Iterated Belief Base Revision: A Dynamic Epistemic Logic Approach
aa r X i v : . [ c s . L O ] F e b Iterated Belief Base Revision: A Dynamic Epistemic Logic Approach
Marlo Souza
Department of Computer Science, UFBASalvador, [email protected] ´Alvaro Moreira
Institute of Informatics, UFRGSPorto Alegre, [email protected]
Renata Vieira
Polytechnic School, PUCRSPorto Alegre, [email protected]
Abstract
AGM’s belief revision is one of the main paradigms in thestudy of belief change operations. In this context, beliefbases (prioritised bases) have been largely used to specifythe agent’s belief state - whether representing the agent’s ‘ex-plicit beliefs’ or as a computational model for her belief state.While the connection of iterated AGM-like operations andtheir encoding in dynamic epistemic logics have been stud-ied before, few works considered how well-known postulatesfrom iterated belief revision theory can be characterised bymeans of belief bases and their counterpart in a dynamic epis-temic logic. This work investigates how priority graphs, asyntactic representation of preference relations deeply con-nected to prioritised bases, can be used to characterise beliefchange operators, focusing on well-known postulates of It-erated Belief Change. We provide syntactic representationsof belief change operators in a dynamic context, as well asnew negative results regarding the possibility of representingan iterated belief revision operation using transformations onpriority graphs.
Introduction
One of the most influential models for be-lief change is the so-called AGM paradigm,named after the authors of the seminal work(Alchourr´on, G¨ardenfors, and Makinson 1985). Althoughthe AGM’s approach has brought profound developmentsfor the problem of belief dynamics, influencing areas suchas Computer Science, Artificial Intelligence, and Philoso-phy, it has not been immune to criticism (Hansson 1992;Rott 2000; Darwiche and Pearl 1997).Particularly, Hansson (1992) criticises the use of deduc-tively closed sets of formulas in the AGM paradigm, provid-ing examples for which not just the meaning, but also thestructure of the beliefs explicitly held by an agent may influ-ence the change. This author proceeds to construct a differ-ent notion of belief revision, which became known as BeliefBase Change, relying on the structure of the information be-lieved by the agent.While many studies (Nebel 1991; Williams 1994;Williams 1995; Rott 2009) propose belief change operatorsbased on the syntactic structure of an agent’s explicit beliefs,or on the syntactic representation of an agent’s belief state, few formal connections have been established between be-lief change postulates and these belief base change operators- some examples of works that investigate such connectionare (Hansson 1994; Ferm´e, Krevneris, and Reis 2008;Ferm´e, Garapa, and Reis 2017). The axiomatic characteri-sation of syntactic-based belief change, however, has beenconcentrated on the one-shot behaviour of these operationsand, thus, are not able to clarify their iterated or dynamicbehaviour.Belief base change operators are iterated, in the sensethat they result in changing the belief state of the agentitself (Nayak, Pagnucco, and Peppas 2003). As such, it isnecessary to establish which formal properties these changeoperations satisfy on a dynamic sense, as studied in the liter-ature of Iterated Belief Change (Darwiche and Pearl 1997;Nayak, Pagnucco, and Peppas 2003;Jin and Thielscher 2007).A step into connecting belief change operators and be-lief base change operators has been achieved by the work ofRott (2009) and Liu (2011) which show that several beliefchange operations can be defined using transformations onsome form of prioritised belief bases called priority graphs.Souza et al. (2016), on the other hand, provide examples ofbelief change operations which cannot be defined in such away.While these latter works focus on using syntactic rep-resentations to construct belief change operations, they donot provide a strong connection between the syntactic-basedtransformations provided and the formal properties of thebelief change operations they represent. To our knowledge,there is no proposed characterisation in the literature of iter-ated belief change postulates based on syntactic representa-tions of an agent’s belief base.In this work, we study the relationship between iteratedbelief change postulates and properties of belief base changeoperators based on the counterpart of these operators astransformations on Liu’s priority graphs (2011). We obtainconstraints on belief base change operators that guaranteesatisfaction of some important postulates in the area of Iter-ated Belief Change.We also show the limits of expressibility of belief changeoperators by means of changes in belief bases. These neg-ative results provide us with a deeper understanding of theconnections between belief base change and iterated beliefhange, helping to delineate the limits of the correspondencebetween semantic and syntactic theories of dynamic beliefchange.This work is structured as follows: we start discussingthe background knowledge in the next cection. Then, westudy how iterated belief change postulates can be charac-terised through transformations on priority graphs. Further,we show how our results can be used to construct syntacticrepresentations of an iterated belief change operation, and inSection
Negative Results we present some impossibility re-sults for this characterisation. Finally, we discuss the relatedliterature and present some final considerations.
Background
Much work has been conducted investigating syn-tactic representations of the explicit beliefs of anagent (Hansson 1992; Williams 1994; Rott 2009;De Jongh and Liu 2009; Baltag, Fiutek, and Smets 2016).It is well-known that an agent’s belief state - usuallyrepresented by a preference relation among worlds - canalso be syntactically represented by means of orders amongsentences, as well-investigated by Lafage and Lang (2005).We start by introducing the syntactic representations wewill use in this work to encode an agent’s belief base. Thisstructure, known as a priority graph, or P-graph for short,was proposed by Liu (2009; 2011) and was further devel-oped by Van Benthem, Grossi and Liu (2014), and Souza etal (2016). We will also present their relation to preferencemodels - a generalisation of the models used in the area ofbelief revision to encode an agent’s belief state. With thisconnection, we will be able to characterise well-known pos-tulates of iterated belief revision using operations on prioritygraphs, thus connecting belief base change operations anditerated belief revision.
Definition 1. (Liu 2011) Let P be a countable set of propo-sitional symbols and L ( P ) the language of classical propo-sitional sentences over the set P . A P-graph is a tuple G = h Φ , ≺i where Φ ⊂ L ( P ) , is a set of propositionalsentences and ≺ is a strict partial order on Φ . Given a non-empty set of possible worlds W and a valua-tion of propositional symbols over worlds of W , the order ≺ of a P-graph can be lifted to an order on worlds. Such an or-dering can be called a preference (or a plausibility) relation. Definition 2. (Liu 2011) Let G = h Φ , ≺i be a P-graph, and v : P → W be a valuation function of propositions over anon-empty set W of possible worlds. The preference relation ≤ G on W induced by P-graph G is defined as follows: w ≤ G w ′ iff ∀ ϕ ∈ Φ : (( w ′ (cid:15) ϕ ⇒ w (cid:15) ϕ ) or ∃ ψ ∈ Φ : ψ ≺ ϕ, w (cid:15) ψ, and w ′ (cid:15) ψ ) More yet, if a P-graph G is finite,Van Benthem, Grossi, and Liu (2014) have shown thata preference relation ≤ G , defined as above, satisfies well-foundedness - a property deeply connected with LewisLimit Assumption, commonly required for semantic modelsof an agent’s belief state.It is worthy of notice that preference relationsare a common semantic representation of an agents belief state. In fact, in Belief Revision Theory(Alchourr´on, G¨ardenfors, and Makinson 1985), an im-portant semantic representation of agents belief states wasproposed by Adam Grove, which is commonly known asGrove’s systems of spheres (SOS) (Grove 1988), or simplyas Grove’s spheres.A generalisation of Grove’s spheres was given by Gi-rard (2008), called preference models or order mod-els, in the context of Preference Logic. Similar mod-els have been proposed before in the context of Qualita-tive Decision Theory (Boutilier 1994), Non-monotonic Rea-soning (Kraus, Lehmann, and Magidor 1990), among oth-ers. These models have been applied to the study ofDynamic Belief Revision by the work of Baltag andSmets (Baltag and Smets 2008), Girard and Rott (2014),Liu (2011), and Souza et al. (2016; 2017) and proved tobe an expressible model for an agent’s belief state. In thiswork, we will adopt Souza’s (2016) definition of preferencemodels - which requires the preference relation ≤ to have awell-founded strict part. Definition 3. (Souza 2016) A preference model is a tuple M = h W, ≤ , v i where W is a set of possible worlds, ≤ isa reflexive, transitive relation over W with a well-foundedstrict part, and v : P → W a valuation function. From the above definitions, it is easy to see that from aP-graph we can construct a preference model by taking thepreference relation induced by such a graph.
Definition 4.
Let G = h Φ , ≺i be a P-graph and let M = h W, ≤ , v i be a preference model. We say M is induced by G iff ≤ = ≤ G . The induction of preference models from P-graphs raisesthe question about the relations between these two struc-tures. Liu (2011) shows that any finite model with a reflex-ive and transitive accessibility relation has an equivalent P-graph.
Theorem 5. (Liu 2011) Let M = h W, R i be a mono-modalKripke structure. The following two statements are equiva-lent:1. The relation R is reflexive and transitive;2. There is a priority graph G = (Φ , ≺ ) and a valuation v s.t. ∀ w, w ′ ∈ W. wRw ′ iff w ≤ G w ′ . It is well-known from work on IteratedBelief Revision (Darwiche and Pearl 1997;Nayak, Pagnucco, and Peppas 2003) that dynamic be-lief change operations can be described by a transformationin the agent’s belief state.As such, we can define dynamic belief change op-erators using the following notion of dynamic opera-tors on preference models, where M ( L ≤ ( P )) is the classof all preference models for a logic language L ≤ ( P ) (Souza, Moreira, and Vieira 2017): Definition 6. (Souza, Moreira, and Vieira 2017) Let ⋆ : M ( L ≤ ( P )) × L ( P ) → M ( L ≤ ( P )) , we say ⋆ is a dy-namic operator on preference models if for any preferencemodel M = h W, ≤ , v i and formula ϕ ∈ L , we have that ⋆ ( M, ϕ ) = h W, ≤ ⋆ , v i . In other words, an operation onreference models is called a dynamic operator iff it onlychanges the relation of preference models. Liu et al (2011; 2014) shows that these dynamic beliefchanges can also be described by means of changes in thepriority graphs representing the agent’s belief base. In thefollowing, G ( P ) denotes the set of all P-graphs constructedover a set P of propositional symbols. Definition 7.
We call a P-graph transformation any function † : G ( P ) × L ( P ) → G ( P ) . A P-graph transformation is, thus, a transformation in theagent’s belief base, as represented by a priority graph.Since P-graphs and preference models are translatableinto one another, it is easy to connect P-graph transforma-tions and dynamic operators as well.
Definition 8.
Let ⋆ : M ( L ≤ ( P )) × L ( P ) → M ( L ≤ ( P )) be a dynamic operator and † : G ( P ) × L ( P ) → G ( P ) bea P-graph transformation. We say ⋆ is induced by † if forany preference model M ∈ M ( L ≤ ( P )) and any P-graph G ∈ G ( P ) , if M is induced by G then the preference model ⋆ ( M, ϕ ) is induced by the P-graph † ( G, ϕ ) , where ϕ is anypropositional formula in L ( P ) , Notice that not all P-graph transformations induce dy-namic operators. We say that syntactically different P-graphsare equivalent if they induce the same preference model.As such, if a P-graph transformation changes equivalent P-graphs in inconsistent ways, no dynamic operator can satisfythe condition of Definition 8.For example, consider a P-graph transformation thatchanges the P-graph p ≺ q into the P-graph p ≺ q , andchanges the P-graph p ∧ q ≺ p ∧ ¬ q ≺ ¬ p ∧ q ≺ ¬ p ∧ ¬ q into p ∧ q ≺ ¬ p ∧ q ≺ p ∧ ¬ q ≺ ¬ p ∧ ¬ q . Such a transforma-tion cannot induce any dynamic operator since the originalP-graphs are equivalent, i.e., induce the same models, butthe resulting P-graphs are not. As such, we define the notionof relevant graph transformation. Definition 9.
We say that a be a P-graph transformation † isrelevant if there is some dynamic operator ⋆ that is inducedby it. Notice that, the existence of relevant P-graph transfor-mations is guaranteed by the previous representation resultsof Liu (2011) and Van Benthem, Grossi, and Liu (2014) onthe characterisation of some dynamic operators by means ofchanges in P-graphs.
Priority Changes Satisfying Postulates forIterated Belief Changes
In this work, we investigate the connection between theproperties satisfied by dynamic belief change operations, fo-cusing on the postulates studied in the field of Iterated BeliefRevision, and the properties satisfied by the P-graph trans-formations that encode these operations. We aim to under-stand better which belief change operations can or cannotbe encoded this way and, thus, the differences between dy-namic belief change based on semantic models and based onsyntactic representations. As such, the main results of our work can be stated as thecharacterisations provided in Propositions 11, 12, 13, 14, 15and 16, as well as the negative results provided in Fact 21and Corollary 23. An interesting application of these resultswill be obtained in the next Section, in which we constructa P-graph transformation to implement the operation oflexicographic revision (Nayak, Pagnucco, and Peppas 2003)and obtain, as a corollary, the harmony result proved byVan Benthem, Grossi, and Liu (2014) stating the correctnessof this transformation.The use of postulates to encode rational constraints inthe way an agent must change her beliefs is a definingcharacteristic of the AGM approach to Belief Revision(Alchourr´on, G¨ardenfors, and Makinson 1985). These pos-tulates, however, are usually defined by means of constraintson changes in the agent’s belief state, thus, in our case, onpreference models. We must, then, define what it means fora P-graph transformation to satisfy some postulate (or prop-erty) for belief change operators.
Definition 10.
We say that a P-graph transformation † sat-isfies a postulate P if (i) † is relevant and (ii) any dynamicoperator ⋆ induced by † satisfies postulate P . AGM belief revision says very little about how to changeone agent’s beliefs repeatedly. In fact, it has been observedthat the AGM approach allows some counter-intuitive be-haviour in the iterated case (Darwiche and Pearl 1997). As aresult, different authors have proposed additional postulatesthat encode rational ways to change one’s beliefs in an iter-ated way.Most famous among them is the work of Darwiche andPearl (1997). They propose a set of postulates known as theDP postulates for iterated revision. Let S = h W, ≤i be aSOS and S ′ = h W, ≤ ∗ ϕ i the result of revising the SOS S bya formula ϕ , the DP postulates can be stated as: (DP-1) If w, w ′ ∈ J ϕ K , then w ≤ ∗ ϕ w ′ iff w ≤ w ′ (DP-2) If w, w ′ J ϕ K , then w ≤ ∗ ϕ w ′ iff w ≤ w ′ (DP-3) If w ∈ J ϕ K and w ′ J ϕ K , then w < w ′ ⇒ w < ∗ ϕ w ′ (DP-4) If w ∈ J ϕ K and w ′ J ϕ K , then w ≤ w ′ ⇒ w ≤ ∗ ϕ w ′ We want to provide a set of constraints on P-graph trans-formations that guarantee that the dynamic operators in-duced by them satisfy these postulates. With that, we wishto study the connections between the classes of belief baserevision operations and iterated belief change operators.Let us start with DP-1. The postulate DP-1 states that forany two worlds w, w ′ satisfying ϕ , there is no reason fortheir relative order to change in the agent’s belief state af-ter revision. In terms of P-graphs, this means that for anyformula that w ′ satisfies in the changed priority graph, ei-ther w must also satisfy it, or there must be a formula that ispreferred to it and that w satisfies. We can ensure this prop-erty guaranteeing that the resulting P-graph is related to theoriginal by a set of constraints in how it must be changed. Proposition 11.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the conditions below, then † satisfies DP-1 :. For all ξ ∈ Φ , there is some ξ ′ ∈ Φ † s.t.(a) ϕ ∧ ξ ≡ ϕ ∧ ξ ′ and(b) ∀ ψ ′ ∈ Φ † , if ψ ′ ≺ † ξ ′ then ψ ′ ≡ ϕ orthere is ψ ∈ Φ s.t. ϕ ∧ ψ ≡ ϕ ∧ ψ ′ and ψ ≺ ξ ;2. For all ξ ∈ Φ † , ξ ≡ ϕ or there is some ξ ′ ∈ Φ s.t.(a) ϕ ∧ ξ ≡ ϕ ∧ ξ ′ and(b) ∀ ψ ′ ∈ Φ , if ψ ′ ≺ ξ ′ then there is ψ ∈ Φ † s.t. ϕ ∧ ψ ≡ ϕ ∧ ψ ′ and ψ ≺ † ξ .Proof. By Definition 10 we have to prove that any dy-namic operator ⋆ induced by † satisfies the postulate DP-1 .Let † be a P-graph transformation satisfying the conditionsabove, ⋆ a dynamic operator induced by † , M = h W, ≤ , v i a preference model, and ϕ ∈ L a propositional formula.Also given M ⋆ϕ = ⋆ ( M, ϕ ) = h W, ≤ ⋆ϕ , v i , take a P-graph G = h Φ , ≺i inducing M s.t. † ( G, ϕ ) = h Φ † , ≺ † i induces ⋆ ( M, ϕ ) . ⇐ :Take w, w ′ ∈ J ϕ K M such that w ≤ ⋆ϕ w ′ , and take ξ ∈ Φ such that M, w ′ (cid:15) ξ - notice that the case in which no suchformula exists is trivial, since necessarily w ≤ w ′ in suchcase. Then M ⋆ϕ , w ′ (cid:15) ξ , since ξ is propositional formula.Since ξ ∈ Φ then, by condition 1(a), there is some ξ ′ ∈ Φ † s.t. ϕ ∧ ξ ≡ ϕ ∧ ξ ′ . As such, M ⋆ϕ , w ′ (cid:15) ξ ′ , given that ⋆ isinduced by † . As w ≤ ⋆ϕ w ′ , either:(i) M ⋆ϕ , w (cid:15) ξ ′ , or(ii) there is some ψ ′ ∈ Φ † s.t. ψ ′ ≺ † ξ ′ , M ⋆ϕ , w (cid:15) ψ ′ and M ⋆ϕ , w ′ (cid:15) ψ ′ . Hence, by condition 1 (b), there is some ψ ∈ Φ s.t. ϕ ∧ ψ ≡ ϕ ∧ ψ ′ and ψ ≺ ξ . As such M, w (cid:15) ψ and M, w ′ (cid:15) ψ .From (i) and (ii) we conclude that w ≤ w ′ ⇒ : Similar to the case before. Take w, w ′ ∈ J ϕ K M s.t. w ≤ w ′ and ξ ∈ Φ † , use condition 2 to conclude that w ≤ ⋆ϕ w ′ .The postulate DP-2 describes the same information asDP-1, only restricted to those worlds that do not satisfy ϕ .As such, we can provide a similar characterisation. Proposition 12.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the conditions below, then † satisfies DP-2 :1. For all ξ ∈ Φ , there is some ξ ′ ∈ Φ † s.t.(a) ¬ ϕ ∧ ξ ≡ ¬ ϕ ∧ ξ ′ and(b) ∀ ψ ∈ Φ , if ψ ≺ ξ then there is ψ ′ ∈ Φ † s.t.i. ¬ ϕ ∧ ψ ≡ ¬ ϕ ∧ ψ ′ andii. ψ ′ ≺ † ξ ′ ;2. For all ξ ∈ Φ † , ξ ≡ ϕ or there is some ξ ′ ∈ Φ s.t.(a) ¬ ϕ ∧ ξ ≡ ¬ ϕ ∧ ξ ′ and(b) ∀ ψ ∈ Φ † , if ψ ≺ † ξ then ψ ≡ ϕ orthere is ψ ′ ∈ Φ s.t. ¬ ϕ ∧ ψ ≡ ¬ ϕ ∧ ψ ′ and ψ ′ ≺ ξ ′ . Proof. Similar to that of Proposition 11.The postulate DP-3 states that for any two worlds w sat-isfying ϕ and w ′ not satisfying it, after revision by ϕ , if w was preferable to w ′ then it must continue to be so in theagent’s belief state. In terms of P-graphs, we can guaranteethis condition requiring that, if there was a formula in theoriginal P-graph that w satisfied and w ′ did not, there mustbe a formula in the revised P-graph s.t. w satisfies and w ′ does not. Therefore, we can characterise DP-3. Proposition 13.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the condition below, then † sat-isfies DP-3 : • For all ξ ∈ Φ or there is some ξ ′ ∈ Φ † s.t.(a) ϕ ∧ ξ ⊢ ξ ′ ,(b) ¬ ϕ ∧ ξ ′ ⊢ ξ , and(c) ∀ ψ ′ ∈ Φ † , if ψ ′ ≺ † ξ ′ then ψ ′ ≡ ϕ or there is ψ ∈ Φ s.t. ϕ ∧ ψ ⊢ ψ ′ , ¬ ϕ ∧ ψ ′ ⊢ ψ and ψ ≺ ξ .Proof. Let † be a P-graph transformation satisfying the con-ditions above, ⋆ a dynamic operator induced by † , M = h W, ≤ , v i a preference model and ϕ ∈ L ( P ) a proposi-tional formula. Also given M ⋆ϕ = ⋆ ( M, ϕ ) = h W, ≤ ⋆ϕ , v i ,take G = h Φ , ≺i be a P-graph inducing M s.t. † ( G, ϕ ) = h Φ † , ≺ † i induces ⋆ ( M, ϕ ) .Take w, w ′ ∈ W s.t. M, w (cid:15) ϕ , M, w ′ (cid:15) ϕ and w < w ′ .From w < w ′ we conclude that there is some ξ ∈ Φ s.t. M, w (cid:15) ξ , M, w ′ (cid:15) ξ and for all ∀ ψ ≺ ξ , if M, w ′ (cid:15) ψ then M, w (cid:15) ψ . Since ξ ∈ Φ , there is some ξ ′ ∈ Φ † s.t. ϕ ∧ ξ ⊢ ξ ′ , ¬ ϕ ∧ ξ ′ ⊢ ξ . As such M ⋆ϕ , w (cid:15) ξ ′ , M ⋆ϕ , w ′ (cid:15) ξ and for all ψ ′ ≺ † ξ ′ , if M ⋆ϕ , w ′ (cid:15) ψ ′ then there is ψ ∈ Φ s.t. ϕ ∧ ψ ⊢ ψ ′ , ¬ ϕ ∧ ψ ′ ⊢ ψ and ψ ≺ ξ . As such M, w ′ (cid:15) ψ and, thus, M, w (cid:15) ψ . Since ϕ ∧ ψ ⊢ ψ ′ , then M ⋆ϕ , w (cid:15) ψ ′ ,thus w ≺ † w ′ .Along the same lines, as we did for DP-3, we can charac-terise postulate DP-4. Proposition 14.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the condition below, then † sat-isfies DP-4 : • For all ξ ∈ Φ † , ξ ≡ ϕ or there is some ξ ′ ∈ Φ s.t.(a) ϕ ∧ ξ ′ ⊢ ξ ,(b) ¬ ϕ ∧ ξ ⊢ ξ ′ , and(c) ∀ ψ ′ ∈ Φ , if ψ ′ ≺ ξ ′ then there is ψ ∈ Φ † s.t. ϕ ∧ ψ ′ ⊢ ψ , ¬ ϕ ∧ ψ ⊢ ψ ′ and ψ ≺ ξ .Proof. Let † be a P-graph transformation satisfying the con-ditions above, ⋆ a dynamic operator induced by † , M = h W, ≤ , v i a preference model, and ϕ ∈ L ( P ) a proposi-tional formula. Also given M ⋆ϕ = ⋆ ( M, ϕ ) = h W, ≤ ⋆ϕ , v i ,take G = h Φ , ≺i be a P-graph inducing M s.t. † ( G, ϕ ) = h Φ † , ≺ † i induces ⋆ ( M, ϕ ) .ake w, w ′ ∈ W s.t. M, w (cid:15) ϕ , M, w ′ (cid:15) ϕ , and w ≤ w ′ .Take ξ ∈ Φ † s.t. M, w ′ (cid:15) ξ . There is some ξ ′ ∈ Φ s.t. ϕ ∧ ξ ′ ⊢ ξ , ¬ ϕ ∧ ξ ⊢ ξ ′ . As such, M, w ′ (cid:15) ξ ′ .Since w ≤ w ′ , either (i) M, w (cid:15) ξ ′ or (ii) there is some ψ ′ ≺ ξ ′ s.t. M, w (cid:15) ψ ′ s.t. M, w ′ (cid:15) ψ ′ . From (ii), we con-clude that there is some ψ ∈ Φ † s.t. ψ ≺ ξ , ϕ ∧ ψ ′ ⊢ ψ ,and ¬ ϕ ∧ ψ ⊢ ψ ′ . Since ¬ ϕ ∧ ψ ⊢ ψ ′ and M, w ′ (cid:15) ψ ′ , weconclude that M, w ′ (cid:15) ψ and, thus, M ⋆ϕ , w ′ (cid:15) ψ , as ψ isa propositional formula. From (i) and (ii) we conclude that w ≤ ⋆ϕ w ′ .We know that the operation of lexicographic revision(Nayak, Pagnucco, and Peppas 2003) can be defined bymeans of transformations on P-graphs (Liu 2011). Also,Nayak et al. (2003) have shown that the operation of lexi-cographic revision is completely characterized by the pos-tulates DP-1, DP-2, and the following postulate known asRecalcitrance (R EC ). (R EC ) If w ∈ J ϕ K and w ′ J ϕ K , then w < ∗ ϕ w ′ .As such, it is expected that we can characterise R EC aswell. Postulate R EC requires for any world satisfying ϕ to bepreferred to those not satisfying it. This requirement can beeasily guaranteed if all the minimal elements of the changedP-graph which express relevant information (i.e., not equiv-alent to ⊥ nor ⊤ ) imply ϕ . Proposition 15.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the condition below, then † sat-isfies R EC : • For all ξ ∈ Φ † , either ξ ≡ ⊤ , ξ ≡ ⊥ , ξ ⊢ ϕ or there issome ψ ∈ Φ † s.t. ψ ≺ † ξ , ψ
6≡ ⊥ and ψ ⊢ ϕ ; • There is some ξ ∈ Φ s.t. ξ ⊢ ϕ .Proof. It is immediate that if a graph transformation satis-fies the condition above, the induced dynamic operator mustsatisfy R EC since any world satisfying ϕ in a model inducedby such a graph would be preferred over any world not sat-isfying ϕ .Since DP-3 and DP-4 can be characterised by means oftransformations on P-graphs, it is expected that a relatedpostulate might be as well. The postulate of Independencebelow, proposed by Jin and Thielscher (2007) and indepen-dently by Booth and Meyer (2006), states that a revision op-eration may not create arbitrary conditional beliefs in theagent’s belief state. (I ND ) If w ∈ J ϕ K and w ′ J ϕ K , then w ≤ w ′ ⇒ w < ∗ ϕ w ′ .The postulate I ND is, in fact, a stronger form of both DP-3and DP-4. As such, we can provide the following charac-terisation for it, based on the characterisation of DP-3 andDP-4. Proposition 16.
Let † : G ( P ) × L ( P ) → G ( P ) be arelevant P-graph transformation. If, for any P-graph G = h Φ , ≺i and propositional formula ϕ ∈ L ( P ) , the P-graph † ( G, ϕ ) = h Φ † , ≺ † i satisfies the condition below, then † sat-isfies I ND : • For all ξ ′ ∈ Φ † , ξ ′ ≡ ϕ or there is some ξ ∈ Φ s.t.(a) ϕ ∧ ξ ⊢ ξ ′ ,(b) ¬ ϕ ∧ ξ ′ ⊢ ξ , and(c) ∀ ψ ′ ∈ Φ † , if ψ ′ ≺ † ξ ′ then there is ψ ∈ Φ s.t. ϕ ∧ ψ ⊢ ψ ′ , ¬ ϕ ∧ ψ ′ ⊢ ψ and ψ ≺ ξ .(d) if ξ ′ ϕ , there is some ψ ′ ≺ † ξ ′ s.t. ψ ′ ≡ ϕ Proof.
Let † be a P-graph transformation satisfying the con-ditions above, ⋆ a dynamic operator induced by † , M = h W, ≤ , v i a preference model and ϕ ∈ L ( P ) a proposi-tional formula. Also given M ⋆ϕ = ⋆ ( M, ϕ ) = h W, ≤ ⋆ϕ , v i ,take G = h Φ , ≺i be a P-graph inducing M s.t. † ( G, ϕ ) = h Φ † , ≺ † i induces ⋆ ( M, ϕ ) .Take w, w ′ ∈ W s.t. M, w (cid:15) ϕ , M, w ′ (cid:15) ϕ and w ≤ w ′ .Take ξ ∈ Φ † s.t. M, w ′ (cid:15) ξ . There is some ξ ′ ∈ Φ s.t. ϕ ∧ ξ ′ ⊢ ξ , ¬ ϕ ∧ ξ ⊢ ξ ′ . As such, M, w ′ (cid:15) ξ ′ .Since w ≤ w ′ , either (i) M, w (cid:15) ξ ′ or (ii) there is some ψ ′ ≺ ξ ′ s.t. M, w (cid:15) ψ ′ and M, w ′ (cid:15) ψ ′ . As such, there is ψ ∈ Φ † s.t. ψ ≺ ξ , ϕ ∧ ψ ′ ⊢ ψ , and ¬ ϕ ∧ ψ ⊢ ψ ′ . Since ¬ ϕ ∧ ψ ⊢ ψ ′ and M, w ′ (cid:15) ψ ′ , we conclude that M, w ′ (cid:15) ψ and, thus, M ⋆ϕ , w ′ (cid:15) ψ , as ψ is a propositional formula.Further, since M, w ′ (cid:15) ξ ′ , then ξ ′ ϕ . As such, there issome ψ ′ ≡ ϕ . We conclude that (iii) M, w ′ (cid:15) ψ ′ and M, w (cid:15) ψ ′ . From (i), (ii), and (iii), we conclude that w < ⋆ϕ w ′ . Deriving P-Graph Transformations fromBelief Revision Policies
The results obtained above can be used to analyse anysyntax-based belief revision policy (P-graph transforma-tions) and derive which belief revision properties (or pos-tulates) it satisfies. On the other hand, these results can alsobe used to derive implementations for a belief revision op-erator based on P-graphs. To illustrate this last point, let usexamine the case of Lexicographic Revision.
Definition 17. (Girard 2008) Let M = h W, ≤ , v i be a pref-erence model and ϕ a formula of L ( P ) . We say the prefer-ence model M ⇑ ϕ = h W, ≤ ⇑ ϕ , v i is the result of the lexico-graphic revision of M by ϕ , where w ≤ ⇑ ϕ w ′ iff w ≤ w ′ if w, w ′ ∈ J ϕ K or w ≤ w ′ if w, w ′ J ϕ K or T rue if w ∈ J ϕ K and w ′ J ϕ K The operation above consists of making each world satis-fying ϕ to be strictly more preferable than those not satisfy-ing it, while maintaining the order otherwise.It is well-known (Nayak, Pagnucco, and Peppas 2003)that lexicographic revision is completely characterized bythe postulates (DP-1), (DP-2) and (R EC ). Hence, usingPropositions 11, 12 and 15, we can construct a P-graphtransformation that satisfies these postulates.A simple P-graph transformation that does satisfy Propo-sitions 11, 12, and 15 is prefixing which was pro-posed by Van Benthem (2009) based on the work ofAndr´eka, Ryan, and Schobbens (2002). Definition 18.
The prefixing of a P-graph G = h Φ , ≺i by a propositional formula ϕ ∈ L ( P ) is the P-graph ; ( G, ϕ ) = h Φ ∪ { ϕ } , ≺ ; ϕ i , usually denoted by ϕ ; G , where ≺ ; ϕ = ≺ ∪ {h ϕ, ψ i | ψ ∈ Φ } bserve that the resulting P-graph maintains all formu-las of Φ , thus satisfying Propositions 11 and 12, and in-cludes a formula ϕ (or equivalent to it) that is preferred toall formulas in Φ , thus satisfying Proposition 15. As such,the dynamic operator induced by P-graph prefixing satisfies(DP-1), (DP-2) and (R EC ). Since these three postulates com-pletely characterize lexicographic revision, we conclude thefollowing. Corollary 19.
Let M be a preference model induced by aP-graph G and ϕ a propositional formula. The model M ⇑ ϕ is induced by the P-graph ϕ ; G . Negative Results
While the previous results are encouraging, Souza etal. (2016) already showed that some belief change opera-tors cannot be defined with P-graphs. As such, it must be thecase that some postulates in the area cannot be representedby means of transformations on P-graphs - or at least not in away in which it is jointly consistent with other postulates. Toprove such a result, those authors show a simple fact aboutpriority graphs: they cannot encode all the information aboutthe models they induce.
Fact 20. (Souza et al. 2016) Let G = h Φ , ≺i a P-graph and ϕ a propositional formula. There is no propositional formula µ ϕ s.t. for every model M = h W, ≤ G , v i induced by G andall w ∈ W , w (cid:15) µ ϕ iff w ∈ Min ≤ G J ϕ K . Fact 20 above provides us with some clues to investigatewhich postulates cannot be characterised through transfor-mations on priority graphs, namely those that refer directlyto the minimal worlds of a model.One trivial example of such a postulate is the property ofan iterated belief change operator to be faithful to AGM’spostulates (Alchourr´on, G¨ardenfors, and Makinson 1985),known as postulate F
AITH below. (F AITH ) If J ϕ K = ∅ then w ∈ Min ≤ J ϕ K iff w ∈ Min ≤ ∗ ϕ W Notice that while F
AITH says something about the mini-mal worlds of a model, it does not characterise this set in anyway. To illustrate it, it suffices to realise that lexicographiccontraction satisfies F
AITH - which describes the change inthe agent’s belief state by changing the preference of all theworlds satisfying a certain propositional formula ϕ . As such,if a P-graph transformation satisfies the postulates DP-1 andREC then it satisfies the postulate FAITH.Let us then consider some belief change operators requir-ing a characterisation of the changes in the belief state whichis completely dependent on some set of minimal worlds. Toconstruct such an operator, let us examine the postulate ofConditional Belief Change Minimisation (CB), proposed byBoutilier (1993). This postulate states that any iterated beliefrevision operation must minimise changes of conditional be-liefs in the belief state of the agent. (CB) If w, w ′ Min ≤ J ϕ K , then w ≤ w ′ iff w ≤ ∗ ϕ w ′ .Together with F AITH , postulate CB characterises a beliefchange operator that is completely defined by the changes inthe minimal worlds satisfying some formula ϕ . As such, it isfairly easy to see that no graph transformation satisfies bothF AITH and CB.
Fact 21.
No relevant P-graph transformation † : G ( P ) ×L ( P ) → G ( P ) satisfies both F AITH and CB .Proof. We suppose that there is a relevant graph transforma-tion † : G ( P ) × L ( P ) → G ( P ) satisfying both F AITH andCB and we will derive a contradiction. Take the preferencemodel M = h{ w , w , w } , ≤ , v i s.t. w < w < w , M, w (cid:15) ¬ p ∧ q , M, w (cid:15) p ∧ ¬ q and M, w (cid:15) p ∧ q . Let G be a P-graph that induces M . Since † satisfies both F AITH and CB, any dynamic operator ⋆ induced by † must satisfythat ⋆ ( M, p ) = h{ w , w , w } , ≤ ′ , v i s.t. w < ′ w < ′ w is induced by † ( G, p ) = h Φ ′ , ≺ ′ i , i.e., for any ξ ∈ Φ ′ s.t. M, w (cid:15) ξ either M, w (cid:15) ξ or there is ψ ∈ Φ ′ s.t. ψ ≺ ′ ξ , M, w (cid:15) ψ and M, w (cid:15) ψ and there is some ξ ∈ Φ ′ s.t. M, w (cid:15) ξ and M, w (cid:15) ξ .Consider now the model M ′ = h{ w , w } , ≤ , v i s.t. w < w and v is the same as before. Clearly, M ′ is in-duced by G as well. Since † satisfies both F AITH and CB,any dynamic operator ⋆ induced by † must satisfy that ⋆ ( M ′ , p ) = h{ w , w } , ≤ ′ , v i s.t. w < ′ w is induced by † ( G, p ) = h Φ ′ , ≺ ′ i . As such, for any ξ ∈ Φ ′ s.t. M, w (cid:15) ξ either M, w (cid:15) ξ or there is ψ ∈ Φ ′ s.t. ψ ≺ ′ ξ , M, w (cid:15) ψ and M, w (cid:15) ψ and there is some ξ ∈ Φ ′ s.t. M, w (cid:15) ξ and M, w (cid:15) ξ . But this is a contradiction with the previ-ous statement, since the valuation v is the same in all mod-els.It is well known, however, that Natural Revision - an it-erated revision operation proposed by Boutilier (1993) - sat-isfies both F AITH and CB and is definable on preferencemodels.
Definition 22.
Let M = h W, ≤ , v i be a preference modeland ϕ a formula of L ( P ) . We say the preference model M ↑ ϕ = h W, ≤ ↑ ϕ , v i is the result of the natural revision of M by ϕ , where w ≤ ↑ ϕ w ′ iff (cid:26) w ∈ Min ≤ J ϕ K , or w ≤ w ′ and w, w ′ Min ≤ J ϕ K As such, we can conclude that Natural Revision cannot berepresented as a P-graph transformation.
Corollary 23.
There is no P-graph transformation that in-duces Natural Revision.
This result shows that some important belief changeoperations are not definable through P-graph transforma-tions. Notice that previous examples, provided by Souza etal (2016), have all been contraction operations. What thesebelief change operations have in common is that their def-inition is intrinsically characterised by the minimal worldssatisfying a certain formula ϕ . In other words, these oper-ations are only well-defined on preference models if we re-quire that preference models be well-founded, a requirementmade in Souza’s (2016) definition for these models but notin Girard’s (2008). As such, this result reinforces our intu-ition that Fact 20 is the cause of the lack of expressibility ofP-graph transformations.Notice that, as F AITH , postulate CB is not solely respon-sible for the impossibility of expressing Natural Revisionthrough P-graph transformations. There is, in fact, a trivialelief change operation that satisfies CB and is expressibleby P-graph transformation: the null change operation.
Definition 24.
Let M = h W, ≤ , v i be a preference modeland ϕ a formula of L ( P ) . We say the preference model M ◦ ϕ = h W, ≤ , v i is the result of the null change of M by ϕ The null operation is the operation of not changing any-thing in the agent’s belief state. It clearly satisfies postulateCB and it is trivially induced by the null change
P-graphtransformation.
Definition 25.
Let G ∈ G ( P ) be a P-graph and ϕ ∈ L ( P ) a propositional formula. We define the null change transfor-mation of G by ϕ as ⊙ ( G, ϕ ) = G . Clearly the only dynamic operator induced by the nullchange transformation ⊙ is the null change operator ◦ . Assuch, ⊙ satisfies postulate CB. Related Work
The AGM approach and the vast literature based on it re-lies mainly on extralogical characterisation of belief changeoperations. The first attempt to integrate belief change op-eration within a logic that we are aware of is the work ofSegerberg (1999), which defines Dynamic Doxastic Logic(DDL).Similar work has focused on embedding specific be-lief change operations within various epistemic log-ics to analyse dynamic phenomena in Formal Episte-mology (Van Benthem 2007; Baltag and Smets 2008). Par-ticularly, Girard (2008) and Van Benthem (2009) pro-pose Dynamic Preference Logic (DPL), a dynamic logicwhich has been used to generalise AGM-like beliefchange operations (Liu 2011; Souza et al. 2016). Aiming tostrengthen the connection between DPL and Belief Change,Souza, Moreira, and Vieira (2017) study how well-knownbelief change postulates can be characterised using DPL ax-ioms.While these studies connect Belief Change with Epis-temic Logic and provide ways to use the results from onearea within the other, their approach is mainly semantic.Research on Belief Base Change, however, stemming fromthe work of Hansson (1992), focus on constructing beliefchange operators based on syntactic representations of theagent’s belief state.Searching for rich syntactic representations of agents’ ex-plicit beliefs, several authors such as Williams (1994; 1995),Rott (1991; 2009) and Benferhat et al. (2002) propose dif-ferent belief base change operations. These works, however,do not explore how the belief change operations constructedover these syntactic representations are connected to the pos-tulates of Belief Change.The work closest to ours is that ofDelgrande, Dubois, and Lang (2006) which con-siders the notion of iterated belief revision,as studied by Darwiche and Pearl (1997) andNayak, Pagnucco, and Peppas (2003), as a special caseof the belief change operation of merging . The authors usesyntactic structures, similar to prioritized bases, to constructmerging operations and show that they satisfy well-known iterated belief revision postulates. More so, the authorspropose codifications of these postulates using the syntacticstructures proposed by their work, differently than previouswork.The main drawback of their codification of postulates, inour opinion, is that they are not general enough. The pro-posed codifications of the postulates are obtained by trans-lating the desired postulates, e.g., Darwiche and Pearl’s(1997) iterated revision postulates, using the operation ofgraph prefixing to stand for revision. However, as we know,graph prefixing does not equate revision but represents aspecific iterated revision policy known as lexicographic re-vision. As such, a more general codification of these pos-tulates by means of syntactic representations of the agent’sbelief state is still an open problem.Liu (2011) has shown that preference models canbe encoded by syntactic structures known as P-graphs.Since preference models have been used to model anagent’s belief state (Girard 2008; Girard and Rott 2014),Liu’s priority graphs can be seen as a syntactic repre-sentation of an agent’s belief state as well. More yet,Van Benthem, Grossi, and Liu (2014) and Souza (2016)have shown that this representation can be used to constructwell-known belief changing operations from iterated beliefchange literature. As before, however, the authors do notconsider how the formal properties of a belief change opera-tor are reflected in its construction based on transformationsof priority graphs.
Conclusion
This work has explored codifications of iterated beliefchange postulates in Dynamic Preference Logic using thesyntactic representation of preference models by means ofLiu’s priority structures (Liu 2011) known as P-graphs. Weprovided conditions on P-graph transformations that enforceadherence to belief change postulates of the induced dy-namic operators.Our work can be seen as a generalisation of previ-ous work on the integration of Belief Revision Theoryand Dynamic Epistemic Logics that allows the use of theDEL framework to reason about classes of belief changeoperators. In some sense, this work is the complementof the characterisation of iterated belief change postu-lates using the proof theory of Dynamic Preference Logic(Souza, Moreira, and Vieira 2017). We point out that, sincepriority graphs can be seen as a syntactic form of rep-resenting evidence, our work can also be connected tothe work on Evidence Logics and Explicit Knowledge(Baltag, Fiutek, and Smets 2016; Lorini 2018).As illustrated in Section
Deriving P-Graph Transforma-tions from Belief Revision Policies , the characterisationsproposed provide a road-map to implement belief changepolicies in computational systems. Besides, in Section
Neg-ative Results we show that a well-known iterated revisionoperator cannot be encoded employing graph transforma-tions, and we indicate which policies cannot be implementedas syntactic transformations in the general case of usingpreference models to reason about belief change. cknowledgements
This study was financed in part by the
Coordenac¸ ˜ao deAperfeic¸oamento de Pessoal de N´ıvel Superior - Brasil(CAPES) - Finance Code 001.
References [Alchourr´on, G¨ardenfors, and Makinson 1985] Alchourr´on,C. E.; G¨ardenfors, P.; and Makinson, D. 1985. On the logic oftheory change: Partial meet contraction and revision functions.
Journal of Symbolic Logic
Journal of Logic and Computation
Textsin logic and games
Advances in Modal Logic , 156–176.[Benferhat et al. 2002] Benferhat, S.; Dubois, D.; Prade, H.; andWilliams, M.-A. 2002. A practical approach to revising priori-tized knowledge bases.
Studia Logica
Journal of Artificial Intelli-gence Research
Proceedings of the Thirteenth Interna-tional Joint Conference on Artificial Intelligence , 519–531.[Boutilier 1994] Boutilier, C. 1994. Toward a logic for qualita-tive decision theory.
Proceedings of the Fourth InternationalConference on Principles of Knowledge Representation andReasoning
Artificial intelligence
Preference Change , volume 42of
Theory and Decision Library . Springer. 85–107.[Delgrande, Dubois, and Lang 2006] Delgrande, J. P.; Dubois,D.; and Lang, J. 2006. Iterated revision as prioritized merging.In
Proceedings of the Tenth International Conference on thePrinciples of Knowledge Representation and Reasoning , 210–220. AAAI Press.[Ferm´e, Garapa, and Reis 2017] Ferm´e, E.; Garapa, M.; andReis, M. D. L. 2017. On ensconcement and contraction.
Jour-nal of Logic and Computation
Journal of Symbolic Logic
Johan van Benthem on Logicand Information Dynamics . Springer International Publishing.203–233.[Girard 2008] Girard, P. 2008.
Modal logic for belief and pref-erence change . Ph.D. Dissertation, Stanford University. [Grove 1988] Grove, A. 1988. Two modellings for theorychange.
Journal of Philosophical Logic
Synthese
TheJournal of Symbolic Logic
Artificial Intelligence
Artificial intelligence
Eu-ropean Journal of Operational Research
Reasoning about preference dynamics ,volume 354. Springer.[Lorini 2018] Lorini, E. 2018. In praise of belief bases: Do-ing epistemic logic without possible worlds. In
Proceedings ofthe Thirty-Second AAAI Conference on Artificial Intelligence ,1915–1922. AAAI Press.[Nayak, Pagnucco, and Peppas 2003] Nayak, A. C.; Pagnucco,M.; and Peppas, P. 2003. Dynamic belief revision operators.
Artificial Intelligence
Proceedings of the SecondInternational Conference on the Principles of Knowledge Rep-resentation and Reasoning , 417–428.[Rott 1991] Rott, H. 1991. Two methods of constructing con-tractions and revisions of knowledge systems.
Journal of Philo-sophical Logic
TheJournal of Philosophy
To-wards Mathematical Philosophy , volume 28 of
Trends in Logic .Springer Netherlands. 269–296.[Segerberg 1999] Segerberg, K. 1999. Two traditions in thelogic of belief: bringing them together. In
Logic, language andreasoning . Springer. 135–147.[Souza et al. 2016] Souza, M.; Moreira, A.; Vieira, R.; andMeyer, J.-J. C. 2016. Preference and priorities: A study basedon contraction. In
Proceedings of the Fifteen InternationalConference on the Principles of Knowledge Representation andReasoning , 155–164. AAAI Press.[Souza, Moreira, and Vieira 2017] Souza, M.; Moreira, ´A.; andVieira, R. 2017. Dynamic preference logic as a logic of beliefchange. In
Proceedings of the First International Workshop onDynamic Logic , 185–200. Springer.[Souza 2016] Souza, M. 2016.
Choices that make you changeyour mind: a dynamic epistemic logic approach to the seman-tics of BDI agent programming languages . Ph.D. Dissertation,Institute of Informatics, Universidade Federal do Rio Grandedo Sul.[Van Benthem, Grossi, and Liu 2014] Van Benthem, J.; Grossi,D.; and Liu, F. 2014. Priority structures in deontic logic.
Theo-ria
Journal of Applied Non-Classical Logics
Preference Change ,volume 42 of
Theory and Decision Library . Springer. 57–84.[Williams 1994] Williams, M.-A. 1994. On the logic of the-ory base change. In
Proceedings of the European Workshop onLogics in Artificial Intelligence (JELIA 94) , 86–105. Springer.[Williams 1995] Williams, M.-A. 1995. Iterated theory basechange: A computational model. In