Ψ ◦ α (cid:75) \ (cid:74) Ψ (cid:75) is anelement of ω ∈ (cid:74) ¬ α (cid:75) .Now we show that every ω ∈ (cid:74) Ψ ◦ α (cid:75) \ (cid:74) Ψ (cid:75) is an element of min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) .Towards a contradiction suppose ω ∈ (cid:74) ¬ α (cid:75) \ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . Then there exists ω (cid:48) ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) such that ω (cid:48) < Ψ ◦ α ω . By (DR9) we can conclude that ω (cid:48) < Ψ ◦ α ω , which is a contradiction to the assumed faithfulness of the assign-ment. (cid:117)(cid:116) Proposition 5.
Let ◦ be a hesitant belief change operator. If there exists andecreasing assignment Ψ (cid:55)→≤ Ψ with respect to ◦ , then we have: (partial success) (cid:74) Ψ (cid:75) ⊆ (cid:74) Ψ ◦ α (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) .4 Sauerwald and Beierle Proof.
This is a direct consequence of Lemma 2 and (DR13). (cid:117)(cid:116)
Lemma 3.
Let ◦ be a belief change operator, Ψ (cid:55)→≤ Ψ a strong faithful assign-ment with respect to ◦ and γ ≺ ◦ Ψ β . Then γ Î ◦ Ψ β if and only if for each ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) and ω ∈ min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) we have either ω (cid:28) Ψ ω or ω (cid:39) Ψ ω .Proof. The "only if" direction. By definition of (cid:22) ◦ Ψ we have min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ γ ∨ ¬ β (cid:75) , ≤ Ψ ) , which implies min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Clearly, it follows that ω ≤ Ψ ω .In the case of ω (cid:39) Ψ ω we are done.For the remaining case of ω < Ψ ω suppose there exists ω / ∈ { ω , ω } suchthat ω < Ψ ω < Ψ ω . This implies that min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ γ ∨ ω (cid:75) , ≤ Ψ ) and ω / ∈ min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) . Thus, by definition we have γ ≺ ◦ Ψ γ ¬ ω . Similarly,we have β ¬ ω ≺ ◦ Ψ β , since ω ∈ min( (cid:74) ¬ β ∨ ω (cid:75) , ≤ Ψ ) and min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) (cid:54)⊆ min( (cid:74) ¬ γ ∨ ω (cid:75) , ≤ Ψ ) . Note that this implies ω (cid:54)| = ¬ β . From the previous obser-vations we conclude min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ γ ∨ ¬ β ∨ ω (cid:75) , ≤ Ψ ) , and therefore γ (cid:22) ◦ Ψ β ¬ ω . This leads to γ ≺ ◦ Ψ β ¬ ω ≺ ◦ Ψ β , which is a contradiction to γ Î ◦ Ψ β .In summary it must be the case that either ω (cid:39) Ψ ω or ω (cid:28) Ψ ω .For the "if" direction suppose that γ ≺ ◦ ψ α ≺ ◦ Ψ β . This im-plies that min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ γ ∨ ¬ α (cid:75) , ≤ Ψ ) and min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) (cid:54)⊆ min( (cid:74) ¬ γ ∨ ¬ α (cid:75) , ≤ Ψ ) . Thus we have ω < Ψ ω for every ω ∈ min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) andsome ω ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . Additionally, we have min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ¬ β ∨¬ α (cid:75) , ≤ Ψ ) and min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) (cid:54)⊆ min( (cid:74) ¬ β ∨ ¬ α (cid:75) , ≤ Ψ ) . Thus we have ω < Ψ ω for every ω ∈ min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) and some ω ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) . Note that ≤ Ψ isa total preorder, and thus, we have ω < Ψ ω < Ψ ω , a contradiction to theassumptions of ω (cid:28) Ψ ω or ω (cid:39) Ψ ω . (cid:117)(cid:116) Theorem 2 (Representation Theorem: Decrement Operators).
Let ◦ bea belief change operator. Then the following items are equivalent:(a) ◦ is a decrement operator(b) there exists a decreasing assignment Ψ (cid:55)→≤ Ψ with respect to ◦ that satisfies: (decrement sucess) , i.e.:there exists n ∈ N such that (cid:74) Ψ ◦ n α (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) and n is the smallest integer such that (cid:74) Ψ ◦ n α (cid:75) (cid:54)⊆ (cid:74) α (cid:75) Proof. (a) to (b)-direction: As ◦ is an hesitant contraction operator, the corre-sponding operator • is defined. We define the total preorder ≤ Ψ as follows: ω ≤ Ψ ω iff ω ∈ (cid:74) Ψ • ¬ ( ω ∨ ω ) (cid:75) By Theorem 1 (and its proof) ≤ Ψ is a strong faithful assignment with respectto ◦ which satisfies (decrement sucess). We show the satisfaction of (DR8) to(DR12). ecrement Operators in Belief Change A.5 (DR8) Let ω , ω ∈ (cid:74) α (cid:75) . Choose β = ¬ ( ω ∨ ω ) and therefore ¬ β | = α . By (D8)we have Bel ( Ψ ◦ α ◦ β ) = α Bel ( Ψ ◦ β ) , which implies:(2) (cid:74) Ψ ◦ β (cid:75) = α (cid:74) Ψ ◦ α • β (cid:75) From (decrement sucess) we obtain (cid:74) Ψ ◦ α • β (cid:75) = (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) (3)and(4) (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Substituting (3) and (4) into Equation (2) leads to (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = α (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Now consider two cases: – Suppose (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) = ∅ . Due to the faithfulness of the assignment,we conclude min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = ¬ β (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . – For (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) (cid:54) = ∅ , from the faithfulness of the assignment we get (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . Then again, we conclude min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = ¬ β (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) .In particular, we can conclude from both cases that:(5) min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) Note that (cid:74) ¬ β (cid:75) has only two elements, (cid:74) ¬ β (cid:75) = { ω , ω } ⊆ (cid:74) α (cid:75) , and thusinformation about the minima provides us the relative order of the two ele-ments ω and ω . So, from Equation (5), we can conclude that ω ≤ Ψ ω ifand only if ω ≤ Ψ ◦ α ω .(DR9) Suppose ω , ω ∈ (cid:74) ¬ α (cid:75) . We choose β = ¬ ( ω ∨ ω ) and therefore, we have ¬ β | = ¬ α . By (D9) we have Bel ( Ψ ◦ α • β ) = ¬ β Bel ( Ψ • β ) , which implies:(6) (cid:74) Ψ ◦ β (cid:75) = ¬ β (cid:74) Ψ ◦ α • β (cid:75) From (decrement sucess) we obtain (cid:74) Ψ ◦ α • β (cid:75) = (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) (7)and(8) (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Substituting (7) and (8) into Equation (6) leads to(9) (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = ¬ β (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Now consider two cases: .6 Sauerwald and Beierle – Suppose (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) = ∅ . Due to the faithfulness of the assignment,we conclude min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = ¬ β (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . – For (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) (cid:54) = ∅ , from the faithfulness of the assignment we get (cid:74) Ψ ◦ α (cid:75) ∩ (cid:74) ¬ β (cid:75) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . Then again, we conclude min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = ¬ β (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) .In both cases we can conclude:(10) min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) Note that (cid:74) ¬ β (cid:75) has only two elements, (cid:74) ¬ β (cid:75) = { ω , ω } ⊆ (cid:74) ¬ α (cid:75) , and thusinformation about the minima provides us the relative order of the two ele-ments ω and ω . So, from Equation (10), we can conclude that ω ≤ Ψ ω ifand only if ω ≤ Ψ ◦ α ω .(DR10) First, observe that the proof of satisfaction of (DR8), (DR9), (DR11)and (DR13) are independent from showing (DR10), and hence we can safelyassume their satisfaction. By Lemma 2, ◦ fulfils (partial success), i.e.: (cid:74) Ψ (cid:75) ⊆ (cid:74) Ψ ◦ α (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) Let ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) and ω < Ψ ◦ α ω and β = ¬ ( ω ∨ ω ) . Then ω ∈ (cid:74) Ψ ◦ α • β (cid:75) and ω / ∈ (cid:74) Ψ ◦ α • β (cid:75) . We show that ω < Ψ ω . This isthe case if ω (cid:54)∈ (cid:74) Ψ • β (cid:75) and ω ∈ (cid:74) Ψ • β (cid:75) . By (partial success) ω is not anelement of (cid:74) Ψ (cid:75) . Since (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) , it remains to show that min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = { ω } . We have two cases:1. For min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = { ω } we conclude directly ω < Ψ ω .2. Now consider the case of ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) , and therefore ω ∈ (cid:74) Ψ • β (cid:75) .Let γ = γ (cid:48) ∨ α , where γ (cid:48) is a formula such that (cid:74) γ (cid:75) = (cid:74) Ψ ◦ α • β (cid:75) . Observenow that ω (cid:54)| = γ and that we have chosen γ and β such that α | = γ .From Ψ ◦ α • β | = γ we conclude Ψ • β | = γ by (D10), a contradiction to ω ∈ (cid:74) Ψ • β (cid:75) .In summary, it must be the case that ω < Ψ ω and thus, we have shownthe satisfaction of (DR10).(DR11) Suppose ω ∈ (cid:74) ¬ α (cid:75) , ω ∈ (cid:74) α (cid:75) and ω < Ψ ω . We want to show ω < Ψ ◦ α ω . For this purpose let β = ¬ ( ω ∨ ω ) . Since Ψ (cid:55)→≤ Ψ is a faithfulassignment it must be the case that ω / ∈ (cid:74) Ψ (cid:75) . By use of (decrement sucess)we can conclude that ω / ∈ (cid:74) Ψ • β (cid:75) and ω ∈ (cid:74) Ψ • β (cid:75) . Now let γ = γ (cid:48) ∨ ¬ α ,where γ (cid:48) is a formula such that (cid:74) Ψ • β (cid:75) ∪ { ω } = (cid:74) γ (cid:75) . Note that ¬ α | = γ and ω (cid:54)| = γ . By using (D11) we conclude Ψ ◦ α • β | = γ . This implies that ω / ∈ (cid:74) Ψ ◦ α • β (cid:75) . Note that by (cid:74) ¬ β (cid:75) = { ω , ω } and (decrement sucess) itmust be the case that ω ∈ (cid:74) Ψ ◦ α • β (cid:75) or ω ∈ (cid:74) Ψ ◦ α • β (cid:75) , leaving the onlyoption ω ∈ (cid:74) Ψ ◦ α • β (cid:75) . In summary, we get ω < Ψ ◦ α ω .(DR12) Let ω (cid:28) Ψ ω with ω | = α and ω | = ¬ α . This means ω < Ψ ω andthere exists no ω such that ω < Ψ ω < Ψ ω . To show that ω ≤ Ψ ◦ α ω , let γ = ¬ ω ∨ ¬ α and β = ¬ ω ∨ α. . Then, we have ¬ α | = γ and α | = β , and min( (cid:74) ω α (cid:75) , ≤ Ψ ) = min( (cid:74) ω (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ω ∨ ω (cid:75) , ≤ Ψ )min( (cid:74) ω ¬ α (cid:75) , ≤ Ψ ) = min( (cid:74) ω (cid:75) , ≤ Ψ ) (cid:54)⊆ min( (cid:74) ω ∨ ω (cid:75) , ≤ Ψ ) . ecrement Operators in Belief Change A.7 Clearly, this is equivalent to ω ≺ Ψ ω . Moreover, by Lemma 3 we have ω Î Ψ ω , and thus by (D12), we have ω (cid:22) Ψ ◦ α ω . By definition we have min( (cid:74) ω (cid:75) , ≤ Ψ ) ⊆ min( (cid:74) ω ∨ ω (cid:75) , ≤ Ψ ) from which it is easy to conclude that ω ≤ Ψ ◦ α ω .(DR13) Let ω (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) such that ω ≤ Ψ ω for all ω . Then ω ∈ (cid:74) Ψ (cid:75) and thus ω ∈ (cid:74) Ψ ◦ α (cid:75) by (D13). Clearly, then we have ω ≤ Ψ ◦ α ω .(b) to (a)-direction: Suppose that Ψ (cid:55)→≤ Ψ is a decreasing assignment with re-spect to ◦ . By Theorem 1, the belief change operator ◦ fulfils (D1) – (D7). Weshow the satisfaction of (D8) to (D13).(D8) Let ¬ β | = α . By (decrement sucess) we have to show (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = α (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . Then, by assumption,Lemma 2 and (DR8) and (DR13), it is easy to see that min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) =min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) .(D9) Let ¬ β | = ¬ α . By (decrement sucess) we have to show (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . Then, by assumption, Lemma2 and (DR9), it is easy to see that min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) .(D10) Let α | = γ and Ψ ◦ α • β | = γ . By Proposition (5) we conclude (cid:74) Ψ (cid:75) ⊆ (cid:74) γ (cid:75) and min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) ⊆ (cid:74) γ (cid:75) . Remember that ◦ satisfies (decrement sucess)and therefore, (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) . Now let ω ∈ (cid:74) ¬ β (cid:75) such that ω / ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . We show that ω / ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) or ω | = γ . Let ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) and thus, ω < Ψ ◦ α ω . We differentiate by case:1. For ω ∈ (cid:74) α (cid:75) we conclude ω | = γ directly from α | = γ .2. In the case of ω ∈ (cid:74) ¬ α (cid:75) and ω ∈ (cid:74) α (cid:75) we conclude ω < Ψ ω bycontraposition of (DR10).3. In the remaining case of ω , ω ∈ (cid:74) ¬ α (cid:75) it is easy to conclude by (DR9)that ω < Ψ ω .This shows that either ω < Ψ ω or ω | = γ , leading to the conclusion that min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ (cid:74) γ (cid:75) . In summary we have (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ (cid:74) γ (cid:75) .(D11) Let ¬ α | = γ and Ψ • β | = γ . We want to show Ψ ◦ α • β | = γ . By satisfactionof (decrement sucess) we have(11) (cid:74) Ψ • β (cid:75) = (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ (cid:74) γ (cid:75) and by Lemma 2 and (decrement sucess) we have (cid:74) Ψ ◦ α • β (cid:75) = (cid:74) Ψ ◦ α (cid:75) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) , (12) (cid:74) Ψ ◦ α • β (cid:75) ⊆ (cid:74) Ψ (cid:75) ∪ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ) ∪ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . (13)We show that every ω ∈ (cid:74) Ψ ◦ α • β (cid:75) is a model of γ . – If ω ∈ (cid:74) Ψ (cid:75) , then by Equation (11) we have ω | = γ . – For ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) assume that ω | = ¬ γ . For ω | = ¬ α , we directlyconclude ω | = γ from ¬ α | = γ . Therefore we can assume ω | = α . Since min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) ⊆ (cid:74) γ (cid:75) , there must be ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) such that .8 Sauerwald and Beierle ω < Ψ ω . If ω , ω ∈ (cid:74) α (cid:75) , then ω < Ψ ◦ α ω by (DR8). For ω ∈ (cid:74) α (cid:75) and ω ∈ (cid:74) ¬ α (cid:75) we conclude ω < Ψ ◦ α ω by (DR11). Thus it must be the casethat ω < Ψ ◦ α ω , which is a contradiction to the minimality of ω . – Suppose that ω ∈ min( (cid:74) ¬ α (cid:75) , ≤ Ψ ◦ α ) . Then, ω | = γ can be directly ob-tained from ¬ α | = γ .From Equation (13) it follows that ω | = γ , and therefore Ψ ◦ α • β | = γ .(D12) Let α | = β and ¬ α | = γ , and γ Î Ψ β . We show now that β (cid:22) Ψ ◦ α γ , whichis the case when min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) ⊆ min( (cid:74) ¬ β ∨ ¬ γ (cid:75) , ≤ Ψ ◦ α ) .First, observe that min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ◦ α ) ⊆ (cid:74) α (cid:75) and min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) ⊆ (cid:74) ¬ α (cid:75) .Thus for every ω ∈ min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) and every ω ∈ min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) we have ω ∈ (cid:74) α (cid:75) and ω ∈ (cid:74) ¬ α (cid:75) . Therefore, by γ Î Ψ β and Lemma 3 we have twocases: – In the case of ω Î Ψ ω we conclude by (DR12) that ω ≤ Ψ ◦ α ω . – In the case of ω (cid:39) Ψ ω we have ω ≤ Ψ ω , and hence, by (DR10), wehave ω ≤ Ψ ◦ α ω .From (DR8) and (DR9) we get min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ) = min( (cid:74) ¬ γ (cid:75) , ≤ Ψ ◦ α ) and min( (cid:74) ¬ β (cid:75) , ≤ Ψ ) = min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) . In summary, we have min( (cid:74) ¬ β (cid:75) , ≤ Ψ ◦ α ) ⊆ min( (cid:74) ¬ β ∨ ¬ γ (cid:75) , ≤ Ψ ◦ α ) , which is equivalent to β (cid:22) Ψ ◦ α γ .(D13) Let ω ∈ (cid:74) Ψ (cid:75) . If ω ∈ (cid:74) α (cid:75) , then by (DR13) and (DR8) we have ω ∈ (cid:74) Ψ ◦ α (cid:75) .In the case of ω ∈ (cid:74) ¬ α (cid:75) we have ω ∈ (cid:74) Ψ ◦ α (cid:75) by (DR9) and (DR10).In summary, Ψ (cid:55)→≤ Ψ is a decreasing assignment.is a decreasing assignment.