Dynamic Preference Logic meets Iterated Belief Change: Representation Results and Postulates Characterization
aa r X i v : . [ c s . L O ] J a n Dynamic Preference Logic meets Iterated BeliefChange: Representation Results and PostulatesCharacterization
Marlo Souza a,1 , Renata Vieira b , ´Alvaro Moreira c a Institute of Mathematics and Statistics, Federal University of Bahia - UFBAAv. Adhemar de Barros, S/N, Ondina - Salvador-BA, Brazil b Faculty of Informatics, Pontifical Catholic University of Rio Grande do Sul - PUCRSAv. Ipiranga, 6681 - Porto Alegre-RS, Brazil c Institute of Informatics, Federal University of Rio Grande do Sul- UFRGSAv. Bento Gon¸calves, 9500 - Porto Alegre-RS, Brazil
Abstract
AGM’s belief revision is one of the main paradigms in the study of belief changeoperations. Recently, several logics for belief and information change have beenproposed in the literature and used to encode belief change operations in rich andexpressive semantic frameworks. While the connections of AGM-like operationsand their encoding in dynamic doxastic logics have been studied before by thework of Segerberg, most works on the area of Dynamic Epistemic Logics (DEL)have not, to our knowledge, attempted to use those logics as tools to investigatemathematical properties of belief change operators. This work investigates howDynamic Preference Logic, a logic in the DEL family, can be used to studyproperties of dynamic belief change operators, focusing on well-known postulatesof iterated belief change.
Keywords:
Dynamic Epistemic Logic, Dynamic Preference Logic, BeliefRevision
1. Introduction
Belief Change is the multidisciplinary area that studies how a doxastic agentcomes to change her mind after acquiring new information. The most influentialapproach to Belief Change in the literature is the AGM paradigm [1]. ˚ Corresponding author
Email addresses: [email protected] (Marlo Souza), [email protected] (RenataVieira), [email protected] ( ´Alvaro Moreira) List of Abbreviations: DEL - Dynamic Epstemic Logic, DPL - Dynamic Preference Logic,SOS- System of Spheres, OCF - Ordinal Conditional Function, m.e. - modally equivalent,PDL - Propositional Dynamic Logic, DDL - Dynamic Doxastic Logic.
Preprint submitted to Elsevier Received: date / Accepted: date
GM defines belief change operations by structural constraints on how thebeliefs of an agent should change. However, it has been argued in the litera-ture, that belief change operations should be defined by means of changes inthe agent’s epistemic state - understood as more than the currently held (un-conditional) beliefs but including also the agent’s dispositions to believe [2, 3].Aiming to extend the AGM framework to account for this idea, several workshave established what became known as
Iterated Belief Change .While the rational constraints for changes in an agent’s mental attitudeshave been well-investigated in areas such as Epistemology and Logic, e.g., [1, 4],the integration of belief change within the logics of beliefs and knowledge isa somewhat recent development. The first of such attempts was proposed bySegerberg [5, 6] with his Dynamic Doxastic Logic (DDL).This shift from the extra-logical characterisation of belief change to its in-tegration within a representation language has important expressiveness con-sequences. It also allows for the exploration of established results from ModalLogic to construct applications of the AGM belief change theory. The work ofSegerberg, Lindstr¨om, and Rabinowicz on DDL [5, 7], for example, show thatthe dynamics of introspective and higher-order beliefs poses problems to theAGM framework as they are not compatible with the AGM postulates. Theiranalysis of the difference between reasoning about the state in which beliefs areheld (the point of evaluation) and reasoning about the state in which certainthings are believed (the point of reference) corresponds directly to the analy-sis of the Ramsey Conditional Test by Baltag and Smets [8] in the context ofDynamic Epistemic Logic. Such integration has also proven to be fruitful inthe area of Interactive Epistemology. Reny [9], for example, shows that if theagent is allowed to revise her beliefs, backward induction is not supported by thecommon belief of rationality. Further, Board [10] shows that the framework ofextensive-form epistemic games extended with belief change is a rich frameworkfor unifying different competing notions in the area.As such, it is clear that incorporating belief change within epistemic anddoxastic logics may provide useful theoretical insights on the phenomena relatedto belief dynamics, as the fact that the variety of doxastic attitudes may giverise to a variety of belief change operations satisfying their formal properties.However, the investigation of the mathematical properties satisfied by thesebelief change operators is rarely pursued by these works.Recently, Girard [11, 12] proposes Dynamic Preference Logic (DPL) to studygeneralisations of belief revision a la
AGM [11, 12]. Some works have integratedwell-known belief change operators within this logic [11, 13, 14, 15] and haveused them to study the dynamic behaviour of attitudes such as Preferences,Beliefs, and Intentions. Often, it is unclear whether these dynamic logics canbe used to express the desirable properties of the belief change operators theystudy.Souza et al. [16] have demonstrated that belief change postulates can beencoded within DPL, showing that any belief change operator satisfying thesepostulates induce the validity of some axioms in their corresponding dynamiclogic. This result allows the use of DPL as a language to reason about classes2f Belief Change operators. However, the authors were not able to completelycharacterise the postulates within DPL, i.e., to provide axioms in the dynamiclogic that guarantee the satisfaction of such postulates by the correspondingbelief change operator.In this work, we study the relationship between the iterated belief changepostulates satisfied by belief change operators and the axioms valid in DPLusing these operators. We provide generalisations of some known iterated beliefchange postulates to the context of relational preference models and providerepresentations of these postulates within DPL, which completely characterisethem.The representation results obtained in our work highlight which relation-changing operators over a given class of preference models can be considered toprovide the semantics of the logic, in a similar sense as to how frame propertiescan be defined by means of modal axioms for relational semantics.We point out that in this work, we choose to employ our preference mod-els as opposed to Grove’s Systems of Spheres to study belief and its changes.The reason for this is that, firstly, preference-like models are more generalstructures and have been extensively studied as models for non-monotonic rea-soning, conditional logics, and mental attitudes, as variant notions of belief[17, 18, 19, 20, 21, 8]. By employing these models to study belief change opera-tors, we may generalise our results and the insights obtained with our study tochange operations for other mental attitudes and diverse dynamic phenomena.The following are the main contributions of our work: (i) Theorem 52, whichproves that we can combine the axiomatisations obtained for each postulatesatisfied by a class of operators, resulting in a sound axiomatisation of the logicinduced by this class of operators, and (ii) the representation results for theiterated belief change postulates provided in Propositions 26, 30, 33, 36, 39, 43,45, and 47.Theorem 52 also establishes a method for deriving axiomatizations of dy-namic operators in DPL. Our method is different from the ones proposed byVan Benthem and Liu [22], and by Aucher [23], which are based on PropositionalDynamic Logic without iteration. It uses the extensively studied postulates fromBelief Change to derive an axiomatisation of the logic with a given operation.As such, our method applies to a wider class of dynamic belief change opera-tors, including those that cannot be encoded using Propositional Dynamic Logicprograms, e.g., Ramachandram et al.’s [24] Lexicographic Contraction.This work is an extended version of the work presented at the 8th BrazilianConference on Intelligent Systems - BRACIS 2019 [25] and includes severaldevelopments that resulted from questions left open in our original work, aswell as those from discussions during the event. As the main new developmentsnot present in the BRACIS paper we list: • the correction of the generalised postulates p DP1a q , p DP1b q , p DP2a q , and p DP2b q to account for the changes in the strict part ă of the preferencerelation; • the correction of the generalised postulate p Rec q to exclude unwanted links3etween ϕ and ϕ worlds, consistent with Nayak et al.’s [26] original p Rec q postulate, in Definitions 25, 29 and 38; • the extension of our analysis to postulates for iterated belief contraction,namely Chopra et al.’s [27] p CR1 q - p CR4 q and Ramachandran et al.’s [24] p LC q , in Propositions 26, 30, 43, 45 and 47; • an extended presentation and discussion of DPL and the results relatedto this logic which are important to our characterisation and discussions,in Section 3; • a better formalisation of the semantics of DPL, in Definition 17, whichallows us to interpret the representation results in Section 4 as constraintson the class of models used to interpret the language; • generalisations of all results showing that the representation results canbe used to define satisfaction of the analysed postulates within any classof preference models, in Section 4; • the proof that the original postulates cannot be represented within DPL,exemplified by Fact 23 which can be restated for all the other postulatesstudied in this work; • a more general soundness result for the resulting axiomatisation, in Theo-rem 52, showing a stronger relationship between proof systems and modelsfor the logic.This work is structured as follows: in Section 2, we discuss AGM beliefchange and the results and postulates in the literature of dynamic and iteratedbelief change; in Section 3, we introduce DPL, a logic in the tradition of DynamicEpistemic Logic recently applied to study belief change; in Section 4 we presentour main results: we investigate the relationship between the iterated beliefchange postulates satisfied by belief change operators and the axioms valid inDPL, as well as providing generalisations of some postulates of belief change forthe context of relational preference models. In Section 5, we discuss the relatedwork, and finally, in Section 6, we present our final considerations. Proofs ofthe main results in this work can be found in the Appendix.
2. Preliminaries
Let us consider a logic L “ x L, Cn y where L is the logical language and Cn : 2 L Ñ L is a consequence operator. In AGM’s approach, the belief stateof an agent is represented by a belief set, i.e. a consequentially closed set B Ď L ,with B “ Cn p B q of L -formulas.In this framework, a belief change operator is any operation ‹ : 2 L ˆ L Ñ L that, given a belief set B and some information ϕ , changes the belief set insome way. AGM investigated three basic belief change operators: expansion,contraction, and revision. Belief expansion blindly integrates a new piece of4nformation into the agent’s beliefs. Belief contraction removes a currently be-lieved sentence from the agent’s set of beliefs, with minimal alterations. Finally,belief revision is the operation of integrating new information into an agent’sbeliefs while maintaining consistency.Among these basic operations, only expansion can be univocally defined.The other two operations are defined by a set of rational constraints or postu-lates, usually referred to as the AGM postulates or the G¨ardenfors postulates.These postulates define a class of suitable change operators representing differ-ent rational ways in which an agent can change her beliefs. Let B Ď L be abelief set and α, β P L L -formulas. For the revision operation ( ˚ ), the authorsintroduce the following postulates: p R1 q B ˚ α “ Cn p B ˚ α qp R2 q α P B ˚ α p R3 q B ˚ α Ď Cn p B Y t α uqp R4 q If α R B , then B ˚ α “ Cn p B Y t α uqp R5 q B ˚ α “ Cn ptKuq iff $ α p R6 q If $ α Ø β then B ˚ α “ B ˚ β p R7 q B ˚ p α ^ β q Ď Cn pp B ˚ α q Y t β uqp R8 q If β R B ˚ α , then Cn pp B ˚ α q Y t β uq Ď B ˚ p α ^ β q Belief expansion, contraction, and revision are interconnected by the pro-perties known as Levi and Harper identities [1].While the AGM approach is independent of the supporting logic’s syntax,it lacks a clear semantic interpretation for its operations. Grove [28] providedone such interpretation using a possible-world semantics, based on Lewis’ [29]spheres. Grove’s model for the operation of belief revision has clarified themeaning of belief change operations and has become a necessary tool for thedevelopment of new methods and operations in the area, such as the iteratedbelief operations we will discuss later.A Grove system of spheres (SOS) is a pair S “ x W, ďy where W is a set ofmodels for the logic L and ď Ď W ˆ W satisfies the following conditions : p i qď is connected, p ii q ď is transitive, and p iii q for any S Ď W , if S ‰ H , thenexists x P S minimal in ď in regards to S .Given a system of spheres S “ x W, ďy , we say that the set of models of aformula ϕ P L is the set J ϕ K “ t w P W | w ( L ϕ u , and for a set of modelsΓ Ď W we say the set of minimal models of Γ is: M in ď Γ “ t w P Γ | E w P Γ s.t. w ď w ^ w ę w u Grove shows that for any belief revision operator ˚ satisfying the AGM pos-tulates p R1 q - p R8 q and any belief set B , there is a system of spheres S B “ x W, ďy such that w P M in ď W iff w ( B and J B ˚ ϕ K “ M in ď J ϕ K . Grove also requires the property of Universality that W is a complete class of models forthe logic L .
5f we take a SOS as representing an agent’s belief state, however, we canview a revision operation ˚ as an operation that changes a system of spheres S B “ x W, ďy into a system S B ˚ ϕ “ x W, ď ˚ ϕ y . In that case, we can thuscharacterise AGM’s revision operators by the p Faith q postulate below [26]: p Faith q if J ϕ K ‰ H , Min ď J ϕ K “ Min ď ˚ ϕ W Similarly, AGM introduce a set of postulates p C1 q - p C8 q to define the be-lief change operation of contraction, which we will omit in this work. Moreimportantly for us, Grove also shows that for any belief contraction opera-tor ´ satisfying the AGM postulates p C1 q - p C8 q and any belief set B , thereis a system of spheres S B “ x W, ďy such that w P M in ď W iff w ( B and J B ´ ϕ K “ Min ď W Y M in ď J ϕ K .As before, if we take a SOS as representing an agent’s belief state, we canview a contraction operation ´ as an operation that changes a system of spheres S B “ x W, ďy into a system S B ´ ϕ “ x W, ď ´ ϕ y . In that case, we can characteriseAGM’s contraction operators by the p GR q postulate below: p GR q M in ď ´ ϕ W “ Min ď W Y Min ď J ϕ K .In the following, we will usually denote by ď ‹ ϕ the resulting relation ofperforming a belief change operation, denoted by some operation symbol ‹ suchas ´ or ˚ , with input ϕ over a relation ď or, more formally, on a SOS S “ x W, ďy .While AGM [1] does not commit to a single logic, in this work, we will focus onpropositional classical logic as an object language. AGM belief change says very little about how to change one agent’s beliefsrepeatedly. In fact, it has been observed that the AGM approach allows somecounter-intuitive behaviour in the iterated case.One example of such behavior is the following situation [3]: suppose we areintroduced to a lady X who sounds smart and looks rich, so we believe thatX is smart and X is rich. Moreover, since we profess to no prejudice, we alsomaintain that X is smart even if found to be poor and, conversely, X is rich evenif found to be not smart. Now, we obtain some evidence that X is not smart,and we remain convinced that X is rich. Still, it would be strange for us to say,“If the evidence turns out false, and X turns out smart after all, we would nolonger believe that X is rich”. If we currently believe that X is smart and rich,then evidence first refuting then supporting that X is smart should not changeour opinion about X being rich. Strangely, the AGM postulates do permit sucha change of opinion.Belief Change is a continuous process in that new information can alwaysbe accommodated into the agent’s epistemic state, and successive pieces of in-formation must be incorporated in a principled and coherent manner. Thisfact is implicitly recognized by AGM, as evidenced by postulates p R7 q and p R8 q ,although the authors do not discuss iteration of change explicitly.To our knowledge, Spohn [2] was one of the firsts to consider this problem.Spohn [2] argues for defining belief changes over the agent’s belief state, and6ot only over the agent’s held beliefs. Further, the result of a change must bean epistemic state in itself, and this is particularly important, argues the authorif we consider repeated epistemic changes. Otherwise, we could not determinethe resulting state of a successive change from the initial belief state and theacquired information. To provide an appropriate account for dynamic beliefchange, the author proposes his semantic framework of Ordinal ConditionalFunctions.Spohn’s work influenced several authors, in particular Darwiche and Pearl [3],who propose the further constraints on AGM’s belief change operators to ac-count for epistemic consistency across repeated changes. The authors’ proposalcan be thought of as encoding Spohn’s explicitly defined conditional statementswithin AGM’s framework and establishes constraints (or postulates) that gov-ern how the agent’s conditional beliefs are changed. Importantly, the authorspresent their postulates in terms of both syntactic structures (sets of formulas)and semantic structures (Grove’s spheres), which allows us to contrast their pos-tulates with those works based on the AGM framework and with those basedon OCF. Here, we focus on the semantic characterisation of these postulates,presented below. p DP1 q If w, w P J ϕ K , then w ď ‹ ϕ w iff w ď w p DP2 q If w, w R J ϕ K , then w ď ‹ ϕ w iff w ď w p DP3 q If w P J ϕ K and w R J ϕ K , then w ă w implies w ă ‹ ϕ w p DP4 q If w P J ϕ K and w R J ϕ K , then w ď w implies w ď ‹ ϕ w Nayak et al. [26] argue that the DP postulates are over-permissible, in thesense that they allow revision operators with undesirable properties, such asBoutilier’s [30] Natural Revision, which is criticized by Darwiche and Pearl [3]themselves as having counter-intuitive behaviour.Then, Nayak et al. [26] propose the operation of Lexicographic Revision,which can be characterised by the postulates p DP1 q , p DP2 q , and p Rec q below.The axiom of recalcitrance states that if two pieces of information ϕ and ψ areconsistent with each other, then if we obtain the information ϕ and, later, theinformation ψ , there are no grounds to discard ϕ . In terms of changes in therelation of a SOS, this can be stated as: p Rec q If w P J ϕ K and w R J ϕ K , then w ă ‹ ϕ w . Definition 1.
Let ď Ď W ˆ W be a total pre-order over W and ϕ a proposi-tional formula. The lexicographic revision of ď by information ϕ is the relation ď ˚ ϕ Ď W ˆ W satisfying postulates p DP1 q , p DP2 q and p Rec q . While iterated belief revision has been extensively studied, iterated beliefcontraction has received far less attention in the literature. AGM [1] show a deepconnection between revision and contraction on the single-shot case, by meansof the Levi and Harper identities, in the sense that these operations are inter-definable. It is not clear, however, how this connection can be extended to the7terated belief change. In fact, several works on iterated belief contraction stemfrom trying to establish a connection between iterated revision and contraction.Particularly, Chopra et al. [27] investigating the role of AGM’s recoveryprinciple [1] for iterated contraction, propose iterated contraction postulates inthe light of Darwich and Pearl’s postulates for iterated revision. The authorsstate their postulates in terms of Grove’s SOS as: p CR1 q If w, w R J β K then w ď w iff w ď ‹ β w p CR2 q If w, w P J β K then w ď w iff w ď ‹ β w p CR3 q If w P J ϕ K and w P J ϕ K , then w ă w implies w ă ‹ ϕ w p CR4 q If w P J ϕ K and w P J ϕ K , then w ď w implies w ď ‹ ϕ w Nayak et al. [31] propose several iterated contraction operators and analysetheir properties. Among them, we highlight Lexicographic Contraction, whichthe authors describe as a dual form of Nayak et al.’s Lexicographic Revision.To define this operator, the authors propose postulate p LC q which changes theagent’s belief state in a manner such that the plausibility attributed to eachpossible world w is determined solely by their relative position according to theworlds satisfying ϕ (or ϕ ), if w satisfies ϕ ( ϕ ). This is equivalent to statethat an agent maintains a conditional belief B p ξ | ψ q if, and only if, this beliefis independent of the agent’s attitude towards ϕ , i.e. she also holds the beliefthat B p ξ | ψ ^ ϕ q and B p ξ | ψ ^ ϕ q . p LC q Let ξ be a member of t ϕ, ϕ u and ξ the other. If w ( ξ and w ( ξ ,then w ď ‹ ϕ w iff there is a chain w , ă w , ¨ ¨ ¨ ă w n of worlds in J ξ K ofmaximal length which ends in w , and there is a chain w , ă w , ¨ ¨ ¨ ă w k of worlds in J ξ K which ends in w and n ď k .We point out that Lexicographic Contraction, as defined by [31, 32] is acontraction operation constructed on the basis of their proposed GeneralisedHarper Identity (GHI). By its relation with GHI, lexicographic contraction is acontraction operation based on degrees of plausibility of the possible worlds -encoded in Grove’s spheres. This is particularly clear in the fact that the ax-iomatic characterisation of this operation in [24] can only be achieved by meansof the richer framework of degrees of belief. This connection will be importantin defining Lexicographic Contraction for preference models in Section 3. Withthe postulate p LC q , the authors define the operation of lexicographic contraction. Definition 2.
Let ď Ď W ˆ W be a total pre-order over W and ϕ a propo-sitional formula. The lexicographic contraction of ď by information ϕ is therelation ď ´ ϕ Ď W ˆ W satisfying postulates p DP1 q , p DP2 q and p LC q .In this work, we explore how the properties (or postulates) discussed in thissection can be encoded inside DPL, i.e. how we can guarantee that a given B p ξ | ψ q stands for the belief that ψ conditionally entails ξ , also denoted as ψ ñ ξ in theliterature related to non-monotonic reasoning.
3. Dynamic Preference Logic
Preference Logic (or Order Logic, as named by Girard [11]) is a modal logiccomplete for the class of transitive and reflexive frames. It has been applied tomodel a plethora of phenomena in Deontic Logic [21], Logics of Preference [34],Logics of Belief [8], etc. Dynamic Preference Logic (DPL) [11] is the resultof “dynamifying” Preference Logic, i.e., extending it with dynamic modalities.This logic is one example among several Dynamic Epistemic Logics, and it isparticularly interesting for its expressiveness, allowing the study of dynamicphenomena of attitudes such as Beliefs, Obligations, Preferences etc.We begin our presentation with the language and semantics of PreferenceLogic, which we will later “dynamify”. Let’s introduce the language of Prefer-ence Logic.
Definition 3.
Let P be a set of propositional letters. We define the language L ď p P q by the following grammar (where p P P ): ϕ :: “ p | ϕ | ϕ ^ ϕ | Aϕ | rďs ϕ | răs ϕ We will often refer to the language L ď p P q simply as L ď , by supposing theset P is fixed. Also, we will denote the language of propositional formulas, i.e.,the language removing all modal formulas from L ď p P q , by L p P q or simply L Definition 4. A well-founded preference model is a tuple M “ x W, ď , v y where W is a set of possible worlds, ď is a a reflexive, transitive relation over W , s.tits strict part ( ă ) is well-founded , and v : P Ñ W a valuation function.In such a model, the accessibility relation ď represents an ordering of thepossible worlds according to the preferences of a certain agent. As such, giventwo possible worlds w, w P W , we say that w is at least as preferred as w if w ď w .The interpretation of the formulas over these models is defined as usual. The A modality is a universal modality satisfied iff all worlds in the model satisfyits argument. The rďs modality is a box modality on the accessibility order ď . A relation R Ď W ˆ W is said well-founded if there is no infinite descending chains, i.e.,for any H ‰ S Ď W , Min ď S ‰ H . In this work, we understand the worlds as epistemically possible worlds, not metaphysi-cally possible. While formally this difference is irrelevant, philosophically it is of importance. răs modality is the strict variant of rďs . They are interpreted as:
M, w ( p iff w P v p p q M, w ( ϕ iff M, w * ϕM, w ( ϕ ^ ψ iff M, w ( ϕ and M, w ( ψM, w ( Aϕ iff @ w P W : M, w ( ϕM, w ( rďs ϕ iff @ w P W : if w ď w then M, w ( ϕM, w ( răs ϕ iff @ w P W : if w ă w then M, w ( ϕ As usual, we will refer as Eϕ to the formula A ϕ , meaning ‘ it is possiblytrue that ϕ ’, and as xďy ϕ ( xăy ϕ ) to the formula rďs ϕ ( răs ϕ ), meaning‘ in a possible situation at least as (strictly more) preferable as the current one, ϕ holds ,’ as commonly done in modal logic.For simplicity, in this work, we will refer to well-founded preference modelsonly as preference models. Notice, however, that in the literature, e.g., [11],preference models need not be well-founded.Given a preference model M and a formula ϕ , we use the notation J ϕ K M ,as also defined for systems of spheres, to denote the set of all the worlds in M satisfying ϕ , or only J ϕ K when the model is clear from the context. The notation M in ď X denotes the ‘most preferred worlds in X ’ in the model, i.e., the minimalelements in a set of possible worlds X , according to the (pre-)order ď .As the concept of most preferred worlds satisfying a given formula ϕ will beof great use in modelling different notions of belief (and different belief changeoperations) in this logic, we define a formula encompassing this exact concept: µϕ ” def ϕ ^ xăy ϕ. It is easy to see that µϕ defines exactly the set of minimal worlds satisfying ϕ . Proposition 5. [13] Let M “ x W, ď , v y be a preference model and ϕ P L ď aformula. For any w P W , M, w ( µϕ iff w P M in ď J ϕ K M . Notice that Preference Logic is highly expressive. It is well known [35, 11,13, 33] that we can define unconditional and conditional preferences, i.e. thenotions of ‘ the agent prefers that ϕ ’ and ‘ in the case of ψ , the agent wouldprefer that ϕ ’. In the context of Belief Change, we can interpret this preferenceas belief and, as such, we can encode in this logic, as previously done by [35, 13],the notion of conditional belief as B p ϕ | ψ q ” def A p µψ Ñ ϕ q and unconditional belief as B p ϕ q ” def B p ϕ |Jq . These encodings preserve the usual notion of conditional belief as the “ themost plausible worlds satisfying ϕ also satisfy ψ ”.10 orollary 6. Let M “ x W, ď , v y be a preference model, w P W be a possibleworld, and ϕ, ψ P L ď be preference formulas. M, w ( B p ψ | ϕ q iff M in ď J ϕ K Ď J ψ K Souza [33] provided the axiomatisation depicted in Figure 1 below for thelogic and showed that it is weak-complete [36] for Preference Logic restricted towell-founded models.
Figure 1: Axiomatization for Preference Logic CP All axioms from Classical Propositional Logic K ď : rďsp ϕ Ñ ψ q Ñ prďs ϕ Ñ rďs ψ q T ď : rďs ϕ Ñ ϕ ď : rďs ϕ Ñ rďsrďs ϕ K ă : răsp ϕ Ñ ψ q Ñ prăs ϕ Ñ răs ψ q W ă : răsprăs ϕ Ñ ϕ q Ñ răs ϕ ăď : rďs ϕ Ñ răs ϕ ăď : răs ϕ Ñ răsrďs ϕ ăď : răs ϕ Ñ rďsrăs ϕ ăď : rďsprďs ϕ _ ψ q ^ răs ψ Ñ ϕ _ rďs ψ K A : A p ϕ Ñ ψ q Ñ p Aϕ Ñ Aψ q T A : Aϕ Ñ ϕ A : Aϕ Ñ AAϕ B A : ϕ Ñ A A ϕA ď : Aϕ Ñ rďs ϕ p Necessitation q $ ϕ ñ $ l ϕ, with l P trďs , răs , A up Modus Ponens q Γ $ ϕ and Γ $ ϕ Ñ ψ ñ Γ $ ψ We will now reintroduce some of the belief change operations presented inSection 2 as transformations of preference models.With this, we will dynamifyPreference Logic by introducing dynamic modalities in the language to representthe execution of a belief change operation in the agent’s epistemic state.We start with lexicographic revision of an epistemic state by a formula ϕ ,which consists of making each world satisfying ϕ strictly more preferable thanthose not satisfying it, while maintaining the order otherwise. Definition 7. [11] Let M “ x W, ď , v y be a preference model and ϕ P L . Wesay the model M ò ϕ “ x W, ď ò ϕ , v y is the result of the lexicographic revision of M by ϕ , where w ď ò ϕ w iff $’&’% w ď w and w, w P J ϕ K , or w ď w and if w, w R J ϕ K , or w P J ϕ K and w R J ϕ K igure 2: Reduction axioms for the Lexicographic Revision rò ϕ s p Ø p rò ϕ s ψ Ø rò ϕ s ψ rò ϕ sp ψ ^ ξ q Ø rò ϕ s ψ ^ rò ϕ s ξ rò ϕ s Aψ Ø A prò ϕ s ψ qrò ϕ srďs ψ Ø ϕ Ñ rďsp ϕ Ñ rò ϕ s ψ q^ ϕ Ñ p A p ϕ Ñ rò ϕ s ψ q ^ rďsp ϕ Ñ rò ϕ s ψ qqrò ϕ srăs ψ Ø ϕ Ñ răsp ϕ Ñ rò ϕ s ψ q^ ϕ Ñ p A p ϕ Ñ rò ϕ s ψ q ^ răsp ϕ Ñ rò ϕ s ψ qq$ ψ ñ $ rò ϕ s ψ We can now introduce the modality rò ϕ s in the language of L ď , where rò ϕ s ψ is read as “after the lexicographic revision by ϕ , ψ holds”. Definition 8.
We define the language of Preference Logic extended with Lex-icographic Revision, denoted by L ď pòq , as the language constituted of all for-mulas of L ď as well as any formula rò ϕ s ψ s.t. ϕ P L and ψ P L ď pòq . Moreyet, let M “ x W, ď , v y be a preference model, w P W and ϕ a formula of L M, w ( rò ϕ s ψ if M ò ϕ , w ( ψ . Liu [13] has shown that Preference Logic extended with Lexicographic Revi-sion is completely axiomatised by the axioms for Preference Logic extended bythe reduction axioms and rules depicted in Figure 2. The authors obtain suchaxiomatization using a Propositional Dynamic Logic (PDL) [37] codification ofLexicographic Revision to derive the axioms [22].A PDL program is a regular expression over some set of symbols representingbasic relations in some interpretation set. Fisher and Laudner [37] introducedPDL programs in their propositional variation of Pratt’s [38] Dynamic Logic torepresent computer programs. Van Benthem and Liu [22] have used them toencode belief change operators. For space constraints, we do not include in thispaper a detailed explanation about PDL programs. We refer the reader to [39]for more details.As for Lexicographic Revision, we can provide axiomatisations for Prefer-ence Logic extended with other belief change operations using their codificationin PDL. However, since some operations cannot be represented as PDL pro-grams, this strategy is not viable for all known belief change operations. Togive an example of such an operation, let’s define Lexicographic Contraction asa transformation on preference models. Later on, in Section 4, we will providea method to derive an axiomatization for Preference Logic extended with Lex-icographic Contraction that does not relies on PDL encoding of belief changeoperators.To encode Lexicographic Contraction in DPL, we need to be able to encodethe notion of a chain of worlds of a given size i all of which satisfy a formula ϕ ,a notion connected to that of degree of belief, denoted by dg ϕ p i q :12 efinition 9. Let M “ x W, ď , v y be a preference model, ϕ a formula of L ,and i P N a natural number. We define the formula dg ϕ p i q as: dg ϕ p i q “ ϕ if i “ ϕ ^ xăy dg ϕ p i ´ q if i ą i , such that dg ϕ p i q is satisfied by some world w in a model M , called the implausibility degree of w in M , is the size of themaximal chain of ϕ -worlds ending with the world w . Lemma 10. [33] Let M “ x W, ď , v y be a preference model, ϕ a formula of L and w P W . M, w ( dg ϕ p i q , i ą iff there is a chain of worlds of w ă w . . . ă w i , such that, w j P J ϕ K , for all j “ ..i , w P M in ď J ϕ K , and w i “ w . Also, if M, w ( µ dg ϕ p i q , then there is no other chain of ϕ -worlds ending in w of greatersize. Notice that the operation of lexicographic contraction, as characterised byRamachandran et al. [24], is not closed for preference models (see Definition 2).The reason for this is that their characterisation assumes the pre-order over thepossible worlds is total. In fact, in the case of preference models, the axioms p DP1 q and p LC q are incompatible, since they may result in loss of transitivity.As such, we propose the following modification for preference models. p LC q : Let ξ, χ be members of t ϕ, ϕ u - not necessarily distinct. If w ( ξ and w ( χ , then w ď ‹ ϕ w iff the maximal length of a chain of ξ -worldswhich ends in w is smaller or equal to the maximal length of a chain of χ -worlds which ends in w .Notice that postulate p LC q is rather strong compared to the original p LC q postulate. The reason for this is that p LC q is an amalgamation of three postu-lates, namely (LC), and the modifications of (DP1), (DP2) below. p DP1 q If w, w P J ϕ K , then w ď ‹ ϕ w iff the maximal length of a chain of ϕ -worlds which ends in w is smaller or equal to the maximal length of achain of ϕ -worlds which ends in w . p DP2 q If w, w R J ϕ K , then w ď ‹ ϕ w iff the maximal length of a chain of ϕ -worlds which ends in w is smaller or equal to the maximal length of achain of ϕ -worlds which ends in w .From a philosophical point of view, lexicographic contraction is a contrac-tion operation based on degrees of plausibility of the possible worlds - encodedin Grove’s spheres. As such, based on degrees of plausibility alone, p DP1 q and p DP2 q (and thus p LC q ) are justified modifications for preference models,since they imply that in the resulting epistemic state comparability between theworlds is determined by their degree of plausibility.With postulate p LC q we can define a lexicographic contraction operator overpreference models - which coincides with Ramachandran et al.’s [24] on Grovemodels. 13 efinition 11. Let M “ x W, ď , v y be a preference model and ϕ a formula of L . We say the model M ó ϕ “ x W, ď ó ϕ , v y is the lexicographic contraction of M by ϕ , where: w ď ó ϕ w iff $’’’&’’’% w P J µdg ϕ p i q K and w P J µdg ϕ p j q K i ď j , or w P J µdg ϕ p i q K and w P J µdg ϕ p j q K i ď j , or w P J µdg ϕ p i q K and w P J µdg ϕ p j q K i ď j , or w P J µdg ϕ p i q K and w P J µdg ϕ p j q K i ď j Again, we can introduce the modality ró ϕ s in the language of L ď , where ró ϕ s ψ is read as “after the lexicographic contraction by ϕ , ψ holds”. Definition 12.
We define the language of Preference Logic extended with Lex-icographic Contraction, denoted by L ď póq , as the language constituted of allformulas of L ď as well as any formula ró ϕ s ψ s.t. ϕ P L and ψ P L ď póq . Moreyet, let M “ x W, ď , v y be a preference model, w P W and ϕ a formula of L M, w ( ró ϕ s ψ if M ó ϕ , w ( ψ While the axiomatisations provided in the literature for the other beliefchange operators were constructed by means of PDL representations of theseoperations, this technique cannot be applied to Lexicographic Contraction dueto the simple fact that this operation is not regular.
Fact 13.
Lexicographic Contraction cannot be encoded by means of PDL pro-grams
To provide an axiomatisation for this operation, Souza [33] shows that if werestrict the logic’s semantics to consider only models with chains of a maximumfinite size then an axiomatisation can be achieved by means of the PDL repre-sentation of Lexicographic Contraction. Later in Section 4, we will employ anew technique, not based in PDL, to derive an axiomatisation for the extendedlanguage, which does not require such a drastic restriction in its semantics.
We define a dynamic operation on a preference model as any operation thattakes a preference model and a propositional formula and changes only thepreference relation of the model. Let
Mod p L ď q denote the class of all preferencemodels for the language L ď , with the syntax given in Definition 3. Definition 14.
Let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q , we say ‹ is a dynamicoperator on preference models if for any preference model M “ x W, ď , v y andpropositional formula ϕ P L , ‹p M, ϕ q “ x W ‹ , ď ‹ , v ‹ y , with W ‹ “ W and v ‹ “ v .We limited our dynamic operators not to change the set of possible worlds orvaluations. This limitation is justified by the fact that we are considering beliefchanging operators, i.e., mental actions which change the plausibility the agent14ttributes to each epistemically possible world, not creating any new epistemiccertainty (knowledge) nor having effects on the world.Given a dynamic operator ‹ , we extend the language L ď with formulas r‹ ϕ s ξ . Definition 15.
Let ‹ be a symbol for dynamic operators. We define the lan-guage L ď p‹q as the smallest set containing L ď and all formulas r‹ ϕ s ξ , with ϕ P L and ξ P L ď p‹q .As for the case of DPL of Lexicographic Revision and DPL of LexicographicContraction (c.f. Definitions 8 and 12) presented before, the formulas of L ď p‹q must be interpreted over a preference model and a dynamic operator that de-scribes the changes in the model. As such, we will introduce the notion ofdynamic preference model containing these two ingredients. Definition 16. A dynamic preference model , or simply a dynamic model , is atuple x M, ‹y , where M P Mod p L ď q is a preference model and ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q is a dynamic operator.In the definition above, ‹ is used both as a symbol denoting a dynamicoperator, seem as a function, and also as a symbol in the object logic language.With that, we can define how formulas in the language L ď p‹q are interpreted. Definition 17.
Let D “ x M, ‹y be a dynamic model, with M “ x W, ď , v y , w P W be a possible world, and ξ P L ď p‹q be a dynamic preference formula.We define the satisfiability of ξ by w in D , denoted by D, w ( ξ , as follows: D, w ( p if w P v p p q D, w ( ϕ ^ ψ if D, w ( ϕ and D, w ( ψD, w ( ϕ if D, w * ϕD, w ( Aϕ if, for all w P W : D, w ( ϕD, w ( rďs ϕ if, for all w P W s.t. w ď w : D, w ( ϕD, w ( răs ϕ if, for all w P W s.t. w ă w : D, w ( ϕD, w ( r‹ ϕ s ψ if D , w ( ψ , where D “ x‹p M, ϕ q , ‹y As usual, we say ξ is valid in D , denoted by D ( ξ , if for all w P W , D, w ( ξ ,and that ξ is valid, denoted ( ξ , if it is valid for any dynamic preference model D . Since in this work we investigate axiomatisations that characterise classes ofdynamic operators, we will need to define the notion of a formula been valid ona class of dynamic models. For that, we define the notion of a class of dynamicmodels, which will be used in Section 4, similarly to how classes of frames areconnected to modal axioms in correspondence theory for Modal Logic [36]. Definition 18.
Let M Ď Mod p L ď q be a class of preference models and C aclass of dynamic operators closed over M , i.e., for any ϕ P L , ‹p M , ϕ q Ď M foreach ‹ P C . We denote by D “ x M , C y the class of dynamic models x M, ‹y s.t. M P M and ‹ P C . 15f D is a class of dynamic models and ξ P L ď p‹q is a dynamic preferenceformula, we will often say ξ is satisfiable in D if there is some dynamic model D P D in which ξ is satisfiable. Similarly, we will say that ξ is valid in D if forany D P D , D ( ξ .
4. Iterated Belief Change and DPL
In this section, we investigate the relationship between the postulates satis-fied by iterated belief change operators discussed in Section 2 and the axiomssatisfied in DPL using these operators. In other words, given an iterated beliefchange postulate P , we investigate which validities are induced in the logic of L ď p‹q if we consider only classes of models in which the dynamic operatorssatisfy postulate P . More yet, we study how to characterise this postulate inDPL, i.e., which axioms should be introduced in the axiomatization of DPLthat imply that the models of this logic must satisfy P .We examine Darwiche and Pearl’s p DP1 q - p DP4 q [3], as well as Nayak et al’s p Rec q [26], for iterated belief revision, and Chopra et al’s p CR1 q - p CR4 q [27] andRamachandran et al’s p LC q [24] for iterated belief contraction. Other postulatesin the literature can be easily encoded in the same way based on our results.The proofs of the results presented in this section are given in the Appendix.First, it is easy to see that for any dynamic operator ‹ , the extended logicmust satisfy some basic principles. Proposition 19.
The following reduction axioms and rule are valid, for anypropositional symbol p P P , propositional formula ϕ P L and formula ξ P L ď p‹q . r‹ ϕ s p Ø p r‹ ϕ sp ξ ^ ξ q Ø r‹ ϕ s ξ ^ r‹ ϕ s ξ r‹ ϕ s ξ Ø r‹ ϕ s ξ r‹ ϕ s Aξ Ø A r‹ ϕ s ξ $ ξ ñ $ r‹ ϕ s ξ As in Souza’s axiomatisation for DPL [33], we do not require Uniform Sub-stitution as a rule in our proof systems. This is an important characteristicof our proof systems, and quite common in dynamic epistemic logics [40]. IfUniform Substitution was included in the system, it would be possible to derive $ r‹ ϕ s ξ Ø ξ for any ξ P L ď p‹q and ϕ P L , i.e., the dynamic modality would befrivolous. Without Uniform Substitution, r‹ ϕ s ξ Ø ξ can only be derived by theaxioms in Proposition 19 if ξ P L , i.e., ξ is a propositional formula. This is con-sistent with the interpretation that our dynamic operators are mental actionsand, thus, do not change the propositional (or ontic) properties of the worlds ina model.We wish to investigate which properties are induced in the dynamic logic,given the postulates satisfied by a given dynamic operator ‹ . First, we study thecharacterisation of iterated belief revision postulates in DPL, then we extendthis study to iterated belief contraction postulates. Further, we show how thesecharacterisations can used to derive axiomatisations for a logic defined overclasses of models satisfying a set of iterated belief change postulates.16 .1. Postulates for Iterated Belief Revision in DPL First, let us consider the basic postulate defining belief revision operations,namely the postulate of p Faith q which indicates that the dynamic operation ‹ is an AGM belief revision operator. Proposition 20.
Let C be a class of dynamic operators ‹ satisfying p Faith q and M Ď Mod p L ď q be a class of preference models. The following axiom schemais valid in x M , C y , for any propositional formula ϕ P L . Eϕ Ñ µϕ Ø r‹ ϕ s µ J Postulate p Faith q states that after a revision by some information ϕ , theminimal elements of the agent’s belief state are exactly the most plausible ϕ -states. As such, the axiom schema in Proposition 20 states that, if a proposition ϕ is satisfiable in a model M , for any minimal ϕ -world w in M , it holds thatafter change by the dynamic operator ‹ , w is a minimal world of the resultingmodel, and vice-versa.We can also show that the axiom schema of Proposition 20 completely char-acterises the postulate p Faith q in DPL. In the results below, we will often denotea class of dynamic model x M , t‹uy simply by x M , ‹y . Proposition 21.
Let D “ x M , ‹y be a class of dynamic models. The axiomschema in Proposition 20 is valid in D , for any propositional formula ϕ P L iff for each M “ x W, ď , v y P M , s.t. ‹p M, ϕ q “ x W, ď ‹ ϕ , v y , if J ϕ K ‰ H , then M in ď J ϕ K “ M in ď ‹ ϕ W , i.e., ‹ satisfies p Faith q in M . While p Faith q is the basic postulate for AGM revision, it does not imply anyconstraints on the iterated properties of the operation. To characterise iteratedbelief revision in DPL, we will consider the logic characterisation of Darwicheand Pearl’s p DP1 q - p DP4 q [3] postulates, as well as Nayak et al.’s p Rec q [26]. Postulate DP1.
Let’s start with postulate p DP1 q , as presented in Section 2. p DP1 q If w, w P J ϕ K , then w ď ‹ ϕ w iff w ď w It is easy to see that this property implies the following validities in the logicdefined by the operators satisfying it.
Proposition 22.
Let C be a class of dynamic operators ‹ satisfying p DP1 q and M Ď Mod p L ď q be a class of preference models. The following axiom schematais valid in x M , C y for any ϕ P L and ξ P L ď p‹q . r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ răsp ϕ Ñ r‹ ϕ s ξ qqrďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qqrăsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srăsp ϕ Ñ ξ qq Notice that p DP1 q establishes a bidirectional relationship between the prefer-ence relations ď and ď ‹ ϕ by a double implication. Each side of this implicationis captured in the axiom schemata in Proposition 22, where the first axiom17tates that if there is two ϕ -world w, w s.t. w ď ‹ ϕ w , and w satisfies ξ , thenthere is some ϕ -world w satisfying r‹ ϕ s ξ and w ď w . This first axiom ofProposition 22 generalises the requirement that if w ď ‹ ϕ w then w ď w in p DP1 q , while the third axiom states the opposite direction of this relation. Thesecond and fourth axioms are variations that express the changes in the strictpart ă of the preference relation.Proposition 22 provides a representation of the postulate p DP1 q as a DPLaxiom schemata induced, in the extended logic, by operations ‹ that satisfythis postulate. However, The logical characterisation of the postulate, i.e., thatif the logic satisfies a certain set of axioms, then the dynamic operator mustsatisfy p DP1 q , as established for p Faith q in Proposition 21, cannot be achievedin DPL. The reason for this is that the language is not expressive enough todistinguish every world in the model. As such, there may be worlds in a modelthat are “modally equivalent” (m.e.) [36], in the sense that they satisfy exactlythe same formulas, and there is no way to express in the logic any relation thatdifferentiates these worlds. As such, it is easy to construct dynamic operatorsthat are equivalent, in the sense that the logics generated by them are the same. Fact 23.
There are two dynamic operators ‹ , ˚ : Mod p L ď q ˆ L Ñ Mod p L ď q s.t. for any class of preference models M , s.t. both ‹ and ˚ are closed over M ,and formula ξ P L ď p‹q , ξ is satisfiable in x M , ‹y iff ξ is satisfiable in x M , ˚y ,but ‹ does not satisfies p DP1 q while ˚ does. It is easy to see from the proof of Fact 23 (c.f. the Appendix) and fromthe axioms presented in Proposition 22 that some dynamic operators may failto satisfy p DP1 q and yet preserve all conditional beliefs regarding the new in-formation ϕ . In fact, they satisfy Darwiche and Pearl’s original formulation ofpostulate p DP1 q based on conditional beliefs [3]. Proposition 24.
Let D “ x M , ‹y be a class of dynamic models satisfying theaxiom schemata in Proposition 22. For any propositional formulas ϕ, ψ P L s.t. D ( ϕ Ñ ψ and any dynamic preference formula ξ P L ď p‹q , it holds that D ( r‹ ψ s B p ξ | ϕ q Ø B pr‹ ψ s ξ | ϕ q The reason for this is that the semantic formulation of p DP1 q is rooted in theidentities of the worlds in the model, not in the information they hold. Moreaccurately, p DP1 q assumes that for every pair of worlds w, w in the model, thereis some proposition ξ that only one of them satisfies. As such, Fact 23 pointsout that the DP postulates based on Grove’s model need to be generalised toour models. We then define the notion of DP1-compliance in a certain class ofmodels, meaning that a dynamic operator ‹ behaves as to preserve the agent’sbeliefs conditioned to the new information ϕ . Definition 25.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is DP1-compliant in regards to M , or M -DP1-compliant, if, for any preference model M “ x W, ď , v y P M , any propositional formula ϕ P L , and any possible worlds w, w P W satisfying ϕ , it holds: 18 DP1a q if w ď ‹ ϕ w ( w ă ‹ ϕ w ) then, for any piece of information ξ P L ď p‹q s.t. D, w ( r‹ ϕ s ξ , there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď w ( w ă w ); p DP1b q if w ď w ( w ă w ) then, for any piece of information ξ P L ď p‹q s.t. D, w ( r‹ ϕ s ξ , there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ , and w ď ‹ ϕ w ( w ă ‹ ϕ w )where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is DP1-compliant.Definition 25 states that no information contained in the worlds satisfying ϕ is lost due to a belief change regarding ϕ , similar to what p DP1 q tries to encodefor Grove models. With this generalisation, we can characterise DP1-compliancein DPL. Proposition 26.
Let D “ x M , ‹y be a class of dynamic models. The axioma-tisation presented in Proposition 22 is valid in D iff ‹ is M -DP1-compliant. In the light of Propositions 24 and 26, it is easy to see that the notion of M -DP1-compliance is, in fact, an adequate generalization of p DP1 q to preferencemodels, as observed in the Fact below. Fact 27.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -DP1-compliant dynamic operator, and D “ x M , ‹y .For any propositional formulas ϕ, ψ P L s.t. D ( ϕ Ñ ψ and any dynamicpreference formula ξ P L ď p‹q , it holds that D ( r‹ ψ s B p ξ | ϕ q Ø B pr‹ ψ s ξ | ϕ q Notice that for models in which every world has a characteristic formula,Definition 25 implies that the dynamic operator ‹ satisfies p DP1 q . In other words,if we consider only Grove models and operators closed over Grove models, p DP1a q and p DP1b q taken together are equivalent to p DP1 q . Fact 28.
Let M be a class of preference models s.t. for any M P M andany possible world w in M there is a characteristic formula ξ w P L ď p P q , s.t. M, w ( ξ w iff w “ w and let ‹ : M od p L ď q ˆ L Ñ M od p L ď q be a dynamicoperator closed over M . It holds that ‹ is M -DP1-compliant iff for any M P M ,any propositional formula ϕ and worlds w, w P J ϕ K it holds that w ď w iff w ď ‹ ϕ w .Postulate DP2. Similar results can be achieved for the other postulates, i.e., wecan provide characterisations of the other postulates by means of DPL axioms.Next, we provide characterisation for DP2.
Definition 29.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is DP2-compliant in M , or M -DP2-compliant, if, for any preference model M “ x W, ď , v y P M , any propositional formula ϕ P L and any possible worlds w, w P W not satisfying ϕ , it holds: 19 DP2a q if w ď ‹ ϕ w ( w ă ‹ ϕ w ) then for any piece of information ξ P L ď p‹q s.t. D, w ( r‹ ϕ s ξ there is some world w R J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď w ( w ă w ); p DP2b q if w ď w ( w ă w ) then for any piece of information ξ P L ď p‹q s.t. D, w ( r‹ ϕ s ξ there is some world w R J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w ( w ă ‹ ϕ w )where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is DP2-compliant.As before, Definition 29 states that no information contained in the worldssatisfying ϕ is lost due to a belief change regarding ϕ , similar to what p DP2 q encodes for Grove models. It is easy to see that for models without proposition-ally indiscernible worlds, this condition is equivalent to p DP2 q , similar to provenin Fact 28. More yet, we can characterise the DP2-compliance using DPL. Proposition 30.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -DP2-compliant. r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ răsp ϕ Ñ r‹ ϕ s ξ qqrďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qqrăsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srăsp ϕ Ñ ξ qq Notice that, given the structural similarities between postulates p DP1 q and p DP2 q , the axioms presented in Proposition 30 are similar to those presented inProposition 22 and also have similar interpretations.Similar to Facts 28 and 27, it is easy to see that our generalisation of p DP2 q implies p DP2 q for Grove-like models, namely those with characterising formulas,and preserves of incompatible conditional beliefs. Fact 31.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -DP2-compliant dynamic operator, and D “ x M , ‹y .For any propositional formulas ϕ, ψ P L s.t. D ( ϕ Ñ ψ and any dynamicpreference formula ξ P L ď p‹q , it holds that D ( r‹ ψ s B p ξ | ϕ q Ø B pr‹ ψ s ξ | ϕ q Postulate DP3.
Similarly, we can generalise the postulate p DP3 q . This postulatestates that no ϕ -world gets promoted after acquiring information that ϕ , orin other words, that all conditional belief consistent with ϕ is maintained. Definition 32.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is DP3-compliantin M , or M -DP3-compliant, if for any preference model M “ x W, ď , v y P M ,any propositional formula ϕ P L , and any possible worlds w, w P W , it holdsthat: 20 DP3a q if w P J ϕ K , w R J ϕ K and w ă w , then for any information ξ P L ď p‹q s.t. D, w ( r‹ ϕ s ξ there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ă ‹ ϕ w .where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is DP3-compliant.As before, we can characterise DP3-compliance using DPL. Proposition 33.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schema is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -DP3-compliant. r‹ ϕ srăsp ϕ Ñ ξ q Ñ ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ q As for p DP1 q and p DP2 q , our generalisation of p DP3 q does encode the ideabehind the original postulate. This property may be observed in Fact 34, aninterpretation of Darwiche and Pearl’s [3] syntactic form of p DP3 q into DPL. Fact 34.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -DP3-compliant dynamic operator, and D “ x M , ‹y . Forany propositional formula ϕ P L and dynamic preference formula ξ P L ď p‹q , itholds that D ( B p ϕ | r‹ ϕ s ξ q Ñ r‹ ϕ s B p ϕ | ξ q Postulate DP4.
Similarly, we can generalise the postulate p DP4 q to the followingcondition: Definition 35.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is DP4-compliantin M , or M -DP4-compliant, if for any preference model M “ x W, ď , v y P M ,any propositional formula ϕ P L , and any possible worlds w, w P W , it holdsthat: p DP4a q if w P J ϕ K , w R J ϕ K and w ď w , then for any information ξ P L ď p‹q s.t. D, w ( r‹s ξ there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w .where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is DP4-compliant.Again, we can characterise DP4-compliance. Proposition 36.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -DP4-compliant. r‹ ϕ srďsp ϕ Ñ ξ q Ñ ϕ Ñ rďsr‹ ϕ sp ϕ Ñ ξ q As before, our generalisation of p DP4 q also encodes the idea behind the orig-inal postulate. Fact 37.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -DP4-compliant dynamic operator, and D “ x M , ‹y . Forany propositional formula ϕ P L and dynamic preference formula ξ P L ď p‹q , itholds that D ( B p ϕ | r‹ ϕ s ξ q Ñ r‹ ϕ s B p ϕ | ξ q ostulate REC. Below we provide a characterisation in DPL for Nayak et al.’s[26] Recalcitrance, or p Rec q . Definition 38.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is M -Rec-compliant, if, for any preference model M “ x W, ď , v y P M , any propositionalformula ϕ P L , and any possible worlds w, w P W , it holds that: p Rec q if w P J ϕ K and w R J ϕ K , then w ę ‹ ϕ w and for any information ξ P L ď s.t. D, w ( r‹ ϕ s ξ there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w .where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is Rec-compliant.From this encoding, we obtain the following characterisation. Proposition 39.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -Rec-compliant. r‹ ϕ srăs ξ Ñ p ϕ Ñ A p ϕ Ñ r‹ ϕ s ξ qq ϕ Ñ pr‹ ϕ srďs ϕ q Similar to Nayak et al. [26]’s p Rec q , our p Rec q postulate guarantees the max-imal preservation of an adopted belief ψ in all conditions that do not contradictit. Fact 40.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -Rec-compliant dynamic operator, and D “ x M , ‹y . Forany propositional formula ϕ P L , and dynamic preference formula ξ P L ď p‹q ,it holds that D ( E p ϕ ^ r‹ ϕ s ξ q Ñ r‹ ϕ s B p ϕ | ξ q Regarding the postulates for iterated belief contraction, we present DPLcharacterisations for Chopra et al.’s [27] p CR1 q - p CR4 q , and Ramachandran etal.’s [24] p LC q . Let us begin with the representation of Grove’s [28] characteri-sation of contraction in DPL, as presented in Section 2. Proposition 41.
Let C be a class of dynamic operators satisfying p GR q and M Ď Mod p L ď q be a class of preference models. The following axiom schema isvalid in x M , C p GR q y , for any propositional formula ϕ P L and formula ξ P L ď p‹q . p µ ϕ _ µ Jq Ø r‹ ϕ s µ J Notice that Chopra et al.’s [27] p CR1 q - p CR4 q are variants of Darwiche andPearl’s [3]’s p DP1 q - p DP4 q . In fact, p CR1 q is the same postulate as p DP2 q and p CR2 q is the same as p DP1 q , while p CR3 q and p DP3 q , and p CR4 q and p DP4 q are structurallysimilar. In fact, we know that p CR3 q and p CR4 q are dual forms of p DP3 q and p DP4 q ,respectively [41]. Since DP1- and DP2-compliance have already been defined, wewill only focus on p CR3 q and p CR4 q . As for p DP4 q and p DP4 q , our generalisationof postulates p CR3 q and p CR4 q also satisfy the original syntactic postulates ofChopra et al.’s [27]. Since these results are very similar to Fact 34 and Fact 37,we omit the results about the suitability of our generalised postulates.22 ontraction Postulate CR3. We can define the notion of CR3-compliance of adynamic operator in regards to a class of preference models based on the notionof DP3-compliance.
Definition 42.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is M -CR3-compliant, if, for any preference model M “ x W, ď , v y P M , any propositionalformula ϕ P L , and any possible worlds w, w P W , it holds that: p CR3a q if w R J ϕ K , w P J ϕ K and w ă w , then for any information ξ P L ď s.t. D, w ( r‹ ϕ s ξ there is some world w R J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ă ‹ ϕ w where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is CR3-compliant.Hence, similarly to DP3-compliance, we can characterise CR3-compliance. Proposition 43.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -CR3-compliant. r‹ ϕ srăsp ϕ Ñ ξ q Ñ ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ q Contraction Postulate CR4. As p CR4 q is similar variation of p DP4 q , it is easy todefine the notion of CR4-compliance. Definition 44.
Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is M -CR4-compliant, if, for any preference model M “ x W, ď , v y P M , any propositionalformula ϕ P L , and any possible worlds w, w P W , it holds that: p CR4a q if w R J ϕ K , w P J ϕ K and w ď w , then for any information ξ P L ď s.t. D, w ( r‹ ϕ s ξ there is some world w R J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w .where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is CR4-compliant.Again, it follows that for models without propositionally indiscernible worlds,this condition is equivalent to p DP4 q . Proposition 45.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -CR4-compliant. r‹ ϕ srďsp ϕ Ñ ξ q Ñ ϕ Ñ rďsr‹ ϕ sp ϕ Ñ ξ q Contraction Postulate LC.
Finally, we can represent Lexicographic Contractionin DPL. Notice that p LC q states how the preference relation must be changesin terms of the maximal chains of worlds in the model that either satisfy ϕ orsatisfy ϕ . As such, we define the notion of LC-compliance. In Definition 46notice that postulate p LC q is the same postulate proposed in Section 3 with aslight difference in presentation to account for the notion of satisfaction of aformula in a dynamic model D . 23 efinition 46. Let M Ď Mod p L ď q be a class of preference models, and let ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a dynamic operator. We say ‹ is M -LC-compliant iff for any preference model M “ x W, ď , v y P M , any propositionalformula ϕ P L and any possible worlds w, w P W , it holds that: p LC q Let ξ, χ be members of t ϕ, ϕ u - not necessarily distinct. If D, w ( ξ and D, w ( χ , then w ď ‹ ϕ w iff the maximal length of a chain of worldsin J ξ K which ends in w is smaller or equal than to the maximal length ofa chain of worlds in J χ K which ends in w .where D “ x M, ‹y . If M “ Mod p L ď q , we say ‹ is LC-compliant.We can, then, provide axioms to encode LC-compliance in DPL. Notice thatthe axioms below simply state that the result of an LC-compliant operator ona model order the worlds based on the length of the maximal ϕ and ϕ chainsin the model , as stated in postulate p LC q . Proposition 47.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D , for all n P N , ϕ P L , and ξ P L ď p‹q if ‹ is M -LC-compliant. Remember that
M, w ( µdg ξ p i q iff there is a maximal chain of ξ -worlds ending in w ,according to Lemma 10. ‹ ϕ srďs ξ Ñ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srăs ξ Ñ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srďs ξ Ð p µdg ϕ p n q _ µdg ϕ p n qq ^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srăs ξ Ð p µdg ϕ p n q _ µdg ϕ p n qq ^ n ´ ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ´ ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q For guidance in comparing our generalised postulates to the original onesproposed for Grove’s systems of spheres, we list all postulates discussed in thiswork with the corresponding generalization in Table 1.
In the following, we investigate how we can derive sound axiomatisations forthe logic L ď p‹q , given the postulates satisfied by ‹ . First, let us properly definethe notion of a logic induced by classes of dynamic preference models and bysets of axioms as proposed earlier. Definition 48.
Let D “ x M , C y be a class of dynamic models and ‹ a symbolfor dynamic operators, we call the logic of D , or the logic defined by D , as theset Log p D q “ t ϕ P L ď p‹q | D ( ϕ u able 1: Original postulates and corresponding postulates for preference models. Postulate Original work Corresponding postulates p Faith q [28] – p DP1 q [3] p DP1a q and p DP1b q (Definition 25) p DP2 q [3] p DP2a q and p DP2b q (Definition 29) p DP3 q [3] p DP3a q (Definition 32) p DP4 q [3] p DP4a q (Definition 35) p Rec q [26] p Rec q (Definition 38) p GR q [28] – p CR1 q [27] p DP2a q and p DP2b q (Definition 29) p CR2 q [27] p DP1a q and p DP1b q (Definition 25) p CR3 q [27] p CR3a q (Definition 42) p CR4 q [27] p CR4a q (Definition 44) p LC q [24] p LC q (Definition 46)It is easy to see that the logic defined by a class of dynamic operators canbe constructed from the logics induced by each operator individually. Moregenerally, we have the following. Proposition 49.
Let M Ď Mod p L ď q be a class of preference models and C “t C i | i P I u a family of classes of dynamic operators indexed by some set I ,which are closed over M . Log px M , ď i P I C i q “ č i P I Log px M , C i yq Proposition 49 implies that the logic defined by a class of dynamic models D “ x M , C y is completely determined by each individual dynamic operator ‹ P C . As such, if we wish to study the properties of, say, DP1-compliantoperators over a class of preference models M using DPL, it suffices to study theproperties of each individual DP1-compliant operator over this class of models.Now, we will concern ourselves with the logic defined by a set of axioms - or aproof system. This will allow us to investigate the properties of soundness andcompleteness for DPL. Definition 50.
Let A Ă L ď p‹q be a set of formulas (or axioms). We define thelogic of A , or defined by the system A , denoted Log p A q , as the smaller set offormulas containing A that is closed by modus ponens and necessitation rules,i.e., (i) if ϕ, ϕ Ñ ψ P Log p A q then ψ P Log p A q and (ii) if ψ P Log p A q then l ϕ P Log p A q , with l P t A, rďs , răs , r‹ ϕ s for any ϕ P L u . We call A an axiomsystem for Log p A q .With that, we can define the notions of an axiomatisation, or of a proofsystem, being sound or complete, as usual.26 efinition 51. Let A Ă L ď p‹q be a set of formulas and D a class of dynamicmodels. We say that • A is sound in regards to D , if Log p A q Ď Log p D q • A is complete in regards to D , if Log p D q Ď Log p A q It is easy to see that the union of sound axiomatisations for a set of classesof dynamic models results in a sound axiomatisation for the intersection of suchclasses.
Theorem 52.
Let M be a class of preference models, C “ t C i | i P I u be a familyof classes of dynamic operators, which are closed over M , and A “ t A i | i P I u afamily of sound axiom systems for C , i.e., Log p A i q Ď Log px M , C i yq , both indexedby some set I . Log p ď i P I A i q Ď Log px M , č i P I C i q In Theorem 52, we prove that we can combine the axiomatisations thatrepresent each postulate (presented in Propositions 19, 22, 30 33 and so on) intoa single axiomatisation that is sound to the class of operators that satisfy allpostulates at once. As such, we can use the results in this section to obtain soundlogics for classes of dynamic operators, such as the logic defined by Darwich andPearl’s Iterated Belief Revision [3] operators or Nayak et al.’s LexicographicRevision operators [26].Notice that, while we were able to characterise Iterated Belief Change pos-tulates in DPL, in Propositions 26, 30, 33, 36, 39, 41, 43 and 45, this does notguarantee that the resulting logic is complete. In fact, it is not easy to obtain acompleteness result for the axiomatisations since, given the semantics of DPLbased on dynamic models, one such proof would require the construction of adynamic model serving as counter-example for any non-theorem of the logic.While the technique of filtrated canonical models, as used by Souza to provecompleteness for Preference Logic [33], does indicate ways to the constructionof such model, it is not clear how to translate the obtained filtrated canonicalmodel into a dynamic model.More yet, even if we obtain complete axiomatisations for some postulates,it is not clear whether we can obtain a general completeness result for theconjoined axiomatisations, such as done in Theorem 52 for soundness.In the light of Theorem 52, we can apply the results obtained in Proposi-tions 19, 22, 30 and 39 to obtain an axiomatisation to DPL of LexicographicRevision.
Corollary 53.
Preference Logic extended with lexicographic revision is soundly xiomatised by the axiomatization of L ď extended by the axioms and rules below. rò ϕ s p Ø p rò ϕ sp ψ ^ ξ q Ø rò ϕ s ψ ^ rò ϕ s ξ rò ϕ s ξ Ø rò ϕ s ξ rò ϕ s Aξ Ø A rò ϕ s ξ r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ răsp ϕ Ñ r‹ ϕ s ξ qqrďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qqrăsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srăsp ϕ Ñ ξ qqr‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ răsp ϕ Ñ r‹ ϕ s ξ qqrďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qqrăsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srăsp ϕ Ñ ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ A p ϕ Ñ r‹ ϕ s ξ qq ϕ Ñ pr‹ ϕ srďs ϕ q$ ξ ñ $ r‹ ϕ s ξ It is not difficult to see that the complete axiomatisation presented in Fig-ure 2 at Section 3 is a simplification of the axiomatisation of Corollary 53 ob-tained using our method. The same stands for axiomatisations for LexicographicContraction (Proposition 47), when compared to the one obtained by Souza [33]for finite models with chains of a bounded size.
5. Related Work
To our knowledge, the work of Segerberg [5] is the first to propose the inte-gration of belief revision operations within an epistemic logic, with his proposalof Dynamic Doxastic Logic (DDL). This integration is important because it al-lows one to analyse the effects of introspection, and other related phenomena, inthe logic of belief change. A famous example of such interaction is the analysisof Moore sentences in the logic of belief change, which shows that AGM’s pos-tulates are incompatible in the face of introspection [7]. In this work, Segerbergprovides a set of axioms , which corresponds to encodings within his logic ofAGM’s postulates for belief change. Our work is linked to that approach byinvestigating these correspondences for dynamic belief change, based on iter-ated belief change postulates, instead of AGM belief change, as pursued bySegerberg.In the context of DDL, Cantwell [42] defines some iterated belief revisionoperators as change operations in hypertheories [7] and he shows how theseoperations can be axiomatically characterised in DDL. Our work differs fromhis in that we analyse how some well known-postulates can be characterisedin our logic and not how to encode specific constructions. Our logic also hasthe advantage of being more expressive because it can encode some notion ofdegrees of belief [33], which cannot be expressed in DDL.Another trend of research on using modal logics to study (and axiomatise)iterated belief change operators or policies is the work on Dynamic Epistemic28ogics. Inspired by Rott [43], Van Benthem [14] proposed the codificationof some iterated belief revision operators within a Dynamic Epistemic Logic(DEL). This work was further extended by Girard [11], Liu [13], and Souza etal. [15] who studied the use of DPL to encode several (relational) belief revisionpolicies. Similarly, Baltag and Smets [8] used a logic similar to DPL to encodedifferent notions for knowledge and belief, based on Board’s work [10]. Theseauthors show how different iterated belief revision operators can be simulatedusing DEL action models and product update.Further, Girard and Rott [12] propose a DPL for studying belief revision.The authors encode several iterated belief revision policies using General Dy-namic Dynamic Logic [44] and show that reduction axioms can be obtained forthem in the same fashion as [22].In the related literature, all these works following the DEL tradition defineoperations semantically in their logic and either provide axiomatisations bymeans of crafting the axioms or by encoding these operations using a variationof dynamic logic to obtain reduction axioms. These works are informed by well-known results in the area of belief change to choose appropriate operations andthen encode these operations in their logics.On the other hand, our work investigates how DPL can be used to char-acterise properties of dynamic belief change operators, instead of applying thecharacterisation in Belief Change theory to construct a logic. To our knowl-edge, our work is the first to do so for dynamic belief change operators. Whileother work, such as that of Darwiche and Pearl’s [3] and of Jin and Thielscher[45] and others, have investigated semantic characterisation of iterated beliefchange postulates, these characterisations have been pursued in an extra-logicalframework, i.e., outside of the object language used to the specify the agent’s be-liefs. As discussed before, these move to specifying belief change has importantconsequences on the expressiveness of the theory.Our work is also connected to that of Souza et al. [46, 47] that uses the con-nection between preference models and Liu’s priority graphs [13] to belief changepostulates as structural properties on transformations in priority graphs. How-ever, These authors show that transformations on priority graphs are a limitedrepresentation for changes of dynamic operators and some of the well-knownpostulates of the area cannot be encoded using them. How our generalisationscan be connected to that work is still an open topic of research. It seemsdoubtful that we can establish significant connections between our axiomaticcharacterisations and the structural constraints imposed by postulates in therealisation of dynamic operators as transformations in priority graphs.As far as we know, the only work that investigates the encoding of postulatesfrom Belief Change as axioms of a logic is the work of Segerberg [5, 6] on DDL.In [6], the author axiomatizes the logic DDL and proves the completeness ofthe logic in regards to an SOS-based semantics, showing thus that their axiomsare accurate representations of AGM’s postulates within their logic. WhileSegerberg’s work is very similar to ours in intent, the authors do not consideriterated belief change postulates and their representation in the logic - the mainfocus of our work. Notice that, while DDL could, in principle, be used as29 foundation for our study, this logic is less expressive than DPL. Also, bychoosing to employ DPL instead of DDL as a foundational logic, our work canbe connected to the work representing different mental attitudes in this logic[34, 21, 20] allowing the application of theoretically founded dynamic operatorsto dynamic phenomena for different mental attitudes.
6. Final Considerations
This work has investigated representation results for well-known iteratedbelief change postulates using the framework of Dynamic Epistemic Logics.We have provided a set of axioms that encode these postulates within DPLand shown that, as a result of the higher expressiveness of preference modelsin comparison to Grove’s SOS, a characterisation of these postulates in DPLcannot be obtained.Further, we have provided a generalisation of the postulates using preferencemodels (Definitions 25, 29, 32, 35, 38, 42, 44, and 46) that coincide with thestudied postulates in the class of models similar to Grove’s spheres (Facts 28,27, 31, 34, 37, and 40), and shown that our encoding characterises these gener-alised postulates (Propositions 26, 30, 33, 36, 39, 43, 19, and 47). Finally, wehave shown (Theorem 52) that we can obtain a sound axiomatisation for DPLinterpreted over a class of dynamic models by aggregating the axioms for eachpostulate satisfied by all dynamic operators in that class.We use the proposed axioms to obtain an axiomatisation of Preference Logicextended with dynamic modalities for Lexicographic Revision (Corollary 53)and Lexicographic Contraction (Proposition 47). We wish to point out that,while our work is concerned with single-agent belief changes, our results can betrivially extended to private changes in the multi-agent case.Our results generalise previous semantic-based postulates by demonstratingare intrinsically linked to the structure of the model on which they are based,i.e., Grove’s system of spheres for classical propositional logic. To avoid ter-minological confusion, we point out that our results are concerned with beliefchange operations defined on well-founded preference models, which we simplycall preference models in this work, and they may not be valid if one considers allpreference models, since some belief change operators may not be well-definedon non-well-founded models [12, 33].A question that may arise from our work is whether the generalisation ofBelief Change postulates from the semantic framework of Grove’s systems ofspheres to preference models is relevant from an epistemological point of view.We point out both that preference models have been extensively studied as mod-els for non-monotonic reasoning and conditional logics [17, 18, 48] and to modeldifferent mental attitudes, such as Preferences [11, 19], Goals and Desires [20],and Obligations [21], and different notions of Belief [8].It is of notice that preference models can be used to represent and reasonabout nested conditionals (or sets of conditionals), as those studied in condi-tional logics [49, 35], which can represent introspective beliefs and their dynam-30cs. These conditionals cannot be expressed by means of Grove models, as thesemodels are injective, in the sense used by Friedman and Halpern [48].The reader may also argue whether Modal Logic is an adequate frameworkto study belief change. We point out that, firstly, modal analysis of belief andother mental attitudes abound in the literature and have proven to be a fruitfuland powerful framework to study attitudes such as beliefs [50, 51, 52]. Also,as De Rijke [53] points out, Dynamic Logic is a standard tool to reason aboutstates and transitions between states, two fundamental notions for any notionof change or dynamics, and can be analysed through well-established tools fromModal Logic. As such, theories based on Dynamic Logic can be easily connectedto well-established modal analysis of mental attitudes in the literature, providinga rich framework to study dynamic phenomena.While we have proved in this work that the axiomatisations provided arecomplete characterisations of the investigated postulates, by means of our pro-posed generalisations, we were not able to provide results about the complete-ness of the logic obtained by extending preference logic with a given dynamicoperator characterised by a set of postulates. Inasmuch as the obtained axioma-tisation is indeed complete for the example studied, i.e., Lexicographic Revision,it is still unclear if this will always be the case. Completeness results are not atrivial topic in Modal Logic and it is not clear that a general result guaranteeingcompleteness of the derived axiomatisation in regards to the extended logic canbe obtained.As future work, we aim to investigate how these characterisations can beconnected to syntactic representations of belief change operations. It is wellknown that dynamic operators can be encoded by means of transformations onpriority graphs, a connection already studied by Liu [13], Souza et al. [15] andothers. Souza et al. [46, 47] have shown that some postulates - as originallydefined in Iterated Belief Change - cannot be encoded by means of transforma-tions on priority graphs unless we restrict our semantics to only consider somespecific classes of preference models. We aim to investigate if the same holdsfor our generalisations and, if so, for which kind of models such a syntacticcharacterisation of iterated belief change postulates can be achieved.
Funding
This study was financed in part by the Coordena¸c˜ao de Aperfei¸coamento dePessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.
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Appendix A. Proofs of selected results in the paper
All proofs related to postulates p DP2 q , p DP4 q , p CR3 q and p CR4 q will be omitted.The proofs of these results employ similar arguments used in the proofs for theanalogous results for postulate p DP1 q , p DP2 q , and p DP3 q .In the proofs below, to simplify the argumentation, we will usually employexistential versions of the axioms discussed in Section 4, obtained by the contra-position of the original axioms. The proof of Fact 13 below requires familiaritywith Propositional Dynamic Logic (PDL). Fact 13.
Lexicographic Contraction cannot be encoded by means of PDL pro-gramsProof.
Let M “ x W, ď , v y be a preference model and ϕ P L a propositionalformula. The preference relation ď ó ϕ resulting from the application of Lexico-graphic Contraction on M can be computed as: ď ó ϕ “ Ť t J p ? µdg ϕ p i q ; J ; ? µdg ϕ p j qq K | i ď j u M Y Ť t J p ? µdg ϕ p i q ; J ; ? µdg ϕ p j qq K | i ď j u M Y Ť t J p ? µdg ϕ p i q ; J ; ? µdg ϕ p j qq K | i ď j u M Y Ť t J p ? µdg ϕ p i q ; J ; ? µdg ϕ p j qq K | i ď j u M As PDL is a logic of regular programs and the construction above corre-sponds with the language L “ t a i cb j | i ď j u , which is well-known not to beregular [54], it is immediate that Lexicographic Contraction cannot be encodedby a PDL progam. 35 roposition 20. Let C be a class of dynamic operators ‹ satisfying p Faith q and M Ď Mod p L ď q be a class of preference models. The following axiom schemais valid in x M , C y , for any propositional formula ϕ P L . Eϕ Ñ µϕ Ø r‹ ϕ s µ J Proof.
Let D “ x M, ‹y be a dynamic model, s.t. ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q is a dynamic operator satisfying p Faith q and M “ x W, ď , v y P M is a preferencemodel, D ´ x M, ‹y , and let ϕ P L be a propositional formula s.t. J ϕ K ‰ H .Let’s call ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y and D “ x M ‹ ϕ , ‹y .Suppose D ( Eϕ , then J ϕ K ‰ H . As M is well-founded, M in ď J ϕ K ‰ H i.e.there is some w P W s.t. D, w ( µϕ . Take one such w , since ‹ satisfies p Faith q ,then w P M in ď ‹ ϕ W . Then, D , w ( µ J , i.e. D, w ( r‹ ϕ s µ J . Proposition 21.
Let D “ x M , ‹y be a class of dynamic models. The axiomschema in Proposition 20 is valid in D , for any propositional formula ϕ P L iff for each M “ x W, ď , v y P M , s.t. ‹p M, ϕ q “ x W, ď ‹ ϕ , v y , if J ϕ K ‰ H , then M in ď J ϕ K “ M in ď ‹ ϕ W , i.e. ‹ satisfies p Faith q in M .Proof. Let D “ x M , ‹y be a class of dynamic models for which the axiom schemain Proposition 20 is valid. Let D “ x M, ‹y P D be a dynamic model with M “x W, ď , v y , ϕ P L be a propositional formula s.t. J ϕ K ‰ H , and D “ x M ‹ ϕ , ‹y with ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y . i q M in ď J ϕ K Ď M in ď ‹ ϕ W :Notice that, since M Ď Mod p L ď q , i.e. it is a well-founded preference model,and J ϕ K ‰ H , by well-foundedness M in ď J ϕ K ‰ H . As such, take w P M in ď J ϕ K ,then D, w ( ϕ , which implies D, w ( Eϕ . Also, since w P M in ď J ϕ K , byProposition 5, it holds that D, w ( µϕ . Hence, M, w ( Eϕ ^ µϕ . Since theaxiom schema is valid in D , it must hold that D, w ( r‹ ϕ s µ J , i.e. D , w ( µ J .By Proposition 5, we conclude that w P M in ď ‹ ϕ J J K “ M in ď ‹ ϕ W . As such, M in ď J ϕ K Ď M in ď ‹ ϕ W . ii q M in ď J ϕ K Ě M in ď ‹ ϕ W :Notice that, since M Ď Mod p L ď q , i.e. it is a well-founded preferencemodel, and W ‰ H , by well-foundedness M in ď ‹ ϕ W ‰ H . As such, take w P M in ď ‹ ϕ W . By Proposition 5, it holds that D , w ( µ J , i.e. D, w ( r‹ ϕ s µ J .Since the axiom schema is valid in D , it must hold that D, w ( µϕ . By Propo-sition 5, w P M in ď J ϕ K . As such, M in ď ‹ ϕ W Ď M in ď J ϕ K . Proposition 22.
Let C be a class of dynamic operators ‹ satisfying p DP1 q and M Ď Mod p L ď q be a class of preference models. The following axiom schematais valid in x M , C y for any ϕ P L and ξ P L ď p‹q . r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qqr‹ ϕ srăs ξ Ñ p ϕ Ñ răsp ϕ Ñ r‹ ϕ s ξ qqrďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qqrăsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srăsp ϕ Ñ ξ qq Proof.
We will only show the case for the axioms regarding the modality rďs (axioms 1 and 3), since for the axioms involving modality răs , it suffices to36bserve that p DP1 q implies that for any w, w P J ϕ K it holds that w ă w iff w ă ‹ ϕ w .Let M “ x W, ď , v y P M be a preference model, ‹ P C be a dynamic oper-ator satisfying p DP1 q , and ϕ P L be a propositional formula. For the sake ofpresentation, let’s call D “ x M, ‹y and D “ x M ‹ ϕ , ‹y , with M ‹ ϕ “ ‹p M, ϕ q “x W, ď ‹ ϕ , v y .(i) Take w P W and ξ P L ď p‹q s.t. D, w ( r‹ ϕ srďs ξ . Clearly, if D, w * ϕ ,it holds that D, w ( ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ q , so we only need to consider the casein which D, w ( ϕ . Since D, w ( r‹ ϕ srďs ξ , by Definition 17, D , w ( rďs ξ ,i.e. for any w P W s.t. w ď ‹ ϕ w , it holds that D , w ( ξ . Then, it holdsthat D, w ( r‹ ϕ s ξ . Notice that, by Proposition 19, for any w P W it holdsthat D , w ( ϕ iff D, w ( ϕ , since ϕ is a propositional formula. As such, take w P W s.t. w ď w . If D, w * ϕ , then D, w ( ϕ Ñ r‹ ϕ s ξ . Otherwise, if D, w ( ϕ , then since ‹ satisfies p DP1 q , it holds that w ď ‹ ϕ w and D , w ( ξ ,thus D, w ( ϕ Ñ r‹ ϕ s ξ . As such, by Definition 17, D, w ( rďsp ϕ Ñ r‹ ϕ s ξ q .Since D, w ( ϕ , it holds that D, w ( ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ q .(ii) Take w P W and ξ P L ď p‹q s.t. D, w ( rďsr‹ ϕ s ξ . Again, if D, w * ϕ , itholds immediately that D, w ( p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qq , so let’s assume D, w ( ϕ . By D, w ( rďsr‹ ϕ s ξ , we conclude that for all w P W s.t. w ď w it holdsthat D, w ( r‹ ϕ s ξ , i.e D , w ( ξ . Since ‹ satisfies p DP1 q , for any world w P W s.t. D, w ( ϕ , it holds that w ď w iff w ď ‹ ϕ w . Take w P W s.t. w ď ‹ ϕ w .Clearly, if D , w * ϕ then D , w ( p ϕ Ñ ξ q . Otherwise, if D , w ( ϕ , then w ď w and, thus, D , w ( ξ , i.e. D , w ( ϕ Ñ ξ . A such, D , w ( rďsp ϕ Ñ ξ q and we conclude, by Definition 17, that D, w ( r‹ ϕ srďsp ϕ Ñ ξ q . Since D, w ( ϕ , then D, w ( ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ q . Fact 23.
There are two dynamic operators ‹ , ˚ : Mod p L ď q ˆ L Ñ Mod p L ď q s.t. for any class of preference models M , s.t. both ‹ and ˚ are closed over M ,and formula ξ P L ď p‹q , ξ is satisfiable in x M , ‹y iff ξ is satisfiable in x M , ˚y ,but ‹ does not satisfies p DP1 q while ˚ does.Proof. Let W “ t w , w u , consider the preference models M “ x W, ď , v y and M “ x W, ď , v y with ď “ tx w , w y , x w , w yu and ď “ tx w , w y , x w , w y , x w , w y , x w , w yu and any propositional valuation s.t. for any p P P , w P v p p q iff w P v p p q .Clearly, M is bissimilar to M [36], and for any w P W and ξ P L ď , M , w ( ξ iff M , w ( ξ . Let’s define the operations ‹ , ˚ : Mod p L ď q ˆ L Ñ Mod p L ď q as ˚p M, ϕ q “ M ‹p M, ϕ q “ M if M ‰ M or ϕ ‰ J M if M “ M and ϕ “ J Notice that, since M and M are bissimilar, for any M P Mod p L ď q and ϕ P L , ˚p M, ϕ q is bissimilar to ‹p M, ϕ q and, thus, modally equivalent [36]. As37uch, for any class of preference models M Ď Mod p L ď q s.t. ‹ is closed over M , it holds that ξ P L ď p‹q is satisfiable in x M , ‹y iff it is satisfiable in x M , ˚y .Also, it is clear that ˚ satisfies p DP1 q and ‹ does not, since for w ď w but w ę w . Fact 27.
Let D “ x M , ‹y be a class of dynamic models satisfying the axiomschemata in Proposition 22. For any propositional formulas ϕ, ψ P L s.t. D ( ϕ Ñ ψ and any dynamic preference formula ξ P L ď p‹q , it holds that D ( r‹ ψ s B p ξ | ϕ q Ø B pr‹ ψ s ξ | ϕ q Proof.
Since D is a class of dynamic models satisfying the axiom schemata inProposition 22, it holds that D ( r‹ ψ s B p ξ | ϕ q ” def r‹ ψ s A p µϕ Ñ ξ q” def r‹ ψ s A pp ϕ ^ xăy ϕ q Ñ ξ qØ P rop A pr‹ ψ s ϕ ^ r‹ ψ srăs ϕ Ñ r‹ ψ s ξ qØ P rop A pp ϕ ^ p ψ Ñ răsp ψ Ñ r‹ ψ s ϕ qqq Ñ r‹ ψ s ξ qØ P rop A pp ϕ ^ p ψ Ñ răsp ψ Ñ ϕ qqq Ñ r‹ ψ s ξ qØ D ( ϕ Ñ ψ A pp ϕ ^ răs ϕ q Ñ r‹ ψ s ξ q” def A pp ϕ ^ xăy ϕ q Ñ r‹ ψ s ξ q” def A p µϕ Ñ r‹ ψ s ξ q” def B pr‹ ψ s ξ | ϕ q Proposition 26.
Let D “ x M , ‹y be a class of dynamic models. The axioma-tisation presented in Proposition 22 is valid in D iff ‹ is M -DP1-compliant.Proof. Let D “ x M, ‹y P D s.t. M “ x W, ď , v y is a preference model and ‹ be adynamic operator. For any ϕ P L , let’s call ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y and D “ x M ‹ ϕ , ‹y . ñ :(i) Take w, w P J ϕ K s.t. w ď ‹ ϕ w ( w ă ‹ ϕ w ) and D, w ( r‹ ϕ s ξ for some ξ P L ď p‹q , then D, w ( ϕ ^ r‹ ϕ sp ϕ ^ ξ q and D, w ( ϕ ^ xďyp ϕ ^ r‹ ϕ sp ϕ ^ ξ qq (or D, w ( ϕ ^ xăyp ϕ ^ r‹ ϕ sp ϕ ^ ξ q , if w ă ‹ ϕ w ). Since the axiomatisationpresented in Proposition 22 is valid in D , it holds by contraposition of the third(fourth) axiom that D, w ( xďyr‹ ϕ sp ϕ ^ ξ q ( D, w ( xăyr‹ ϕ sp ϕ ^ ξ q ). As such,there is a w P J ϕ K s.t. w ď w ( w ă w ) and D, w ( r‹ ϕ s ξ . In other words, ‹ satisfies p DP1a q in M .(ii) Take w, w P J ϕ K s.t. w ď w ( w ă w ) and D, w ( r‹ ϕ s ξ for some ξ P L ď p‹q . Then D, w ( r‹ ϕ sp ϕ ^ ξ q and D, w ( ϕ ^ xďyr‹ ϕ sp ϕ ^ ξ q (sim-ilarly, D, w ( ϕ ^ xăyr‹ ϕ sp ϕ ^ ξ q if w ă w ). Since the axiomatisation inProposition 22 is valid in D , by contraposition of the first (second) axiom,then D, w ( r‹ ϕ sxďyp ϕ ^ ξ q ( D, w ( r‹ ϕ sxăyp ϕ ^ ξ q ). In other words, there ‹ is clearly closed for any class M s.t. either M R M or M , M P M .
38s some w P J ϕ K , s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w ( w ă ‹ ϕ w ). Thus, ‹ satisfies p DP2a q in M . ð :As before, we will omit the proof for the axioms regarding the modality răs since they are similar to that of rďs .(i) Take w P W s.t. D, w ( ϕ ^ r‹ ϕ sxďyp ϕ ^ ξ q , for some ϕ P L and ξ P L ď p‹q . Then, w P J ϕ K and there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w . By p DP1a q , there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď w . Then D, w ( xďyr‹ ϕ s ξ . As such, we conclude that D ( p ϕ ^ r‹ ϕ sxďyp ϕ ^ ξ qq Ñ xďyr‹ ϕ s ξ. By the contrapositive, D ( rďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qq . Thus, D ( rďsr‹ ϕ s ξ Ñ p ϕ Ñ r‹ ϕ srďsp ϕ Ñ ξ qq . (ii) Take w P W s.t. D, w ( ϕ ^ xďyp ϕ ^ r‹ ϕ sp ϕ ^ ξ qq , for some ϕ P L and ξ P L ď p‹q . Then w P J ϕ K and there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď w . By p DP1b q , there is some world w P J ϕ K s.t. D, w ( r‹ ϕ s ξ and w ď ‹ ϕ w . As such, D, w ( r‹ ϕ sxďy ξ and, as such, D ( p ϕ ^ xďyp ϕ ^ r‹ ϕ sp ϕ ^ ξ qqq Ñ r‹ ϕ sxďy ξ. By the contrapositive, D ( r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qq and thus, D ( r‹ ϕ srďs ξ Ñ p ϕ Ñ rďsp ϕ Ñ r‹ ϕ s ξ qq . Fact 28.
Let M be a class of preference models s.t. for any M P M and anypossible world w in M there is a characteristic formula ξ w P L ď , s.t. M, w ( ξ w iff w “ w and let ‹ : M od p L ď q ˆ L Ñ M od p L ď q be a dynamic operatorclosed over M . It holds that ‹ is M -DP1-compliant iff for any M P M , anypropositional formula ϕ and worlds w, w P J ϕ K it holds that w ď w iff w ď ‹ ϕ w .Proof. Notice that the implication that satisfaction of the p DP1 q condition im-plies DP1-compliance is trivial and holds for any class of models, so we willfocus on implication that for adequate classes of models M , which includesthe class of all Grove spheres (or concrete models in the terminology of [47]),DP1-compliance implies satisfaction of the p DP1 q condition.Take M a class of preference models with characteristic formulas and ‹ bea M -DP1-compliant dynamic operator closed over M . Let M “ x W, ď , v y P M be a preference model and w, w P W be possible worlds s.t. w, w P J ϕ K . Let’scall ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y . We need to show that w ď w iff w ď ‹ ϕ w .39uppose w ď w . As ‹ is closed over M , then M ‹ ϕ P M , thus there is ξ w P L ď s.t. M ‹ ϕ , w ( ξ w iff w “ w . Clearly, M ‹ ϕ , w ( ξ w . By p DP1b q weconclude that there is w P W s.t. M, w ( r‹ ϕ s ξ w and w ď ‹ ϕ w , but thisimplies M ‹ ϕ , w ( ξ w and w ď ‹ ϕ w . Since ξ w is the characteristic formula of w in M ‹ ϕ , we conclude that w ď ‹ ϕ w .Now, suppose w ď ‹ ϕ w . As ‹ is closed over M , then M ‹ ϕ P M , thus thereis ξ w P L ď s.t. M ‹ ϕ , w ( ξ w iff w “ w . Clearly, M ‹ ϕ , w ( ξ w . By p DP1a q we conclude that there is w P W s.t. M, w ( r‹ ϕ s ξ w and w ď w , but thisimplies M ‹ ϕ , w ( ξ w and w ď w . Since ξ w is the characteristic formula of w in M ‹ ϕ , we conclude that w ď w . Proposition 33.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schema is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -DP3-compliant. r‹ ϕ srăsp ϕ Ñ ξ q Ñ ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ q Proof.
Let D “ x M, ‹y P D s.t. M “ x W, ď , v y is a preference model and ‹ be adynamic operator. For any ϕ P L , let’s call ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y and D “ x M ‹ ϕ , ‹y . ñ :Take w, w P W s.t. w P J ϕ K , w R J ϕ K , and w ă w , and ξ P L ď p‹q s.t. D , w ( ξ . Then, D, w ( r‹ ϕ sp ϕ ^ ξ q , therefore, D, w ( ϕ ^ xăyr‹ ϕ sp ϕ ^ ξ q . By hypothesis, it holds D ( r‹ ϕ srăsp ϕ Ñ ξ q Ñ ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ q . Then,by contraposition, D, w ( r‹ ϕ sxăyp ϕ ^ ξ q , i.e. D , w ( xăyp ϕ ^ ξ q . As such,there is some w P W s.t. w P J ϕ K and D , w ( ξ , hence p DP3a q holds. Assuch, we conclude that ‹ is M -DP3-compliant. ð :Take w P W s.t. D, w ( ϕ ^ xăyr‹ ϕ sp ϕ ^ ξ q for some ξ P L ď p‹q . Thus, w R J ϕ K and there is some w P W s.t. w P J ϕ K , D , w ( ξ . Since ‹ is M -DP3-compliant, it holds that there is w P W s.t. w ă ‹ ϕ w and D , w ( ξ . Assuch, D, w ( r‹ ϕ sxăyp ϕ ^ ξ q . Hence, we conclude that D ( p ϕ ^ xăyr‹ ϕ sp ϕ ^ ξ qq Ñ r‹ ϕ sxăyp ϕ ^ ξ q . By contraposition, D ( pr‹ ϕ srăsp ϕ Ñ ξ qq Ñ p ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ qq . Thus, D ( pr‹ ϕ srăsp ϕ Ñ ξ qq Ñ p ϕ Ñ răsr‹ ϕ sp ϕ Ñ ξ qq . Fact 34.
Let M Ď Mod p L ď q be a class of preference models, ‹ : Mod p L ď q ˆ L Ñ Mod p L ď q be a M -DP3-compliant dynamic operator, and D “ x M , ‹y . Forany propositional formula ϕ P L and dynamic preference formula ξ P L ď p‹q , itholds that D ( B p ϕ | r‹ ϕ s ξ q Ñ r‹ ϕ s B p ϕ | ξ q roof. Let D “ x M, ‹y P D s.t. M “ x W, ď , v y is a preference model and ‹ be a dynamic operator and D ( B p ϕ | r‹ ϕ s ξ q . For any ϕ P L , let’s call ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y and D “ x M ‹ ϕ , ‹y . We have to show that M in ď ‹ ϕ J ξ K D Ď J ϕ K D .Take w P M in ď ‹ ϕ J ξ K D and suppose D , w * ϕ , by definition D , w ( răs ξ ,thus by Proposition 19, D, w ( r‹ ϕ srăs ξ . Since ‹ is M -DP3-compliant,by Proposition 33, D, w ( răsr‹ ϕ sp ξ Ñ ϕ q . As D ( B p ϕ | r‹ ϕ s ξ q , then M in ď J r‹ ϕ s ξ K D Ď J ϕ K D and thus either (i) w P J r‹ ϕ s ξ K D which contradicts D , w * ϕ , or (ii) M in ď J r‹ ϕ s ξ K D Ę J ϕ K D , which contradicts D ( B p ϕ | r‹ ϕ s ξ q .Thus it must hold that D , w ( ϕ , i.e. M in ď ‹ ϕ J ξ K D Ď J ϕ K D and, thus, D ( r‹ ϕ s B p ϕ | ξ q . Proposition 39.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D for any ϕ P L and ξ P L ď p‹q iff ‹ is M -Rec-compliant. r‹ ϕ srăs ξ Ñ p ϕ Ñ A p ϕ Ñ r‹ ϕ s ξ qq ϕ Ñ pr‹ ϕ srďs ϕ q Proof.
Let D “ x M, ‹y P D s.t. M “ x W, ď , v y is a preference model and ‹ be adynamic operator. For any ϕ P L , let’s call ‹p M, ϕ q “ M ‹ ϕ “ x W, ď ‹ ϕ , v y and D “ x M ‹ ϕ , ‹y . ñ :Take w, w P W s.t. w P J ϕ K and w R J ϕ K and ξ P L p‹q s.t. D , w ( ξ , i.e. D, w ( r‹ ϕ s ξ .Firstly, let’s show that w ę ‹ ϕ w . As w P J ϕ K , then D, w ( ϕ . Since theaxiom schemata is valid in D , then D, w ( r‹ ϕ srďs ϕ , i.e. for any w P W if w ď ‹ ϕ w then D , w ( ϕ . Since ϕ is propositional formula, then it must holdthat for any w P W if w ď ‹ ϕ w then D, w ( ϕ . As w R J ϕ K , then D, w * ϕ and, thus, w ę ‹ ϕ w .Now let’s show that all information is preserved. As ϕ is a propositionalformula, by Proposition 19, it holds that D, w ( ϕ ^ r‹ ϕ sp ϕ ^ ξ q . Since thereis a world in W that satisfies ϕ ^ r‹ ϕ sp ϕ ^ ξ q , by Definition 17, we can concludethat D, w ( ϕ ^ E p ϕ ^r‹ ϕ sp ϕ ^ ξ qq . Since the axiom schemata is valid in D byhypothesis, it must hold that D, w ( r‹ ϕ sxăyp ϕ ^ ξ q , i.e. D , w ( xăyp ϕ ^ ξ q .As such, there is some w P W s.t. D , w ϕ ^ ξ and w ă ‹ ϕ w . Hence, w P J ϕ K , D, w ( r‹ ϕ s ξ and w ă ‹ ϕ w .In other words, ‹ satisfies p Rec q . Since it holds for any D P D , ‹ is M -Rec-compliant. ð :(i) Take w P W s.t. D, w ( ϕ ^ E p ϕ ^ r‹ ϕ s ξ q for some ξ P L ď p‹q . Then w P J ϕ K and there is some w P W s.t. w P J ϕ K and D, w ( r‹ ϕ s ξ , i.e. D , w ( ξ .By p Rec q , there is some w P W s.t. w ă ‹ ϕ w and D , w ( ξ , as such D , w ( xăy ξ , i.e. D, w ( r‹ ϕ sxăy ξ . Since it holds for any w P W , we canconclude that D ( p ϕ ^ E p ϕ ^ r‹ ϕ s ξ qq Ñ r‹ ϕ sxăy ξ.
41y contraposition, D ( r‹ ϕ srăs ξ Ñ p ϕ Ñ A p ϕ Ñ r‹ ϕ s ξ qq . Since it holds for any D P D , D ( r‹ ϕ srăs ξ Ñ p ϕ Ñ A p ϕ Ñ r‹ ϕ s ξ qq . (ii) Take w, w P W s.t. D, w ( ϕ and w ď ‹ ϕ w . Clearly, D, w ( ϕ ,otherwise by p Rec q w ă ‹ ϕ w . As such, D , w ( rďs ϕ , i.e. D, w ( r‹ ϕ srďs ϕ .Since it holds for any w P W and D P D , we conclude that D ( ϕ Ñ r‹ ϕ srďs ϕ Proposition 47.
Let D “ x M , ‹y be a class of dynamic models. The followingaxiom schemata is valid in D , for all n P N , ϕ P L , and ξ P L ď p‹q if ‹ is M -LC-compliant. r‹ ϕ srďs ξ Ñ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srăs ξ Ñ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i ` µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srďs ξ Ð p µdg ϕ p n q _ µdg ϕ p n qq ^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ qr‹ ϕ srăs ξ Ð p µdg ϕ p n q _ µdg ϕ p n qq ^ n ´ ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ´ ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q roof. We will only show it holds for the schemata related to modality rďs ,since the proof for the others is similar.Let D “ x M , ‹y be a class of dynamic models and ϕ P L be a propositionalformula. Take D P D s.t. D “ x M, ‹y and D “ x M ‹ ϕ , ‹y , with M “ x W, ď , v y and M ‹ ϕ “ x W, ď ‹ ϕ , v y .(i) Let w P W be a possible world s.t. D, w ( r‹ ϕ srďs ξ for some ξ P L ď p‹q and D, w ( µdg χ p n q for χ P t ϕ, ϕ u and n P N - notice that one such n alwaysexists since the models are well-founded. As D, w ( µdg χ p n q , by Lemma 10,there is a maximal chain of χ -worlds of size n , starting in a minimal χ -world andending in w . Since D, w ( r‹ ϕ srďs ξ , for all w P W s.t. w ď ‹ ϕ w , then D , w ( ξ . Since ‹ is M -LC-compliant, by p LC q , w ď ‹ ϕ w iff there is a maximal chainof χ -worlds of size n , with χ P t ϕ, ϕ u , starting in a minimal χ -world andending in w and n ď n . From that we conclude that for any i ď n and w P W ,if D, w ( µdg χ p i q , then it must hold that D , w ( ξ , i.e. D, w ( r‹ ϕ s ξ . Thus, D ( r‹ ϕ srďs ξ Ñ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ n ľ j “ i µdg ϕ p j q Ñ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q , for n P N (ii) Let w P W be a possible world s.t. D, w ( µdg χ p n q for χ P t ϕ, ϕ u and n P N and for any i ď n it holds that D, w ( A p µdg χ p i q Ñ r‹ ϕ s ξ q , for χ P t ϕ, ϕ u . As such, for any w P W , if D, w ( µdg χ p i q then D, w ( r‹ ϕ s ξ ,for any i ď n and χ P t ϕ, ϕ u . As D, w ( µdg χ p n q there is a maximalchain of χ -worlds of size n , starting in a minimal χ -world and ending in w .Since ‹ is M -LC-compliant, by p LC q , w ď ‹ ϕ w iff there is a maximal chainof χ -worlds of size n , with χ P t ϕ, ϕ u , starting in a minimal χ -world andending in w and n ď n . From that we conclude that for all w P W s.t. w ď ‹ ϕ w it must hold that D, w ( r‹ ϕ s ξ . As such, D, w ( rďsr‹ ϕ s ξ . Thus, D ( r‹ ϕ srďs ξ Ð p µdg ϕ p n q _ µdg ϕ p n qq ^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q^ n ľ i “ A p µdg ϕ p i q Ñ r‹ ϕ s ξ q Theorem 52.
Let M be a class of preference models, C “ t C i | i P I u be a familyof classes of dynamic operators which are closed over M , and A “ t A i | i P I u afamily of sound axiom systems for C , i.e. Log p A i q Ď Log px M , C i yq , both indexedby some set I . Log p ď i P I A i q Ď Log px M , č i P I C i q Proof.
Let’s call Ş C “ t‹ : M od p L ď q ˆ L Ñ M od p L ď q | ‹ P C i for all i P I u .43otice that if Ş C “ H , then Log ptx M, ‹y | M P M and ‹ P C i for all i P I uq “ Log px M , Ş C yq “ Log pHq “ L ď p‹q and, clearly, Log p Ť i P I A i q Ď L ď p‹q , so let’sassume Ş C ‰ H .Take ‹ P Ş C , then for all i P I , it must be the case that Log p A i q Ď Log px M , ‹yq , since Log p Ť i P I A i q Ď Log px M , C i yq Ď Log px M , ‹yq , by Proposi-tion 49. Particularly, Ť i P I A i Ď Log px M , ‹yq .It is easy to see that for any class of preference models M and dynamicoperator ‹ , Log px M , ‹yq is closed over modus ponens and necessitation rules.Hence, by Definition 50, Log p Ť i P I A i q Ď Log px M , ‹yq . Since, for any ‹ P Ş C , Log p Ť i P I A i q Ď Log px M , ‹yq , then Log p ď i P I A i q Ď Log ptx M, ‹y | M P M and ‹ P C i for all i P I uq ..