yet more modal logics of preference change and belief revision

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Yet More Modal Logics of Preference Change

and Belief Revision

Jan van Eijck

Centre for Mathematics and Computer Science (CWI)Kruislaan 4131098 SJ Amsterdam, The Netherlands

[email protected]

Abstract

We contrast Bonanno’s ‘Belief Revision in a Temporal Framework’(Bonanno, 2008) with preference change and belief revision from theperspective of dynamic epistemic logic (DEL). For that, we extend thelogic of communication and change of van Benthem et al. (2006b) withrelational substitutions (van Benthem, 2004) for preference change,and show that this does not alter its properties. Next we move to amore constrained context where belief and knowledge can be definedfrom preferences (Grove, 1988; Board, 2002; Baltag and Smets, 2006,2008b), prove completeness of a very expressive logic of belief revision,and define a mechanism for updating belief revision models using acombination of action priority update (Baltag and Smets, 2008b) andpreference substitution (van Benthem, 2004).

1 Reconstructing AGM style belief revision

Bonanno’s paper offers a rational reconstruction of Alchourron GardenforsMakinson style belief revision (AGM belief revision) (Alchourron et al.,1985; see also Gardenfors, 1988 and Gardenfors and Rott, 1995), in a frame-work where modalities B for single agent belief and I for being informedare mixed with a next time operator © and its inverse ©−1.

Both the AGM framework and Bonanno’s reconstruction of it do notexplicitly represent the triggers that cause belief change in the first place.Iϕ expresses that the agent is informed that ϕ, but the communicativeaction that causes this change in information state is not represented. Also,ϕ is restricted to purely propositional formulas. Another limitation thatBonanno’s reconstruction shares with AGM is that it restricts attention toa single agent: changes of the beliefs of agents about the beliefs of otheragents are not analyzed. In these respects (Bonanno, 2008) is close toDynamic Doxastic Logic (DDL), as developed by Segerberg (1995, 1999).

AGM style belief revision was proposed more than twenty years ago, andhas grown into a paradigm in its own right in artificial intelligence. In the

Krzysztof R. Apt, Robert van Rooij (eds.). New Perspectives on Games and Interaction.

Texts in Logic and Games 4, Amsterdam University Press 2008, pp. 81–104.

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82 J. van Eijck

meanwhile rich frameworks of dynamic epistemic logic have emerged thatare quite a bit more ambitious in their goals than AGM was when it wasfirst proposed. AGM analyzes operations +ϕ for expanding with ϕ, −ϕfor retracting ϕ and ∗ϕ for revising with ϕ. It is formulated in a purelysyntactic way, it hardly addresses issues of semantics, it does not proposesound and complete axiomatisations. It did shine in 1985, and it still shinesnow, but perhaps in a more modest way.

Bonanno’s paper creates a nice link between this style of belief revisionand epistemic/doxastic logic. While similar in spirit to Segerberg’s work,it addresses the question of the rational reconstruction of AGM style beliefrevision more explicitly. This does add quite a lot to that framework: clearsemantics, and a sound and complete axiomatisation. Still, it is fair to saythat this rational reconstruction, nice as it is, also inherits the limitationsof the original design.

2 A broader perspective

Meanwhile, epistemic logic has entered a different phase, with a new focuson the epistemic and doxastic effects of information updates such as publicannouncements (Plaza, 1989; Gerbrandy, 1999). Public announcements areinteresting because they create common knowledge, so the new focus oninformation updating fostered an interest in the evolution of multi-agentknowledge and belief under acts of communication.

Public announcement was generalized to updates with ‘action models’that can express a wide range of communications (private announcements,group announcements, secret sharing, lies, and so on) in (Baltag et al., 1998)and (Baltag and Moss, 2004). A further generalization to a complete logicof communication and change, with enriched actions that allow changingthe facts of the world, was provided by van Benthem et al. (2006b). Thetextbook treatment of dynamic epistemic logic in (van Ditmarsch et al.,2007) bears witness to the fact that this approach is by now well established.

The above systems of dynamic epistemic logic do provide an account ofknowledge or belief update, but they do not analyse belief revision in thesense of AGM. Information updating in dynamic epistemic logic is mono-tonic: facts that are announced to an audience of agents cannot be unlearnt.Van Benthem (2004) calls this ‘belief change under hard information’ or‘eliminative belief revision’. See also (van Ditmarsch, 2005) for reflectionon the distinction between this and belief change under soft information.

Assume a state of the world where p actually is the case, and where youknow it, but I do not. Then public announcement of p will have the effectthat I get to know it, but also that you know that I know it, that I knowthat you know that I know it, in short, that p becomes common knowledge.But if this announcement is followed by an announcement of ¬p, the effect

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Yet More Modal Logics of Preference Change and Belief Revision 83

will be inconsistent knowledge states for both of us.It is clear that AGM deals with belief revision of a different kind: ‘be-

lief change under soft information’ or ‘non-eliminative belief revision’. In(van Benthem, 2004) it is sketched how this can be incorporated into dy-namic epistemic logic, and in the closely related (Baltag and Smets, 2008b)a theory of ‘doxastic actions’ is developed that can be seen as a further stepin this direction.

Belief revision under soft information can, as Van Benthem observes, bemodelled as change in the belief accessibilities of a model. This is differentfrom public announcement, which can be viewed as elimination of worldswhile leaving the accessibilities untouched.

Agent i believes that ϕ in a given world w if it is the case that ϕ istrue in all worlds t that are reachable from w and that are minimal for asuitable plausibility ordering relation ≤i. In the dynamic logic of belief revi-sion these accessibilities can get updated in various ways. An example from(Rott, 2006) that is discussed by van Benthem (2004) is ⇑A for so-called‘lexicographic upgrade’: all A worlds get promoted past all non-A worlds,while within the A worlds and within the non-A worlds the preference re-lation stays as before. Clearly this relation upgrade has as effect that itcreates belief in A. And the belief upgrade can be undone: a next updatewith ⇑¬A does not result in inconsistency.

Van Benthem (2004) gives a complete dynamic logic of belief upgradefor the belief upgrade operation ⇑A, and another one for a variation onit, ↑A, or ‘elite change’, that updates a plausibility ordering to a new onewhere the best A worlds get promoted past all other worlds, and for therest the old ordering remains unchanged.

This is taken one step further in a general logic for changing prefer-ences, in (van Benthem and Liu, 2004), where upgrade as relation changeis handled for (reflexive and transitive) preference relations ≤i, by meansof a variation on product update called product upgrade. The idea is tokeep the domain, the valuation and the epistemic relations the same, butto reset the preferences by means of a substitution of new preorders for thepreference relations.

Treating knowledge as an equivalence and preference as a preorder, with-out constraining the way in which they relate, as is done by van Benthemand Liu (2004), has the advantage of generality (one does not have to spec-ify what ‘having a preference’ means), but it makes it harder to use thepreference relation for modelling belief change. If one models ‘regret’ aspreferring a situation that one knows to be false to the current situation,then it follows that one can regret things one cannot even conceive. And us-ing the same preference relation for belief looks strange, for this would allowbeliefs that are known to be false. Van Benthem (private communication)

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84 J. van Eijck

advised me not to lose sleep over such philosophical issues. If we follow thatadvice, and call ‘belief’ what is true in all most preferred worlds, we can stilltake comfort from the fact that this view entails that one can never believeone is in a bad situation: the belief-accessible situations are by definitionthe best conceivable worlds. Anyhow, proceeding from the assumption thatknowledge and preference are independent basic relations and then study-ing possible relations between them has turned out very fruitful: the recenttheses by Girard (2008) and Liu (2008) are rich sources of insight in whata logical study of the interaction of knowledge and preference may reveal.

Here we will explore two avenues, different from the above but related toit. First, we assume nothing at all about the relation between knowledge onone hand and preference on the other. We show that the dynamic logic ofthis (including updating with suitable finite update models) is complete anddecidable: Theorem 3.1 gives an extension of the reducibility result for LCC,the general logic of communication and change proposed and investigatedin (van Benthem et al., 2006b).

Next, we move closer to the AGM perspective, by postulating a closeconnection between knowledge, belief and preference. One takes prefer-ences as primary, and imposes minimal conditions to allow a definition ofknowledge from preferences. The key to this is the simple observation inTheorem 4.1 that a preorder can be turned into an equivalence by takingits symmetric closure if and only if it is weakly connected and converselyweakly connected. This means that by starting from weakly and converseweakly connected preorders one can interpret their symmetric closures asknowledge relations, and use the preferences themselves to define conditionalbeliefs, in the well known way that was first proposed in (Boutilier, 1994)and (Halpern, 1997). The multi-agent version of this kind of conditionalbelief was further explored in (van Benthem et al., 2006a) and in (Baltagand Smets, 2006, 2008b). We extend this to a complete logic of regular dox-astic programs for belief revision models (Theorem 4.3), useful for reasoningabout common knowledge, common conditional belief and their interaction.Finally, we make a formal proposal for a belief change mechanism by meansof a combination of action model update in the style of (Baltag and Smets,2008b) and plausibility substitution in the style of (van Benthem and Liu,2004).

3 Preference change in LCC

An epistemic preference model M for set of agents I is a tuple (W,V,R, P )where W is a non-empty set of worlds, V is a propositional valuation, R isa function that maps each agent i to a relation Ri (the epistemic relationfor i), and P is a function that maps each agent i to a preference relationPi. There are no conditions at all on the Ri and the Pi (just as there are

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no constraints on the Ri relations in LCC (van Benthem et al., 2006b)).We fix a PDL style language for talking about epistemic preference mod-

els (assume p ranges over a set of basic propositions Prop and i over a setof agents I):

ϕ ::= ⊤ | p | ¬ϕ | ϕ1 ∧ ϕ2 | [π]ϕ

π ::= ∼i | ≥i |?ϕ | π1;π2 | π1 ∪ π2 | π∗

This is to be interpreted in the usual PDL manner, with [[[π]]]M giving therelation that interprets relational expression π in M = (W,V,R, P ), where∼i is interpreted as the relation Ri and ≥i as the relation Pi, and wherethe complex modalities are handled by the regular operations on relations.We employ the usual abbreviations: ⊥ is shorthand for ¬⊤, ϕ1 ∨ ϕ2 isshorthand for ¬(¬ϕ1∧¬ϕ2), ϕ1 → ϕ2 is shorthand for ¬(ϕ1∧ϕ2), ϕ1 ↔ ϕ2

is shorthand for (ϕ1 → ϕ2)∧ (ϕ2 → ϕ1), and 〈π〉ϕ is shorthand for ¬[π]¬ϕ.The formula [π]ϕ is true in world w of M if for all v with (w, v) ∈ [[[π]]]M

it holds that ϕ is true in v. This is completely axiomatised by the usualPDL rules and axioms (Segerberg, 1982; Kozen and Parikh, 1981):

Modus ponens and axioms for propositional logicModal generalisation From ⊢ ϕ infer ⊢ [π]ϕ

Normality ⊢ [π](ϕ→ ψ)→ ([π]ϕ→ [π]ψ)Test ⊢ [?ϕ]ψ ↔ (ϕ→ ψ)Sequence ⊢ [π1;π2]ϕ↔ [π1][π2]ϕChoice ⊢ [π1 ∪ π2]ϕ↔ ([π1]ϕ ∧ [π2]ϕ)Mix ⊢ [π∗]ϕ↔ (ϕ ∧ [π][π∗]ϕ)Induction ⊢ (ϕ ∧ [π∗](ϕ→ [π]ϕ))→ [π∗]ϕ

In (van Benthem et al., 2006b) it is proved that extending the PDL languagewith an extra modality [A, e]ϕ does not change the expressive power of thelanguage. Interpretation of the new modality: [A, e]ϕ is true in w in M ifsuccess of the update of M with action model A to M ⊗ A implies that ϕis true in (w, e) in M ⊗ A. To see what that means one has to grasp thedefinition of update models A and the update product operation ⊗, whichwe will now give for the epistemic preference case.

An action model (for agent set I) is like an epistemic preference modelfor I, with the difference that the worlds are now called events, and thatthe valuation has been replaced by a precondition map pre that assigns toeach event e a formula of the language called the precondition of e. Fromnow on we call the epistemic preference models static models.

Updating a static model M = (W,V,R, P ) with an action model A =(E,pre,R,P) succeeds if the set

{(w, e) | w ∈W, e ∈ E,M, w |= pre(e)}

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86 J. van Eijck

is non-empty. The result of this update is a new static model M ⊗ A =(W ′, V ′, R′, P ′) with

• W ′ = {(w, e) | w ∈W, e ∈ E,M, w |= pre(e)},

• V ′(w, e) = V (w),

• R′i is given by {(w, e), (v, f)) | (w, v) ∈ Ri, (e, f) ∈ Ri},

• P ′i is given by {(w, e), (v, f)) | (w, v) ∈ Pi, (e, f) ∈ Pi}.

If the static model has a set of distinguished states W0 and the action modela set of distinguished events E0, then the distinguished worlds of M ⊗ Aare the (w, e) with w ∈ W0 and e ∈ E0.

Figure 1. Static model and update.

Figure 1 gives an example pair of a static model with an update action.The distinguished worlds of the model and the distinguished event of theaction model are shaded grey. Only the R relations are drawn, for threeagents a, b, c. The result of the update is shown in Figure 2, on the left.This result can be reduced to the bisimilar model on the right in the samefigure, with the bisimulation linking the distinguished worlds. The result ofthe update is that the distinguished “wine” world has disappeared, withoutany of a, b, c being aware of the change.

In LCC, action update is extended with factual change, which is handledby propositional substitution. Here we will consider another extension, withpreference change, handled by preference substitution (first proposed byvan Benthem and Liu, 2004). A preference substitution is a map fromagents to PDL program expressions π represented by a finite set of bindings

{i1 7→ π1, . . . , in 7→ πn}

where the ij are agents, all different, and where the πi are program ex-pressions from our PDL language. It is assumed that each i that does

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Figure 2. Update result, before and after reduction under bisimulation.

not occur in the lefthand side of a binding is mapped to ≥i. Call the set{i ∈ I | ρ(i) 6= ≥i} the domain of ρ. If M = (W,V,R, P ) is a preferencemodel and ρ is a preference substitution, then Mρ is the result of changingthe preference map P of M to P ρ given by:

P ρ(i) := Pi for i not in the domain of ρ,

P ρ(i) := [[[ρ(i)]]]M for i = ij in the domain of ρ.

Now extend the PDL language with a modality [ρ]ϕ for preference change,with the following interpretation:

M, w |= [ρ]ϕ :⇐⇒ Mρ, w |= ϕ.

Then we get a complete logic for preference change:

Theorem 3.1. The logic of epistemic preference PDL with preference changemodalities is complete.

Proof. The preference change effects of [ρ] can be captured by a set of re-duction axioms for [ρ] that commute with all sentential language constructs,and that handle formulas of the form [ρ][π]ϕ by means of reduction axiomsof the form

[ρ][π]ϕ ↔ [Fρ(π)][ρ]ϕ,

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88 J. van Eijck

with Fρ given by:

Fρ(∼i) := ∼i

Fρ(≥i) :=

{

ρ(i) if i in the domain of ρ,≥i otherwise,

Fρ(?ϕ) := ?[ρ]ϕ,Fρ(π1;π2) := Fρ(π1);Fρ(π2),Fρ(π1 ∪ π2) := Fρ(π1) ∪ Fρ(π2),Fρ(π

∗) := (Fρ(π))∗.

It is easily checked that these reduction axioms are sound, and that for eachformula of the extended language the axioms yield an equivalent formulain which [ρ] occurs with lower complexity, which means that the reductionaxioms can be used to translate formulas of the extended language to PDLformulas. Completeness then follows from the completeness of PDL. q.e.d.

4 Yet another logic. . .

In this section we look at a more constrained case, by replacing the epis-temic preference models by ‘belief revision models’ in the style of Grove(1988), Board (2002), and Baltag and Smets (2006, 2008b) (who call them‘multi-agent plausibility frames’). There is almost complete agreement thatpreference relations should be transitive and reflexive (preorders). But tran-sitivity plus reflexivity of a binary relation R do not together imply thatR ∪ Rˇ is an equivalence. Figure 3 gives a counterexample. The two extra

Figure 3. Preorder with a non-transitive symmetric closure.

conditions of weak connectedness for R and for Rˇ remedy this. A binaryrelation R is weakly connected (terminology of Goldblatt, 1987) if the fol-lowing holds:

∀x, y, z((xRy ∧ xRz)→ (yRz ∨ y = z ∨ zRy)).

Theorem 4.1. Assume R is a preorder. Then R ∪Rˇ is an equivalence iffboth R and Rˇ are weakly connected.

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Proof. ⇒: immediate.⇐: Let R be a preorder such that both R and Rˇ are weakly connected.

We have to show that R∪Rˇ is an equivalence. Symmetry and reflexivity areimmediate. For the check of transitivity, assume xR ∪ R y and yR ∪ R z.There are four cases. (i) xRyRz. Then xRz by transitivity of R, hencexR ∪ R z. (ii) xRyR z. Then yR x and yR z, and by weak connectednessof R , either (xR z or zR x), hence xR ∪ R z, or x = z, hence xRz byreflexivity of R. Therefore xR ∪ R z in all cases. (iii) xR yRz. Similar.(iv) xR yR z. Then zRyRx, and zRx by transitivity of R. ThereforexR ∪R z. q.e.d.

Call a preorder that is weakly connected and conversely weakly con-nected locally connected. The example in Figure 4 shows that locally con-nected preorders need not be connected. Taking the symmetric closure of

Figure 4. Locally connected preorder that is not connected.

this example generates an equivalence with two equivalence classes. Moregenerally, taking the symmetric closure of a locally connected preorder cre-ates an equivalence that can play the role of a knowledge relation definedfrom the preference order. To interpret the preference order as conditionalbelief, it is convenient to assume that it is also well-founded: this makes fora smooth definition of the notion of a ‘best possible world’.

A belief revision model M (again, for a set of agents I) is a tuple(W,V, P ) where W is a non-empty set of worlds, V is a propositional valua-tion and P is a function that maps each agent i to a preference relation ≤ithat is a locally connected well-preorder. That is, ≤i is a preorder (reflexiveand transitive) that is well-founded (in terms of <i for the strict part of≤i, this is the requirement that there is no infinite sequence of w1, w2, . . .

with . . . <i w2 <i w1), and such that both ≤i and its converse are weaklyconnected.

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90 J. van Eijck

In what follows we will use <i with the meaning explained above, ≥i forthe converse of ≤i, >i for the converse of <i, and ∼i for ≤i ∪ ≥i.

The locally connected well-preorders≤i can be used to induce accessibil-ity relations→P

i for each subset P of the domain, by means of the followingstandard definition:

→Pi := {(x, y) | x ∼i y ∧ y ∈MIN≤i

P},

where MIN≤iP , the set of minimal elements of P under ≤i, is defined as

{s ∈ P : ∀s′ ∈ P (s′ ≤ s⇒ s ≤ s′)}.

This picks out the minimal worlds linked to the current world, accord-ing to ≤i, within the set of worlds satisfying [[ϕ]]M. The requirement ofwellfoundedness ensures that MIN≤P will be non-empty for non-empty P .Investigating these→P relations, we see that they have plausible propertiesfor belief:

Proposition 4.2. Let ≤ be a locally connected well-preorder on S and letP be a non-empty subset of S. Then→P is transitive, euclidean and serial.

Proof. Transitivity: if x→P y then y ∼ x and y ∈MIN≤P . If y →P z thenz ∼ y and z ∈ MIN≤P . It follows by local connectedness of ≤ that z ∼ x

and by the definition of →P that x→P z.Euclideanness: let x→P y and x→P z. We have to show y →P z. From

x →P y, y ∼ x and y ∈ MIN≤P . From x →P z, z ∼ x and z ∈ MIN≤P .From local connectedness, y ∼ z. Hence y →P z.

Seriality: Let x ∈ P . Since ≤ is a preorder there are y ∈ P withy ≤ x. The wellfoundedness of ≤ guarantees that there are ≤ minimal suchy. q.e.d.

Transitivity, euclideanness and seriality are the frame properties correspond-ing to positively and negatively introspective consistent belief (KD45 belief,Chellas, 1984).

Figure 5 gives an example with both the ≤ relation (shown as solidarrows in the direction of more preferred worlds, i.e., with an arrow from x

to y for x ≥ y) and the induced → relation on the whole domain (shown asdotted arrows). The above gives us in fact knowledge relations ∼i togetherwith for each knowledge cell a Lewis-style (1973) counterfactual relation: aconnected well-preorder, which can be viewed as a set of nested spheres, withthe minimal elements as the innermost sphere. Compare also the conditionalmodels of Burgess (1981) and Veltman (1985) (linked to Dynamic DoxasticLogic in Girard, 2007).

Baltag and Smets (2008b,a) present logics of individual multi-agent be-lief and knowledge for belief revision models, and define belief update for

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Figure 5. Preference (solid arrows) and belief (dotted arrows).

this as a particular kind of action update in the style of (Baltag et al., 1998),called action priority update. Here we sketch the extension to a system thatalso handles common knowledge and common conditional belief, and wherethe action update has belief change incorporated in it by means of relationalsubstitution.

The set-up will be less general than in the logic LCC: in LCC no assump-tions are made about the update actions, and so the accessibility relationscould easily deteriorate, e.g., as a result of updating with a lie. Since inthe present set-up we make assumptions about the accessibilities (to wit,that they are locally connected well-preorders), we have to ensure that ourupdate actions preserve these relational properties.

Consider the following slight modification of the PDL language (againassume p ranges over a set of basic propositions Prop and i over a set ofagents I):

ϕ ::= ⊤ | p | ¬ϕ | ϕ1 ∧ ϕ2 | [π]ϕ

π ::= ∼i | ≤i | ≥i | →ϕi | ←

ϕi | G |?ϕ | π1;π2 | π1 ∪ π2 | π

∗

Call this language LPref. This time, we treat ∼i as a derived notion, byputting in an axiom that defines ∼i as ≤i∪≥i. The intention is to let ∼i beinterpreted as the knowledge relation for agent i, ≤i as the preference rela-tion for i, ≥i as the converse preference relation for i,→ϕ

i as the conditionalbelief relation defined from ≤i as explained above, ←ϕ

i as its converse, andG as global accessibility. We use →i as shorthand for →⊤

i .We have added a global modalityG, and we will set up things in such way

that [G]ϕ expresses that everywhere in the model ϕ holds, and that 〈G〉ϕexpresses that ϕ holds somewhere. It is well-known that adding a globalmodality and converses to PDL does not change its properties: the logic re-

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92 J. van Eijck

mains decidable, and satisfiability remains EXPTIME-complete (Blackburnet al., 2001).

The semantics of LPref is given relative to belief revision models as indi-cated above. Formula meaning [[ϕ]]M and relational meaning [[[π]]]M are han-dled in the usual way. The interpretation of the knowledge relation of agenti is given by [[[∼i]]]

M := ≤M

i ∪ ≥M

i , that for the preference relation of agenti by [[[≤i]]]

M := ≤M

i , that for the converse preference relation of agent i by

its converse, that for the conditional belief of agent i by [[[→ϕi ]]]M :=→

[[ϕ]]M

i ,that for ←ϕ

i by its converse. The global modality is interpreted as the uni-versal relation, and test, sequential composition, choice and Kleene star areinterpreted as usual.

The interplay between the modalities [∼i] (knowledge) and [≥i] (safebelief) is analysed by Baltag and Smets (2008b), where they remark thatthe converse preference modality [≥i] in belief revision models behaves likean S4.3 modality (reflexive, transitive and not forward branching), and liveshappily together with the S5 modality for [∼i].

To see how this all works out, let us have a look at the truth conditionsfor [→ϕ

i ]ψ. This is true in a world w in model M if in all worlds v withv ∼i w and v minimal in [[[ϕ]]]M under ≤i it holds that ψ is true. This isindeed conditional belief, relative to ϕ. Compare this with [≥i]ψ. This istrue in a world w if in all worlds that are at least as preferred, ψ is true.Finally, [∼i]ψ is true in w if ψ is true in all worlds, preferred or not, that ican access from w.

As a further example, consider a situation where Alexandru is drinkingwine, while Jan does not know whether he is drinking wine or beer, andSonja thinks that he is drinking tea. The actual situation is shaded grey,Jan’s preference relation has solid lines, that of Sonja dotted lines. Reflexivearrows are not drawn, so Alexandru’s preferences are not visible in thepicture.

In the actual world it is true that Jan knows that Alexandru knows whatAlexandru is drinking: [∼j ]([∼a]w∨ [∼a]b), and that Sonja believes Alexan-dru is drinking tea and that Alexandru knows it: [s][∼a]t. Under condi-tion ¬t, however, Sonja has the belief in the actual world that Alexandruis drinking beer: [→¬t

s ]b. Moreover, Jan and Sonja have a common be-lief under condition ¬t that Alexandru is drinking wine or beer: [→¬t

j ∪→¬ts ; (→¬t

j ∪→¬ts )∗](w ∨ b). As a final illustration, note that [←s]⊥ is true

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in a world if this is not among Sonja’s most preferred worlds. Notice thatif Sonja conditionalizes her belief to these worlds, she would believe that

Alexandru is drinking beer: [→[sˇ]⊥s ]b is true in the actual world.

It should be clear from the example that this language is very expressive.To get at a complete logic for it, we need axioms and rules for propositionallogic, S5 axioms for the global modality (Goranko and Passy, 1992), axiomsfor forward connectedness of ≥ and of ≤ (see Goldblatt, 1987), axioms forconverse, relating ≤ to ≥ and → to ←, as in temporal logic (Prior, 1967),and the general axioms and rules for PDL (Segerberg, 1982). Finally, addthe following definition of conditional belief in terms of knowledge and safebelief that can already be found in (Boutilier, 1994) as an axiom:

[→ϕi ]ψ :≡ 〈∼i〉ϕ→ 〈∼i〉(ϕ ∧ [≥i](ϕ→ ψ)).

This definition (also used in Baltag and Smets, 2008b) states that condi-tional to ϕ, i believes in ψ if either there are no accessible ϕ worlds, orthere is an accessible ϕ world in which the belief in ϕ→ ψ is safe. The fullcalculus for LPref is given in Figure 6.

Theorem 4.3. The axiom system for LPref is complete for belief revisionmodels; LPref has the finite model property and is decidable.

Proof. Modify the canonical model construction for modal logic for the caseof PDL, by means of Fischer-Ladner closures (Fischer and Ladner, 1979)(also see Blackburn et al., 2001). This gives a finite canonical model withthe properties for ≤i and ≥i corresponding to the axioms (since the axiomsfor ≤i and ≥i are canonical). In particular, each≥i relation will be reflexive,transitive and weakly connected, each relation ≤i will be weakly connected,and the ≤i and ≥i relations will be converses of each other. Together thisgives (Theorem 4.1) that the ≤i ∪ ≥i are equivalences. Since the canonicalmodel has a finite set of nodes, each ≤i relation is also well-founded. Thus,the canonical model is in fact a belief revision model. Also, the →i and←i relations are converses of each other, and related to the ≥i relationsin the correct way. The canonical model construction gives us for eachconsistent formula ϕ a belief revision model satisfying ϕ with a finite setof nodes. Only finitely many of the relations in that model are relevantto the satisfiability of ϕ, so this gives a finite model (see Blackburn et al.,2001, for further details). Since the logic has the finite model property it isdecidable. q.e.d.

Since the axiomatisation is complete, the S5 properties of ∼i are deriv-able, as well as the principle that knowledge implies safe belief:

[∼i]ϕ→ [≥i]ϕ.

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94 J. van Eijck

Modus ponens and axioms for propositional logic

Modal generalisation From ⊢ ϕ infer ⊢ [π]ϕ

Normality ⊢ [π](ϕ→ ψ)→ ([π]ϕ→ [π]ψ)

Inclusion of everything in G ⊢ [G]ϕ→ [π]ϕ

Reflexivity of G ⊢ [G]ϕ→ ϕ

Transitivity of G ⊢ [G]ϕ→ [G][G]ϕ

Symmetry of G ⊢ ϕ→ [G]〈G〉ϕ

Knowledge definition ⊢ [∼i]ϕ↔ [≤i ∪ ≥i]ϕ

Truthfulness of safe belief ⊢ [≥i]ϕ→ ϕ

Transitivity of safe belief ⊢ [≥i]ϕ→ [≥i][≥i]ϕ

≥ included in ≤ˇ ⊢ ϕ→ [≥i]〈≤i〉ϕ

≤ included in ≥ˇ ⊢ ϕ→ [≤i]〈≥i〉ϕ

Weak connectedness of ≤ ⊢ [≤i]((ϕ ∧ [≤i]ϕ)→ ψ) ∨ [≤i]((ψ ∧ [≤i]ψ)→ ϕ)

Weak connectedness of ≥ ⊢ [≥i]((ϕ ∧ [≥i]ϕ)→ ψ) ∨ [≥i]((ψ ∧ [≥i]ψ)→ ϕ)

Conditional belief definition ⊢ [→ϕi ]ψ ↔ (〈∼i〉ϕ→ 〈∼i〉(ϕ ∧ [≥i](ϕ→ ψ)))

→ included in ←ˇ ⊢ ϕ→ [→ψi ]〈←ψ

i 〉ϕ

← included in →ˇ ⊢ ϕ→ [←ψi ]〈→ψ

i 〉ϕ

Test ⊢ [?ϕ]ψ ↔ (ϕ→ ψ)

Sequence ⊢ [π1;π2]ϕ↔ [π1][π2]ϕ

Choice ⊢ [π1 ∪ π2]ϕ↔ ([π1]ϕ ∧ [π2]ϕ)

Mix ⊢ [π∗]ϕ↔ (ϕ ∧ [π][π∗]ϕ)

Induction ⊢ (ϕ ∧ [π∗](ϕ→ [π]ϕ))→ [π∗]ϕ

Figure 6. Axiom system for LPref.

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The same holds for the following principles for conditional belief given in(Board, 2002):

Safe belief implies belief ⊢ [≥i]ϕ→ [→ψi ]ϕ

Positive introspection ⊢ [→ψi ]ϕ→ [∼i][→ψi]ϕ

Negative introspection ⊢ ¬[→ψi ]ϕ→ [∼i]¬[→

ψi ]ϕ

Successful revision ⊢ [→ϕi ]ϕ

Minimality of revision ⊢ 〈→ϕi 〉ψ → ([iϕ∧ψ]χ↔ [→ϕ

i ](ψ → χ))

We end with an open question: is →ϕi definable from ≥i and ≤i using only

test, sequence, choice and Kleene star?

5 Combining update and upgrade

The way we composed knowledge and belief by means of regular operationsmay have a dynamic flavour, but appearance is deceptive. The resultingdoxastic and epistemic ‘programs’ still describe what goes on in a staticmodel. Real communicative action is changing old belief revision modelsinto new ones. These actions should represent new hard information thatcannot be undone, but also soft information like belief changes that canbe reversed again later on. For this we can use update action by meansof action models, with soft information update handled by means of actionpriority update (Baltag and Smets, 2008b,a), or preference substitution asin (van Benthem and Liu, 2004). Here we will propose a combination ofthese two.

Action models for belief revision are like belief revision models, butwith the valuation replaced by a precondition map. We add two extraingredients. First, we add to each event a propositional substitution, to beused, as in LCC, for making factual changes to static models. Propositionalsubstitutions are maps represented as sets of bindings

{p1 7→ ϕ1, . . . , pn 7→ ϕn}

where all the pi are different. It is assumed that each p that does not occur inthe lefthand side of a binding is mapped to p. The domain of a propositionalsubstitution σ is the set {p ∈ Prop | σ(p) 6= p}. If M = (W,V, P ) is a beliefrevision model and σ is an LPref propositional substitution, then V σ

Mis the

valuation given by λwλp · w ∈ [[pσ]]M. In other words, V σM

assigns to w theset of basic propositions p such that pσ is true in world w in model M. Mσ

is the model with its valuation changed by σ as indicated. Next, we addrelational substitutions, as defined in Section 3, one to each event. Thus,an action model for belief revision is a tuple A = (E,pre,P,psub, rsub)with E a non-empty finite set of events, psub and rsub maps from E topropositional substitutions and relational substitutions, respectively, andwith rsub subject to the following constraint:

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96 J. van Eijck

If e ∼i f in the action model, then rsub(e) and rsub(f) have thesame binding for i.

This ensures a coherent definition of the effect of relational substitution ona belief structure. The example in Figure 7 illustrates this. But note thatthe substitutions are subject to further constraints. In the action model in

Figure 7. Unconstrained relational substitution creates havoc.

the example, a single agent has a preference for ¬p over p. In the updatemodel, a substitution reverses the agent’s preferences, but the agent cannotdistinguish this from an action where nothing happens. What should theresult of the update look like? E.g., is there a preference arrow from (2, 3)to (1, 4)? This is impossible to answer, as action 3 asks us to reverse thepreference and action 4 demands that we keep the initial preference. Theconstraint on substitutions rules out such dilemmas.

The relational substitution ρ = rsub(e) at event e in action model A ismeant to to be interpreted ‘locally’ at each world w in input model M. IfP is the preference map of M, then let P ρw be given by:

P ρw(i) := Pi ∩ |w|2[[[∼i]]]M

for i not in the domain of ρ,

P ρw(i) := [[[ρ(i)]]]M ∩ |w|2[[[∼i]]]M

for i = ij in the domain of ρ.

Thus, P ρw is the result of making a change only to the local knowledge cellat world w of agent i (which is given by the equivalence class |w|[[[∼i]]]M).Let

P ρ(i) :=⋃

w∈W

P ρw(i)

Then P ρ(i) gives the result of the substitution ρ on P (i), for each knowledgecell |w|[[[∼i]]]M for i, and P ρ gives the result of the substitution ρ on P , foreach agent i.

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Now the result of updating belief revision model M = (W,V, P ) with ac-tion model A = (E,pre,P,psub, rsub) is given by M⊗A = (W ′, V ′, P ′),where

• W ′ = {(w, e) | w ∈W, e ∈ E,M, w |= pre(e)},

• V ′(w, e) = V σ(w),

• P ′(i) is given by the anti-lexicographical order defined from P ρ(i) andP(i) (see Baltag and Smets, 2008b,a).

With these definitions in place, what are reasonable substitutions? A pos-sible general form for a preference change could be a binding like this:

≥i 7→ [ϕ1, ϕ2, . . . , ϕn].

This is to be interpreted as an instruction to replace the belief preferencesof i in the local knowledge cells by the new preference relation that prefersthe ϕ1 states above everything else, the ¬ϕ1 ∧ ϕ2 above the ¬ϕ1 ∧ ¬ϕ2

states, and so on, and the ¬ϕ1 ∧ ¬ϕ2 ∧ · · · ∧ ¬ϕn−1 ∧ ϕn states above the¬ϕ1 ∧ ¬ϕ2 ∧ · · · ∧ ¬ϕn states. Such relations are indeed connected well-preorders.

Note that we can take [ϕ1, ϕ2, . . . , ϕn] as an abbreviation for the followingdoxastic program:

(∼i; ?ϕ1) ∪ (?¬ϕ1;∼i; ?¬ϕ1; ?ϕ2)

∪ (?¬ϕ1; ?¬ϕ2;∼i; ?¬ϕ1; ?¬ϕ2; ?ϕ3)

∪ · · ·

∪ (?¬ϕ1; · · · ; ?¬ϕn;∼i; ?¬ϕ1; · · · ; ?¬ϕn−1; ?ϕn)

In general we have to be careful (as is also observed in (van Benthem andLiu, 2004)). If we have a connected well-preorder then adding arrows toit in the same knowledge cell may spoil its properties. Also, the union oftwo connected well-preorders need not be connected. So here is a question:what is the maximal sublanguage of doxastic programs that still guaranteesthat the defined relations are suitable preference relations? Or should beliefrevision models be further constrained to guarantee felicitous preferencechange? And if so: how?

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98 J. van Eijck

6 Examples

Global amnesia: the event of agent a (Jason Bourne) forgetting all hisbeliefs, with everyone (including himself) being aware of this, is representedby the following action model (for the case of three agents a, b, c):

Alzheimer: the event of agent a forgetting everything, with the othersbeing aware of this, while a wrongly believes that nothing has happened. Itis tempting to model this with the following update model:

Note however that this does not satisfy the constraint on relation update(the two actions are connected, but the substitution for a is not the same),so it may result in incoherent models.

Lacular amnesia (specific forgetting): forgetting everything about p.One way to model this is by means of an action model with a single action,accessible to all, with the relational substitution

≥i 7→ (≥i ∪ ∼i; ?¬p)∗

This will effectively add best-world arrows from everywhere in the knowl-edge cell to all ¬p worlds.

Confession of faith in p, or publicly accepting p: an action modelwith a single action, accessible to all, with the relational substitution

≥i 7→ (≥i ∪ (?¬p;∼i; ?p))∗.

This will make the p worlds better than the ¬p worlds everywhere.

Submission to a guru: the act of adopting the belief of someone else,visible to all. A problem here is that the guru may know more than I do, sothat the guru’s preferences within my knowledge cell may not be connected.This means that the substitution ≤i 7→ ≤j—the binding that expressesthat i takes over j’s beliefs—may involve growth or loss of knowledge for i.Consider the example of the wine-drinking Alexandru again: if Jan were totake over Sonja’s beliefs, he would lose the information that Alexandru isdrinking an alcoholic beverage.

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Conformism: adopting the common beliefs of a certain group, visible toall: an action model with a single action accessible to all, with the followingsubstitution for conformist agent i:

≥i 7→ (≥i ∪ ≥j); (≥i ∪ ≥j)∗.

Belief coarsening: the most preferred worlds remain the most preferred,the next preferred remain the next preferred, and all further distinctionsare erased. An action model with a single action accessible to all, and thefollowing substitution for agent i:

≥i 7→ →i ∪ ?⊤.

The union with the relation ?⊤ has the effect of adding all reflexive arrows,to ensure that the result is reflexive again.

Van Benthem’s ⇑ϕ is handled by a substitution consisting of bindingslike this:

≥i 7→ (?ϕ;∼i?¬ϕ) ∪ (?ϕ;≥i?ϕ) ∪ (?¬ϕ;≥i?¬ϕ).

This is an alternative for an update with an action model that has ¬ϕ <B ϕ.The example shows that conservative upgrade is handled equally well byaction priority updating and by belief change via substitution. But beliefchange by substitution seems more appropriate for ‘elite change’. For thiswe need a test for being in the best ϕ world that i can conceive, by meansof the Panglossian formula 〈←ϕ

i 〉⊤. The negation of this allows us to defineelite change like this:

≥i 7→ →ϕi ∪ (≥i; ?[←

ϕi ]⊥).

This promotes the best ϕ worlds past all other worlds, while leaving therest of the ordering unchanged. Admittedly, such an operation could alsobe performed using action priority updating, but it would be much morecumbersome.

7 Further connections

To connect up to the work of Bonanno again, what about time? Notethat perceiving the ticking of a clock can be viewed as information update.A clock tick constitutes a change in the world, and agents can be awareor unaware of the change. This can be modelled within the frameworkintroduced above. Let t1, . . . , tn be the clock bits for counting ticks inbinary, and let C := C + 1 be shorthand for the propositional substitutionthat is needed to increment the binary number t1, . . . , tn by 1. Then publicawareness of the clock tick is modelled by:

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100 J. van Eijck

Thus, perception of the ticking of a clock can be modelled as ‘being intune with change in the world’. Still, this is not quite the same as the ‘nextinstance’ operator©, for the DEL framework is specific about what happensduring the clock tick, while © existentially quantifies over the change thattakes place, rather in the spirit of (Balbiani et al., 2007).

In belief revision there is the AGM tradition, and its rational reconstruc-tion in dynamic doxastic logic a la Segerberg. Now there also is a modalversion in Bonanno style using temporal logic. It is shown in (van Benthemand Pacuit, 2006) that temporal logic has greater expressive power thanDEL, which could be put to use in a temporal logic of belief revision (al-though Bonanno’s present version does not seem to harness this power). Asan independent development there is dynamic epistemic logic in the Amster-dam/Oxford tradition, which was inspired by the logic of public announce-ment, and by the epistemic turn in game theory, a la Aumann. Next to this,and not quite integrated with it, there is an abundance of dynamic logicsfor belief change based on preference relations (Spohn, Shoham, Lewis), andagain the Amsterdam and Oxford traditions. I hope this contribution hasmade clear that an elegant fusion of dynamic epistemic logic and dynamiclogics for belief change is possible, and that this fusion allows to analyzeAGM style belief revision in a multi-agent setting, and integrated within apowerful logic of communication and change.

Acknowledgements

Thanks to Alexandru Baltag, Johan van Benthem and Floor Sietsma forhelpful comments on several earlier versions of the paper, and for construc-tive discussions. Hans van Ditmarsch sent me his last minute remarks. Iam grateful to the editors of this volume for their display of the right mixof patience, tact, flexibility and firmness (with Robert van Rooij exhibitingthe first three of these qualities and Krzysztof Apt the fourth). The sup-port of the NWO Cognition Programme to the ‘Games, Action and SocialSoftware’ Advanced Studies project is gratefully acknowledged.

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yet more modal logics of preference change and belief revision

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