DEL-sequents for progression

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DEL-sequents for progression

Guillaume Aucher

To cite this version:

Guillaume Aucher. DEL-sequents for progression. Journal of Applied Non-Classical Logics, Taylor &

Francis, 2011, 21 (3-4), pp.289-321. �10.3166/jancl.21.289-321�. �hal-00674150�

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DEL-sequents for Progression

Guillaume Aucher

INRIA – Universit´e de Rennes 1 Campus Beaulieu

35042 Rennes cedex (France) guillaume.aucher@irisa.fr

Abstract

Dynamic Epistemic Logic (DEL) deals with the representation and the study in a multi-agent setting of knowledge and belief change. It can express in a uniform way epistemic statements about:

(i) what is true about an initial situation

(ii) what is true about an event occurring in this situation

(iii) what is true about the resulting situation after the event has occurred.

We axiomatize within the DEL framework what we can infer about (iii) given (i) and (ii). Given three formulasφ,φ⁰andφ⁰⁰describing respectively (i), (ii) and (iii), we also show how to build a formulaφ⊗φ⁰which captures all the information which can be inferred about (iii) fromφandφ⁰. We show how our results extend to other modal logics thanK. In our proofs and definitions, we resort to a large extent to the normal form formulas for modal logic originally introduced by Kit Fine. In a companion paper [Aucher, 2012], we axiomatize what we can infer about (ii) given (i) and (iii), and what we can infer about (i) given (ii) and (iii), and show how to build two formulasφφ⁰⁰andφ⁰φ⁰⁰which capture respectively all the information which can be inferred about (ii) fromφandφ⁰⁰, and all the information which can be inferred about (i) fromφ⁰andφ⁰⁰.

1 Introduction

Dynamic Epistemic Logic (DEL) deals with the representation and the study in a multi-agent setting of knowledge and belief change, and more generally of information change [van Ditmarsch et al., 2007]. The core idea of DEL is to split the task of representing the agents’ beliefs into three parts: first, one represents their beliefs about an initial situation; second, one represents their beliefs about an event taking place in this situation; third, one represents the way the agents update their beliefs about the situation after (or during) the occurrence of the event. Consequently, within the logical framework of DEL, one can express uniformly epistemic statements about:

(i) what is true about an initial situation,

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(ii) what is true about an event occurring in this situation,

(iii) what is true about the resulting situation after the event has occurred.

From a logical point of view, this trichotomy begs the following three questions (which were already raised in [Kooi, 2007]). In these questions,φ,φ⁰andφ⁰⁰ are three epistemic formulas describing respectively (i), (ii) and (iii).

Question 1:

1. Given (i) and (ii), what can we infer about (iii):φ, φ⁰ φ⁰⁰?

2. How can we build a single formulaφ⊗φ⁰which captures all the information which can be inferred about (iii) fromφandφ⁰?

Question 2:

1. Given (i) and (iii), what can we infer about (ii):φ, φ⁰⁰ φ⁰?

2. How can we build a single formulaφφ⁰⁰ which captures all the information which can be inferred about (ii) fromφandφ⁰⁰?

Question 3:

1. Given (ii) and (iii), what can we infer about (i):φ⁰, φ⁰⁰ φ?

2. How can we build a single formulaφφ⁰⁰ which captures all the information which can be inferred about (i) fromφ⁰andφ⁰⁰?

Providing formal tools that answer these questions leads to applications in artificial intelligence and theoretical computer science, and as it turns out, some of them have already been addressed in DEL and other logical formalisms.¹

Question 1: Progression. Answering the first question leads to the development of tools that can be used by (artificial) agents to compute autonomously their representation of situations as events occur or to reason about the effects of these events. This question has been addressed in the situation calculus, where it is related to the notion ofprogression[Reiter, 2001]. In the logics of programs, our DEL-sequentφ, φ⁰ φ⁰⁰correspond to the partial correctness specifications {φ}π{φ⁰⁰}of Hoare’s logic [Hoare, 1969] which read as “after every successful execution of programπstarting from a state where preconditionφholds, post- conditionφ⁰⁰holds in the final state”. Likewise, our formulaφ⊗φ⁰corresponds to the strongest post-condition of Propositional Dynamic Logic [Pratt, 1976]. That the product update of DEL is in fact the same as the strongest post-condition has been elaborated on and proved in an algebraic setting in [Baltag et al., 2005]. A sequent calculus is also provided in this algebraic setting.

1The question of determining the exact relationship of DEL with other logical formalims has recently started to be investigated. The interested reader can consult [van Benthem, 2011, van Ditmarsch et al., 2009]

for its relationship with the situation calculus and [van Benthem et al., 2009a] for its relationship with epistemic temporal logic.

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Question 2: Epistemic planning. Answering the second question also leads to applications in artificial intelligence in the area ofepistemic planning: (artificial) agents often need to determine autonomously which actions they need to per- form in order to achieve a given epistemic goal. This second question is also related to the notion of explanation and has been dealt with in the event calculus of [Shanahan, 1997] for instance, where it is shown that planning prob- lems can be handled via abduction (using logic programming). In computer science, this second question is also related to thesynthesis problemraised by Church in its full generality [Church, 1957]. He asked whether, given a desired relation between a set of inputs and a set of outputs, we can construct a function that produces the desired outputs from arbitrary inputs. This problem has been declined as the problem of program synthesis: given a specification, can we construct a program that is guaranteed to satisfy this specification? It was extensively studied in the 1980s and 1990s for temporal logic specifications. The synthesis problem is more challenging when the input is incom- plete [Kupferman & Vardi, 1999]. Open (reactive) environments can be a reason of incompleteness of the input, and epistemic logic is a natural formalism to resort to model such situations, as argued in [Halpern & Moses, 1990]. For single-agent temporal epistemic logic, this synthesis problem has been solved in [van der Meyden & Vardi, 1998]. However, this problem has not been addressed so far within the DEL approach, although its methodology and formal setting lend itself rather naturally to address it.

Question 3: Regression. Answering the third question is related to the notion ofre- gressionintroduced in the situation calculus [Reiter, 2001]. This technique is used to determine whether a statement holds after a sequence of events (called theprojectionproblem) by reducing (regressing) this statement about the resulting situation to a statement about the initial situation. In DEL, regression corresponds to the classical reduction method used to prove completeness of an axiomatization: a formula with dynamic operator(s) is ‘reduced’ equivalently to a formula without dynamic operator by pushing the dynamic operator through the logical connectives, performing some kind of regression of the initial formula with dynamic operator. In the companion paper [Aucher, 2012], our inductive definition of the regression ofφ⁰⁰byφ⁰,i.e.φ⁰φ⁰⁰, is based on the reduction axioms of DEL [Baltag & Moss, 2004]. Note that in Propositional Dynamic Logic,

¬(φ¬φ⁰⁰) also corresponds to the weakest precondition.

In this paper, we will answer the first question within the DEL framework. The second and third questions are dealt with in a companion paper [Aucher, 2012]. In a third paper [Aucher et al., 2011], we provide a tableau method to decide whether an inference of one of the three kinds above holds and show that this decision problem is NEXPTIME-complete. The tableau method is also implemented in LOTRECscheme.

From a conceptual perspective, axiomatizing the first inference relationφ, φ⁰ φ⁰⁰ leads to a natural characterization of the notion of (belief) update. This notion can therefore be studied rigorously and systematically, as it has been done in the restricted setting of a single agent in the theory of belief revision of [Alchourr´on et al., 1985]

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and the theory of belief update of [Katsuno & Mendelzon, 1992]. In particular, this axiomatic/postulational approach permits a natural comparison, analysis and introduction of new revision and update operations. In DEL, [van Benthem, 2007] proposes another approach to analyse updates. The idea is to apply the same technique used in modal logic for modal frame correspondance to the study of the modal operator of update. Dynamic axioms can characterize classes of operations on models in the way that classical modal axioms characterize classes of frames. This approach is at the same time less restrictive than ours, but also more abstract. Less restrictive in the sense that the language used can combine within a single formula expressions dealing with (i), (ii) and (iii) altogether, unlike our DEL-sequents where these expressions are clearly separated. More abstract in the sense that what is really studied is the inference relation from (i) to (iii), the second item (ii) being somehow abstracted away. However, in its current form, this approach cannot account explicitly for specific classes of updates induced by classes of event models as the ones typically used in DEL, simply because the language cannot refer to classes of event models.

The paper is organized as follows. In Section 2, we introduce our logical formalism and show how one can naturally express epistemic statements about (i), (ii) and (iii) within this framework. In Section 3, we introduce some mathematical objects needed in the subsequent proofs, namely normal form formulas and the notion of refinement. We also characterize this notion of refinement syntactically by means of our normal form formulas. In Section 4, we provide two equivalent sequent calculi which axiomatize the inference relation of Question 1) a), both for epistemic and ontic events. In Section 5, we propose a constructive definition of the formulaφ⊗φ⁰ of Question 1) b). In Section 6, we show how our results extend to other modal logics thanK, and in Section 7 we give some examples of derivations of valid inferences in our calculi. We end the paper in Section 8 by some concluding remarks.

2 Dynamic Epistemic Logic

Following the DEL methodology described above, we split the exposition of our logical formalism into three subsections. In the rest of the paper,Agtis a finite set of agents andΦis a set of propositional letters calledatomic facts.

2.1 Representation of the initial situation: L-model

A (pointed)L-model (M,w) represents how the actual world represented bywis perceived by the agents. Atomic facts are used to state properties of this actual world.

Definition 1(L-model). AL-modelis a tupleM=(W,R,V) whereWis a non-empty set of possible worlds,R:Agt→2^W×Wis a function assigning to each agent j∈Agtan accessibility relation onW, andV :Φ→2^Wis a function called avaluationassigning to each propositional letter ofΦa subset ofW.

We writew∈M forw∈W, and (M,w) is called apointedL-model. Ifw,v∈W, we writewR_jvforR(j)(w,v) andR_j(w) for{v∈W |wR_jv}.

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Intuitively, in the definition above,v∈Rj(w) means that in worldwagent jconsiders worldvas being possibly the worldw.

Now, we define the epistemic languageLwhich can be used to describe and state properties ofL-models. In particular, the formulaBjφreads as “agent j Believesφ”.

Its truth conditions are defined in such a way thatBjφholds in a possible world when φholds in all the worlds agent jconsiders possible. Dually, the formulahBjiφreads as

“agent jconsiders possible thatφholds”.

Definition 2(LanguageL). We define the languageLinductively as follows:

L:φ ::= p | ¬φ | φ∧φ | Bjφ

where p ranges over Φ and j over Agt. The formulaφ∨ψ is an abbreviation for

¬(¬φ∧ ¬ψ);φ → ψis an abbreviation for¬φ∨ψ; andhB_jiφis an abbreviation for

¬B_j¬φ.² Ifφ∈ L, then we noteP(φ) the set of atomic events appearing inφ.

Let M be aL-model, w ∈ M andφ ∈ L. Thesatisfaction relation M,w |= φis defined inductively as follows:

M,w|=p iff w∈V(p) M,w|=¬φ iff notM,w|=φ

M,w|=φ∧ψ iff M,w|=φandM,w|=ψ M,w|=Bjφ iff for allv∈Rj(w),M,v|=φ

We writeM |=φwhenM,w|=φfor allw∈M, and|=φwhenM |=φfor allL-model

M.

Example 1. Assume that agents A, B and C play a card game with three cards: a white one, a red one and a blue one. Each of them has a single card but they do not know the cards of the other players. At each step of the game, some of the players show their/her/his card to another player or to both other players, either privately or publicly. We want to study and represent the dynamics of the agents’ beliefs in this game. The initial situation is represented by the pointedL-model (M,w) of Figure 1.

In this example,Φ ={rj,bj,wj| j∈ {A,B,C}}whererjstands for ‘agent jhas thered card’,bjstands for ‘agent j has theblue card’ andwjstands for ‘agent jhas thewhite card’. The boxed possible world corresponds to the actual world. The propositional letters not mentioned in the possible worlds do not hold in these possible worlds. The accessibility relations are represented by arrows indexed by agents between possible worlds. Reflexive arrows are omitted in the figure, which means that for all worlds v∈ Mand all agents j∈ {A,B,C},v∈ R_j(v). In this model, we have for example the following statement: M,w|=(w_B∧ ¬B_Aw_B)∧B_C¬B_Aw_B. It states that player A does not ‘know’ that player B has the white card and player C ‘knows’ it.

Theorem 1(Soundness and completeness ofK). [Blackburn et al., 2001] The logicK is defined by the following axiom schemata and inference rules:

(Propositional) All propositional axiom schemata and inference rules (Bj-distribution) `Bj(φ→ψ)→(Bjφ→Bjψ)

(Bj-necessitation) If `φthen `Bjφ

2The degreedeg(φ) of a formulaφ ∈ Lis defined inductively as follows: deg(p) = 0,deg(¬φ) = deg(φ),deg(φ∧ψ)=max{deg(φ),deg(ψ)},deg(Bjφ)=1+deg(φ). We define similarly the degreedeg(φ⁰) of a formulaφ⁰from the languageL⁰of Definition 5.

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rC,bB,wA C

&&

B

ww

rA,bB,wC

77

A

C

++

rC,bA,wB

ff

B

ssw:r_A,b_C,w_B

OO

C '' 33

r_B,b_A,w_C

A

OOkk

xx B

rB,bC,wA A

OO

gg 88

Figure 1: Cards Example

A formulaφ ∈ Lis aK-theorem, writtenφ ∈ Kor ` φ, whenφcan be derived by successively applying (some of) the inference rules on (some of) the axioms. We say that φisK-inconsistentwhen¬φis derivable inK, andK-consistentotherwise. Then, for allφ∈ L,`φimplies that|=φ(soundness), and|=φimplies that`φ(completeness).

2.2 Representation of the event: L

⁰

-model

The propositional lettersp⁰describing events are calledatomic eventsand range over an infinite setΦ⁰. To each atomic eventp⁰, we assign a formulaPre(p⁰) of the language L, which is called theprecondition of p⁰. This precondition corresponds to the property that should be true at any worldwof aL-model so that the atomic event p⁰can

‘physically’ occur in this worldw. Note that the definition below constrains indirectly the definition of the infinite setΦ⁰.

Definition 3 (Precondition function). Aprecondition function Pre : Φ⁰ → L is a function which assigns to each propositional letterp⁰a formula ofLsuch that for all ψ∈ L, there isp⁰∈Φ⁰such thatPre(p⁰)=ψ.

A pointedL⁰-model (M⁰,w⁰) represents how the actual event represented byw⁰is perceived by the agents.

Definition 4(L⁰-model). AL⁰-modelis a tupleM⁰=(W⁰,R⁰,V⁰) whereW⁰is a non- empty set of possible events,R⁰: Agt→2^W⁰^×W⁰ is a function assigning to each agent j ∈ Agt an accessibility relation onW⁰, andV⁰ : Φ⁰ → 2^W⁰ is a function called a valuationassigning to each propositional letter ofΦ⁰a subset ofW⁰such that

for allw⁰∈W⁰, there isat most one p⁰such thatw⁰∈V(p⁰). (Exclusivity) We writew⁰∈M⁰forw⁰∈W⁰, and (M⁰,w⁰) is called apointedL⁰-model. Ifw⁰,v⁰∈W⁰, we writew⁰R⁰_jv⁰forR⁰(j)(w⁰,v⁰) andR⁰_j(w⁰) for{v⁰∈W⁰|w⁰R⁰_jv⁰}.

Intuitively,v⁰∈Rj(w⁰) means that while the possible event represented byw⁰is occurring, agent jconsiders possible that the possible event represented byv⁰is actually

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occurring. The condition (Exclusivity) expresses in our framework the fact that a single precondition is assigned to each possible event, as in the standard BMS framework of [Baltag & Moss, 2004].

In fact, the definition of a L⁰-model is equivalent to the definition of an action signature in the logical formalism of [Baltag & Moss, 2004]. Indeed, anaction sig- natureis a tuple Σ = (W⁰,R⁰,(w⁰₁, . . . ,w⁰_n)) where: 1) W⁰ is a non-empty and finite set of action types (possible events are called “action types” in the BMS formalism), 2)R⁰ : Agt → 2^W⁰^×W⁰ is a function assigning to each agent j ∈ Agtan accessibility relation onW⁰, and 3){w⁰₁, . . . ,w⁰_n}is a subset ofW⁰such that for alli,j∈ {1, . . . ,n}, ifi , jthenw⁰_i , w⁰_j. If we consider an action signatureΣ = (W⁰,R⁰,(w⁰₁, . . . ,w⁰_n)) together with a set of formulasφ₁, . . . , φ_n ∈ L, then we can get back anL⁰-model:

theL⁰-model associated to(Σ, φ₁, . . . , φ_n) is the tuple M⁰=(W⁰,R⁰,V⁰) where for all p⁰ ∈Φ⁰,V⁰(p⁰) is equal to{w⁰_i}ifPre(p⁰)=φ_ifor somei, and equal to the empty set otherwise.

Just as we defined a languageLfor epistemic models, we also define a language L⁰forL⁰-models whose truth conditions are identical to the ones of the languageL.

This language was already introduced in [Baltag et al., 1999]. In the sequel, formulas ofL⁰will always be indexed by the quotation mark ’, unlike formulas ofL.

Definition 5(LanguageL⁰). We define the languageL⁰inductively as follows:

L⁰:φ⁰ ::= p⁰ | ¬φ⁰ | φ⁰∧φ⁰ | Bjφ⁰

wherep⁰ranges overΦ⁰and jover Agt. The formulaφ⁰∨ψ⁰is an abbreviation for

¬(¬φ⁰∧ ¬ψ⁰);φ⁰→ ψ⁰is an abbreviation for¬φ⁰∨ψ⁰; andhBjiφ⁰is an abbreviation for¬Bj¬φ⁰. Ifφ⁰∈ L⁰, then we noteP⁰(φ⁰) the set of atomic events appearing inφ⁰.

LetM⁰be aL⁰-model,w⁰∈M⁰andφ⁰∈ L⁰. Thesatisfaction relation M⁰,w⁰|=φ⁰ is defined inductively as follows:

M⁰,w⁰|=p⁰ iff w⁰∈V⁰(p⁰) M⁰,w⁰|=¬φ⁰ iff notM⁰,w⁰|=φ⁰

M⁰,w⁰|=φ⁰∧ψ⁰ iff M⁰,w⁰|=φ⁰andM⁰,w⁰|=ψ⁰ M⁰,w⁰|=B_jφ⁰ iff for allv⁰∈R⁰_j(w⁰),M⁰,v⁰|=φ⁰.

We writeM⁰ |=φ⁰when M⁰,w⁰ |=φ⁰for allw⁰ ∈ M⁰, and|=φ⁰whenM⁰ |=φ⁰for all

L⁰-modelM⁰.

Example 2. Let us resume Example 1 and assume that players A and B show their card to each other. As it turns out, C noticed that A showed her card to B but did not notice that B did so to A. Players A and B know this. This event is represented in the L⁰-model (M⁰,w⁰) of Figure 2. The boxed possible eventw⁰corresponds to the actual event. The atomic eventp⁰stands for ‘player A shows herred card’,q⁰stands for the atomic event ‘player A shows herwhite card’ andr⁰stands for the atomic event ‘players A and B show theirred andwhite cards respectively to each other’. The precondition Pre(p⁰) ofp⁰isr_A, the preconditionPre(q⁰) ofq⁰isw_A, and the preconditionPre(r⁰) of r⁰isr_A∧w_B. We mention in the possible events of theL⁰-model only the atomic events

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w⁰:r⁰

C

||

C

""

A,B

p⁰

A,B,C 88 oo C //q⁰ff A,B,C

Figure 2: Players A and B show their cards to each other in front of player C

that hold in these possible events. The following statement holds in our example:

M⁰,w⁰|=r⁰∧ hBAir⁰∧BAr⁰∧ hBBir⁰∧BBr⁰

∧ hB_Cip⁰∧ hB_Ciq⁰∧B_C p⁰∨q⁰ (1) It states that players A and B show their cards to each other, players A and B ‘know’

this and consider it possible, while player C considers possible that player A shows her white card and also considers possible that player A shows herred card, since he does not know her card. In fact, that is all that player C considers possible since he believes that either player A shows herred card or herwhite card.

Remark 1. Note that the ontological nature of events (such as ‘looking at the card’

versus ‘being shown the card’) is irrelevant as far as change of beliefsabout the world is concerned, because the epistemic change potential of epistemic events is fully deter- mined by their preconditions and their epistemic indistinguishability. However, when it comes to account for change of beliefsabout events themselves, their ontological nature does play a role, because this ontological nature may be the subject of preconditions of some other ‘meta’-events. The change of beliefsabout eventsdue to the occurrence of some other ‘meta’-events is studied within the DEL framework in [Aucher, 2009].

Now, we introduce the notion ofP⁰-complete models which will play a technical role in the axiomatization of our DEL-sequents in the next sections.

Definition 6(P⁰-completeL⁰-model). LetP⁰be a finite subset ofΦ⁰. AP⁰-complete L⁰-modelis aL⁰-modelM⁰such that

for allw⁰∈M⁰, there is aunique p⁰∈P⁰such thatw⁰∈V⁰(p⁰). (P⁰-complete) AcompleteL⁰-modelis aΦ⁰-completeL⁰-modelM⁰. Theorem 2(Soundness and completeness ofK⁰andK^P’). The logicK⁰is defined by the following axiom schemata and inference rules:

(Propositional) All propositional axiom schemata and inference rules (B_j-distribution) `⁰B_j(φ⁰→ψ⁰)→(B_jφ⁰→B_jψ⁰)

(B_j-necessitation) If `⁰φ⁰then `⁰B_jφ⁰

(Exclusivity) `⁰ p⁰→ ¬q⁰ for all p⁰,q⁰

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Let P⁰be a finite subset ofΦ⁰. The logicK^P’is defined by adding to the logicK⁰the following axiom:

(P⁰-Complete) `⁰ _

p⁰∈P⁰

p⁰

We say that a formulaφ⁰∈ L⁰is aK⁰-theorem, writtenφ⁰∈K⁰or`⁰φ⁰, whenφ⁰can be derived by successively applying (some of) the inference rules on (some of) the axioms ofK⁰. We say thatφ⁰isK⁰-inconsistentwhen¬φ⁰is derivable inK⁰, andK⁰-consistent otherwise. Then, for allφ⁰∈ L⁰,`⁰φ⁰implies that|=⁰φ⁰(soundness) and|=⁰φ⁰implies

`⁰φ⁰(completeness). Similar definitions and results hold forK^P’.

Proof(Sketch). Soundness is standard. Completeness can easily be proved by adapting the canonical model construction forKgiven in [Blackburn et al., 2001].

2.3 Update of the initial situation by the event: product update

The precondition function of Definition 3 induces a precondition function for L⁰- models, which assigns to each possible eventw⁰of aL⁰-model a formula ofL. This formula corresponds to the property that should be true at any worldwof aL-model so that the possible eventw⁰can ‘physically’ occur in the worldw.

Definition 7(Precondition function of aL⁰-model). Let M⁰ = (W⁰,R⁰,V⁰) be aL⁰- model. Theprecondition function of M⁰is the function Pre : W⁰ → Ldefined as follows:

Pre(w⁰) =

( Pre(p⁰) if there isp⁰such thatM⁰,w⁰|=p⁰

> otherwise. (2)

where>is anyK-theorem.

We then redefine equivalently in our setting the BMS product update of [Baltag & Moss, 2004].

This product update takes as argument a pointedL-model (M,w) and a pointed L⁰- model (M⁰,w⁰) representing respectively how an initial situation is perceived by the agents and how an event occurring in this situation is perceived by them, and yields a new pointedL-model (M,w)⊗(M⁰,w⁰) representing how the new situation is perceived by the agents after the occurrence of the event.

Definition 8(Product update). Let (M,w) = (W,R,V,w) be a pointed L-model and (M⁰,w⁰) = (W⁰,R⁰,V⁰,w⁰) be a pointedL⁰-model such that M,w |= Pre(w⁰). The product update of (M,w) and (M⁰,w⁰) is the pointed L-model (M,w)⊗(M⁰,w⁰) = (W^⊗,R^⊗,V^⊗,(w,w⁰)) defined as follows:

W^⊗= (

(v,v⁰)∈W×W⁰

M,v|=Pre(v⁰) )

(3) R^⊗_j(v,v⁰)=

(

(u,u⁰)∈W^⊗

u∈R_j(v) andu⁰∈R⁰_j(v⁰) )

(4) V^⊗(p)=(

(v,v⁰)∈W^⊗

M,v|=p )

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(w,w⁰) :rA,bC,wB

vvC ^C ((

A,B

r_A,b_C,w_B

C //

r_B,b_C,w_A

oo

A

rA,bB,wC A

OO

rC,bB,wA

OO

Figure 3: Situation after the update of the situation represented in Figure 1 by the event represented in Figure 2

Example 3. As a result of the event described in Example 2, the agents update their beliefs. We get the situation represented in theL-model (M,w)⊗(M⁰,w⁰) of Figure??.

In this model, we have for example the following statement:

(M,w)⊗(M⁰,w⁰)|=(wB∧BAwB)∧BC¬BAwB.

It states that player A ‘knows’ that player B has the white card but player C believes

that it is not the case.

3 Mathematical Intermezzo

In this section, we introduce some mathematical objects that will play a central role from a technical point of view in the sequel.

3.1 Kit Fine’s formulas

We will resort in our proofs to particular kinds of modal formulas which capture the structure of epistemic models (modulo bisimulation) up to a given modal depth. These formulas were defined in [Moss, 2007] and are very similar to the normal form formulas for modal logic which were originally introduced in [Fine, 1975]. In Section 3.1.1, we introduce them for the logicK. We will adapt these definitions for the logicsK⁰and K^P’in Section 3.1.2. Note that a similar work has been done in [Moss, 2007] for other logics and languages, notably for the modal language with the ‘star’ operator.

3.1.1 Kit Fine’s formulas forK

A Kit Fine formulaδn+1provides a complete syntactic representation of a pointedL- model up to modal depthn+1. So, intuitively, if we view a Kit Fine formulaδn+1

ofS_n^P₊₁as the syntactic representation up to modal depthn+1 of a possible worldw where it holds, a formulaδ_nofSn^jcan also be viewed as a syntactic representation up to

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modal depthnof a possible world accessible byRjfromw. This justifies our notations in Equation 7.

Definition 9(SetsS^P_n). [Moss, 2007] LetPbe a finite subset ofΦ. We define inductively the setsS^P_n forn∈Nas follows:

S₀^P=











^

p∈S₀

p∧^

p<S0

¬p

S0⊆P











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S^P_n₊₁=











δ₀∧ ^

j∈Agt







^

δn∈S_n^j

hB_jiδ_n∧B_j





 _

δn∈S_n^j

δ_n













δ₀∈S^P₀,S_n^j⊆S^P_n









 .

A formula ofδ∈S_n^Pfor somen>0 will often be written as follows:

δ=δ0∧ ^

j∈Agt







^

γ∈R_j(δ)

hBjiγ∧Bj





 _

γ∈R_j(δ)

γ













. (7)

The following proposition not only tells us that a formulaδncompletely character- izes the structure up to modal depthnof any pointed epistemic model where it holds (first item), but also that the structure ofanyepistemic model up to modal depthncan be characterized by such a formulaδn(second item). If (M,w) is a pointedL-model, thenδn(w) will denote the unique element ofS^P_n such thatM,w|=δn(w).

Proposition 3. [Moss, 2007] Let n∈Nand let P be a finite set of propositional letters.

Letφ∈ Lbe such that deg(φ)≤n and such that P(φ)⊆P.

1. For allδn∈S^P_n, eitherδn→φ∈Korδn→ ¬φ∈K.

2. _

δn∈S^P_n

δn∈K.

The following corollary will play an important role in the sequel. It states that any formula (of degreen) can be reduced to a disjunction ofδ_ns. This explains why these formulas are callednormal form formulas. The decomposition of a formulaφ into δs somehow captures completely and syntactically the relevant structure of the set of pointedL-models which makeφtrue: eachδcan be seen as a syntactic description of the modal structure (up to depthnand modulo bisimulation) of a pointedL-model which makesφtrue.

Corollary 1. Let n ∈ Nand let P be a finite subset of Φ. Let φ ∈ Lbe such that deg(φ) ≤ n and P(φ) ⊆ P. Then, there is S ⊆ S^P_n (possibly empty) such that φ ↔ _

δ∈S

δ∈K.

Proof. Letδ¹_n, . . . , δ^k_nbe the formulas ofS^P_n such thatδⁱ_n→φ∈Kfor alli∈ {1, . . . ,k}.

Thenδ¹_n ∨. . .∨δ^k_n → φ ∈ K. Then, by Proposition 3, for all δn ∈ S_n^P such that

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δn , δ¹_n, . . . , δn , δ^k_n, we have thatδn → ¬φ ∈ K. Thereforeφ → ¬δn ∈ K for all δn ∈ S^P_n such that δn , δ¹_n, . . . , δn , δ^k_n. However, because _

δ_n∈S^P_n

δn ∈ K, we have φ→δ¹_n∨. . .∨δ^k_n ∈K. Finally,φ↔(δ¹_n∨. . .∨δ^k_n)∈K.

3.1.2 Kit Fine’s formulas forK⁰andK^P’

In this section, we adapt the definitions and propositions of the previous section for the logicK⁰andK^P’. We also define the notion of precondition of a Kit Fine formula for K⁰andK^P’.

Definition 10(SetsS_n^P⁰andE^P_n⁰). LetP⁰be a finite subset ofΦ⁰. We define inductively the setsS^P_n⁰andE_n^P⁰forn∈Nas follows:

S₀^P⁰=P⁰∪











^

p⁰∈P⁰

¬p⁰











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S_n^P₊⁰₁=











δ⁰₀∧ ^

j∈Agt







^

δ⁰n∈S_n^j

hB_jiδ⁰_n∧B_j





 _

δ⁰n∈S_n^j

δ⁰_n













δ⁰₀∈S₀^P⁰,S_n^j⊆S_n^P⁰











E₀^P⁰=P⁰ (9)

E_n^P₊⁰₁=











δ⁰₀∧ ^

j∈Agt







^

δ⁰_n∈S_n^j

hB_jiδ_n∧B_j





 _

δ⁰_n∈S_n^j

δ⁰_n













δ⁰₀∈E₀^P⁰,S_n^j⊆E_n^P⁰











A formula ofδ⁰∈S^P_n⁰∪E_n^P⁰for somen>0 will often be written as follows:

δ⁰=δ⁰₀∧ ^

j∈Agt







^

γ⁰∈R_j(δ⁰)

hBjiγ⁰∧Bj





 _

γ⁰∈R_j(δ⁰)

γ⁰











 .

Note that the only difference between formulas ofS^P_n⁰andE^P_n⁰lies in the definition ofS^P₀⁰ andE₀^P⁰: the latter only contains formulas of the form p⁰whereas the former also contains the formula ^

p⁰∈P⁰

¬p⁰.

Example 4. Formula 1 of Example 2 belongs toE^P₁⁰, withP⁰=p⁰,q⁰,r⁰ . The following Propositions 4 and 5 are the counterpart of Proposition 3 for the log- icsK⁰andK^P’, respectively. Both propositions are proved by induction onn, similarly to Proposition 3.

Proposition 4. Let n∈Nand let P⁰be a finite subset ofΦ⁰. Letφ⁰∈ L⁰be such that deg(φ⁰)≤n and such that P⁰(φ⁰)⊆P⁰.

1. For allδ⁰_n∈S^P_n⁰, eitherδ⁰_n→φ⁰∈K⁰orδ⁰_n→ ¬φ⁰∈K⁰.

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2. _

δ⁰_n∈S^P_n⁰

δ⁰_n∈K⁰.

Proposition 5. Let n∈Nand let P⁰be a finite subset ofΦ⁰. Letφ⁰∈ L⁰be such that deg(φ⁰)≤n.

1. For allδ⁰_n∈E_n^P⁰, eitherδ⁰_n→φ⁰∈K^P’orδ⁰_n→ ¬φ⁰∈K^P’. 2. _

δ⁰_n∈E^P_n⁰

δ⁰_n∈K^P’.

The following Corollaries 2 and 3 are the counterpart of Corollary 1 for the logics K⁰andK^P’, respectively. Both corollaries are proved similarly to Corollary 1.

Corollary 2. Let n ∈ Nand let P⁰be a finite subset ofΦ⁰. Letφ⁰ ∈ L⁰be such that deg(φ⁰)≤n and such that P⁰(φ⁰)⊆P⁰. Then, there is S⁰ ⊆S^P_n⁰(possibly empty) such thatφ⁰↔ _

δ⁰∈S⁰

δ⁰∈K⁰.

Corollary 3. Let n ∈ Nand let P⁰be a finite subset ofΦ⁰. Letφ⁰ ∈ L⁰be such that deg(φ⁰)≤n. Then, there is E⁰⊆E_n^P⁰(possibly empty) such thatφ⁰↔ _

δ⁰∈E⁰

δ⁰∈K^P’.

The following proposition relatesK⁰-consistency withK^P’-consistency.

Proposition 6. Letφ⁰ ∈ L⁰. The formulaφ⁰isK⁰-consistent if and only ifφ⁰isK^P’- consistent, and so for any finite subset P⁰ofΦ⁰such that P⁰(φ⁰)⊂P⁰.

Proof. The right to left direction is trivial. Assume thatφ⁰isK⁰-consistent. Then, by completeness ofK⁰, there is a pointedL⁰-model (M⁰,w⁰) such thatM⁰,w⁰ |= φ⁰. The L⁰-model (M⁰,w⁰) can easily be modified using an atomic eventq⁰ ∈ P⁰−P⁰(φ⁰) so that the resulting pointedL⁰-model (M⁰_P0,w⁰_P0) still makesφ⁰true and isP⁰-complete:

it suffices to assign the atomic eventq⁰to any possible eventw⁰of M⁰which does not make anyp⁰∈P⁰(φ⁰) true. So, by completeness ofK^P’,φ⁰isK^P’-consistent, and so for any finite subsetP⁰ofΦ⁰such thatP⁰(φ⁰)⊂P⁰.

The precondition of a Kit Fine formulaδ⁰is naturally defined as follows:

Definition 11(Precondition ofδ⁰,Pre(δ⁰)). Letδ⁰=δ⁰₀∧^

j∈Agt







^

γ⁰∈R_j(δ⁰)

hB_jiγ⁰∧B_j





 _

γ⁰∈R_j(δ⁰)

γ⁰













∈ S_n^P⁰∪E^P_n⁰for somen≥0. We define theprecondition of δ⁰, writtenPre(δ⁰), as follows:

Pre(δ⁰)=

( Pre(p⁰) ifδ⁰₀=p⁰

> otherwise. (10)

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3.2 Refinement

A refinement is the dual of a simulation, the classical notion of bisimulation being both a refinement and a simulation at the same time [Blackburn et al., 2001].

Definition 12(Refinement). LetM1 =(W1,R1,V1) andM2 =(W2,R2,V2) be twoL- models (orL⁰-models). A non-empty relationR ⊆W1×W2is arefinementifffor all (w1,w2)∈ R:

• for allp∈Φ,w1∈V1(p) if and only ifw2∈V2(p)

• for allv2 ∈Rj(w2), there isv1 ∈Rj(w1) such that (v1,v2)∈ R.

We say that (M2,w2) is arefinementof (M1,w1), which we write M1,w1←M2,w2, if and only if there exists a refinement relationR ⊆ W1×W2 such that (w1,w2) ∈ R.

We say that (M2,w2) is arefinement up to modal depth nof (M1,w1), which we write M1,w1←_nM2,w2, if and only if there exists a refinement relationRon the restriction of M1to





 [

j∈Agt

Rj







≤n

(w1) and the restriction ofM2to





 [

j∈Agt

Rj







≤n

(w2) such that (w1,w2)∈ R.³

The following result will turn out to play a central role in the definition of our notion of progression ofφbyφ⁰in Section 5.

Proposition 7.[van Ditmarsch&French, 2008] Let M1and M2be twofiniteL-models.

M1←M2if and only if there is aL⁰-model M⁰such that M1⊗M⁰=M2.

Now, we provide a syntactic characterization of the notion of refinement up to modal depthn. Intuitively, the refinement of aL-model is another L-model where some accessibility relations have been removed, modulo bisimulation. This cutting of accessibility relations is reflected in the subset condition of Equation 12.

Definition 13(FunctionRe f). Letn ∈Nand letPbe a finite subset ofΦ. We define the functionRe f :S^P_n →2^S^Pⁿ inductively as follows:

n=0: for allδ₀∈S^P₀,

Re f(δ0)={δ0} (11)

n+1: ifδ=δ₀∧ ^

j∈Agt







^

δn−1∈Rj(δn)

hBjiδ_n∧Bj

_

δn−1∈Rj(δn)

δ_n





 , then

Re f(δ)= (

δ0∧ ^

j∈Agt







^

δ_j∈Re f(R_j(δ))

hBjiδj∧Bj

_

δ_j∈Re f(R_j(δ))

δj







Re f(R_j(δ))⊆ (

Re f(δ_j)

δ_j∈R_j(δ) ))

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3IfRis a relation andn∈N,R^≤nis defined byR^≤n(w)={v|there isw=w₀, . . . ,wk=vsuch thatw_iRw_i+1 andk≤n}ifn>0, andR^≤0(w)={w}.

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Proposition 8. For all pointedL-models(M1,w1)and(M2,w2), M1,w1←_nM2,w2 if and only ifδn(w2)∈Re f(δn(w1)).

Proof. By induction on n. The casen = 0 is clear. We prove the induction step.

M1,w1←_n₊₁M2,w2

iffδ0(w1)=δ0(w2) and

for allv2 ∈Rj(w2) there isv1∈Rj(w1) such thatM1,v1←_nM2,v2

iffδ₀(w₁)=δ₀(w₂) and

for allv2 ∈Rj(w₂) there isv1∈Rj(w₁) such thatδ_n(v₂)∈Re f(δ_n(v₁)) by induction hypothesis

iffδ₀(w₁)=δ₀(w₂) and

for allδ_n∈R_j(δ_n₊1(w2)) there isδ^∗_n∈R_j(δ_n₊1(w1)) such thatδ_n∈Re f(δ^∗_n) iffδ0(w1)=δ0(w2) and

Rj(δn+1(w2))⊆ {Re f(δn)|δn∈Rj(δn+1(w1))}

iffδn+1(w2)∈Re f(δn+1(w1)).

4 Definition and axiomatization of φ, φ

⁰

φ

⁰⁰

Definition 14(Inference relationφ, φ⁰ φ⁰⁰). Letφ, φ⁰⁰∈ Landφ⁰∈ L⁰. Theinference relationφ, φ⁰ φ⁰⁰is defined as follows:

φ, φ⁰ φ⁰⁰ iff for all pointedL-model (M,w), andL⁰-model (M⁰,w⁰) such that M,w |= Pre(w⁰), if M,w |= φ and M⁰,w⁰ |= φ⁰ then (M,w)⊗(M⁰,w⁰)|=φ⁰⁰

We first provide in Section 4.1 a sequent calculus for the case of epistemic events, i.e.events which do not change atomic facts in a situation. The case of ontic events, i.e.events which change atomic facts, will be dealt with in Section 4.2.

4.1 DEL-Sequent Calculus for epistemic events

Definition 15(DEL-Sequent CalculusSC). TheDEL-Sequent CalculusSCis defined by the following axiom schemata and inference rules. Below,⊥(resp.>) stands for any K-inconsistent formula (resp. K-theorem), and ⊥⁰ (resp.>⁰) stands for anyK⁰- inconsistent formula (resp.K⁰-theorem).

⊥, φ⁰ φ⁰⁰ A₁ φ,⊥⁰ φ⁰⁰ A₂ φ, φ⁰ > A₃ p,>⁰ p A₄ ¬p,>⁰ ¬p A₅ ¬Pre(p⁰),p⁰ ⊥ A₆ φ, φ⁰ φ⁰⁰ φ, φ⁰ φ⁰⁰→ψ⁰⁰

φ, φ⁰ ψ⁰⁰ R₁ φ∧ψ, φ⁰ φ⁰⁰ ¬ψ, φ⁰ φ⁰⁰ φ, φ⁰ φ⁰⁰ R₂ φ, φ⁰∧ψ⁰ φ⁰⁰ φ,¬ψ⁰ φ⁰⁰

φ, φ⁰ φ⁰⁰ R₃ φ, φ⁰ φ⁰⁰

Bjφ,Bjφ⁰ Bjφ⁰⁰ R₄

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φ, φ⁰ φ⁰⁰

hB_ji(φ∧Pre(p⁰)),hB_ji(φ⁰∧p⁰) hB_jiφ⁰⁰ R₅

The key axiom schemata and inference rules areA₄,A₅,R₄andR₅. They permit to build all the valid DEL-sequents by induction on the degree of the formula. Axiom schemataA₄andA₅can be seen as the base case, and rulesR₄andR₅(together with the rest of the axiom schemata) can be seen as the induction steps allowing to build DEL-sequents of higher degree. Axiom schemataA₄andA₅ also illustrates the fact that we deal as in the standard framework of DEL with epistemic events,i.e.events which do not change atomic facts. We will see in the next section that axiom schemata A₄ andA₅can be adapted in order to deal with ontic events, i.e. events that change atomic facts. Axiom schemaA₆illustrates the fact that an atomic event can occur only in a possible world where its precondition holds.

These axiom schemata and inference rules provide a formal and accurate analysis of the product update. It turns out that the informal motivations for the definition of the product update in [Baltag & Moss, 2004] are somehow formalized by ruleR₅. Here is how the product update was informally motivated in this paper (the notations in this quotation are replaced by our notations):

“The update product restricts the full Cartesian product W ×W⁰ to the smaller set W ⊗W⁰ in order to insure that states surviveactions in the appropriate sense. [...] The components of ourL⁰-models are “simple actions”, so the uncertainty regarding the action is assumed to be indepen- dent of the uncertainty regarding the current (input) state. This indepen- dence allows us to “multiply” these two uncertainties in order to compute the uncertainty regarding the output state: if whenever the input state isw, agent jthinks the input might be some other statev, and if whenever the current action happening isw⁰, agent jthinks the current action might be some other actionv⁰, and ifvsurvivesv⁰, then whenever the output state (w,w⁰) is reached, agent jthinks the alternative output state (v,v⁰) might have been reached.”

[Baltag & Moss, 2004]

Now, if one thinks of formulasφ, φ⁰andφ⁰⁰in ruleR₅as representing respectively the input statev, the actionv⁰and the output state (v,v⁰), then the conclusion of this rule somehow formalizes these informal motivations.

We propose below a second DEL-Sequent Calculus SC^∗, which will be proved equivalent to the first DEL-Sequent CalculusSC in Section 4.3. Note that RulesR₇ andR₈below are similar to the introduction rules of disjunction of Gentzen’s sequent calculus, and ruleR₆(and in a sense also rulesR₉andR₁₀) is similar to the introduction rules of conjunction of Gentzen’s sequent calculus.

Definition 16(DEL-Sequent CalculusSC^∗). TheDEL-Sequent CalculusSC^∗is defined by the following axiom schemata and inference rules, together with the axiom schemataA₂andA₆and inference rulesR₄andR₅of the DEL-Sequent CalculusSC.