Dynamic Epistemic Temporal Logic

Dynamic Epistemic Temporal Logic∗

Bryan Renne† Joshua Sack‡§ Audrey Yap¶

March 25, 2010

Abstract

We propose Dynamic Epistemic Temporal Logic, a dynamic-protocol frameworkthat overcomes the Problem of Synchronicity in the popular Dynamic Epistemic Logicapproach to reasoning about multi-agent belief change. Dynamic Epistemic TemporalLogic not only extends the domain of applicability of standard Dynamic EpistemicLogic, but it also clarifies how certain structural properties (such as synchronicity)arise from the inherent structure of the standard “update frames” (or “action models”)of Dynamic Epistemic Logic.

1 Introduction

Epistemic Logic [5] is a modal-logic approach to reasoning about beliefs, including beliefsabout basic assertions (“Alice believes that the train stops at Chongqing”) and higher-orderbeliefs (beliefs about others’ beliefs: “Bob believes that Alice believes that the train stops atChongqing”). Dynamic Epistemic Logic [1, 2, 3, 13, 14] is an extension of Epistemic Logicused for reasoning about changes in belief that arise as a result of certain “informationalevents” such as public announcements or private communications. An informational eventmay change an agent’s beliefs about which basic assertions are true or false in the world:after a public announcement that “the train stops at Chongqing,” Alice will believe that thetrain stops at Chongqing. But informational events may also change higher-order beliefs:after the public announcement that “the train stops at Chongqing,” Bob will believe thatAlice believes that the train stops at Chongqing.

The characteristic feature of the Dynamic Epistemic Logic approach is its use of updatemodals, which are modal operators [U, s] that describe operations on Kripke models. Theseoperations, called updates, represent “informational events” in which the agents receive infor-mation that may bring about changes in their beliefs. The basic idea is that an update modal

∗This paper is a revised and extended version of [9].†Faculty of Philosophy, University of Groningen, The Netherlands. http://bryan.renne.org/‡School of Computer Science, Reykjavık University, Iceland. http://www.joshuasack.info/§Joshua Sack was partly supported by the projects “New Developments in Operational Semantics”

(nr. 080039021) and “Processes and Modal Logic” of The Icelandic Research Fund along with a grantfrom Reykjavık University’s Development Fund.¶Department of Philosophy, University of Victoria, Canada. http://web.uvic.ca/∼ayap/

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http://bryan.renne.org/

http://www.joshuasack.info/

http://web.uvic.ca/~ayap/

[U, s] describes a specific partial function f[U,s] that maps a pointed Kripke model (M,w) inthe domain of f[U,s] to another pointed Kripke model that we write as

(M [U ], (w, s)

). This

allows us to view a sequence

(M0, w0), (M1, w1), (M2, w2), . . . , (Mn, wn) (1)

of pointed Kripke models, with (Mi+1, wi+1) generated from (Mi, wi) by the update f[Ui+1,si+1]

described by update modal [Ui+1, si+1], as a discrete-time distributed multi-agent system inwhich the state of the system at time i is described by (Mi, wi). Defining the time of a world win Mi within the sequence (1) to be the index i, we obtain a notion of time that is external tothe pointed Kripke model (Mi, wi). One consequence of adopting this external notion of timeis that all of the worlds that an agent considers possible relative to a world w in Mi have timei. This implies that at every world, every agent knows the current time. Systems in whichthe current time is known at every world are called synchronous [11, 12]. Dynamic EpistemicLogic frameworks that adopt this external notion of time are consequently restricted to thestudy of synchronous multi-agent systems [11, 12]. Such frameworks are therefore unableto represent systems in which agents can become mistaken or uncertain about time; we callthis limitation the Problem of Synchronicity.

An important multi-agent system framework that avoids the Problem of Synchronicityis Epistemic Temporal Logic [11, 12], which has close connections with the knowledge-basedmessage semantics of [8] and the related interpreted systems approach of [5]. In fact, theProblem of Synchronicity is avoided even in a very basic Epistemic Temporal Logic whoseonly temporal modality is a discrete one-step–past operator. We call this basic logic SimpleEpistemic Temporal Logic. Like Epistemic Temporal Logic, Simple Epistemic TemporalLogic uses epistemic temporal models, which are Kripke models in which one of the relationalcomponents is designated as a time-keeping relation. We label the time-keeping relationusing the symbol Y (a mnemonic for “yesterday”). When a world w is related to a world w′

according to the time-keeping relation (so there is a “Y -arrow” from w to w′) the intendedinterpretation is that w′ is a possible way the system might have been “yesterday,” meaningone time-step before w. Defining the time of a world w in an epistemic temporal model M tobe the maximum number of Y -arrows that can be taken from w, we obtain an internal notionof time because the time of a world is determined solely based on the time-keeping relation,which is internal to the model. Since Simple Epistemic Temporal Logic is arguably the mostbasic Epistemic Temporal Logic in which the time of a world can be expressed, we will focuson this basic Epistemic Temporal Logic in this paper. And in order to make clear whether agiven Kripke model is one that has a designated time-keeping relation (called the Y -relation),we adopt the following terminology: epistemic temporal models are Kripke models with adesignated Y -relation—these have an internal notion of time—whereas epistemic models areKripke models without a designated Y -relation—these have an external notion of time.

One key strength of epistemic temporal models is that the internal notion of time canbe used to describe multi-agent systems that need not be synchronous. Since DynamicEpistemic Logic uses epistemic models (having external time) and hence suffers from theProblem of Synchronicity, we see that even Simple Epistemic Temporal Logic can handle awider class of multi-agent systems than that which can be handled by Dynamic EpistemicLogic. This observation might suggest that Epistemic Temporal Logic is more useful thanDynamic Epistemic Logic. But in Epistemic Temporal Logic, every possible way in which

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the system can evolve must be determined in advance within the model using the internaltime-keeping relation; said informally, the protocol is fixed in Epistemic Temporal Logic.In contrast, the protocol in Dynamic Epistemic Logic is dynamic, as the way the systemprogresses is determined on-the-fly by the choice of the update modal that is used to producethe next pointed Kripke model appearing in the sequence (1). And although DynamicEpistemic Logic is expressively equivalent to Epistemic Logic, the succinctness result ofLutz [7] showed that an important multi-agent fragment of Dynamic Epistemic Logic isexponentially more succinct than Epistemic Logic in handling this dynamics. To summarize,there is an important trade-off between the two approaches: Epistemic Temporal Logic canhandle asynchronicity but is limited to fixed protocols, whereas Dynamic Epistemic Logichas (succinct) dynamic protocols but suffers from the Problem of Synchronicity.

In this paper, we propose a solution to the Problem of Synchronicity in a dynamic-protocol framework. Our solution is the framework we call Dynamic Epistemic TemporalLogic. Dynamic Epistemic Temporal Logic keeps track of time internally using epistemictemporal models, thereby permitting the possibility of asynchronous multi-agent systems.To maintain dynamic protocols, Dynamic Epistemic Temporal Logic extends the updatesof Dynamic Epistemic Logic from the class of epistemic models (having external time) tothe class of epistemic temporal models (having internal time). To achieve this, DynamicEpistemic Temporal Logic adds a new structural component to update modals [U, s]: theY -arrow. Y -arrows are used to specify exact positions in which the update f[U,s] is to insertY -arrows in the updated model M [U ]. Whereas standard Dynamic Epistemic Logic updatesensure that each world in M [U ] is one time-step ahead of any world in M , our Y -arrowsallow for greater flexibility in modeling the passage of time. In particular, using the internalnotion of time associated with the Y -relation, Y -arrows allow us to give worlds in M [U ] anynatural-number time. For example, in certain updates that embed M into M [U ], each worldin M [U ] can be seen either as a world in M or else as an arbitrarily distant possible futureof a world in M . This flexibility is essential in the study of asynchronous systems and is thekey to our solution to the Problem of Synchronicity.

After describing Dynamic Epistemic Temporal Logic and its use of Y -arrows, will showthat it is possible to identify sufficient conditions on our new update modals [U, s] thatwill guarantee that the update f[U,s] embeds M into M [U ] or preserves properties such assynchronicity in the resulting epistemic temporal model M [U ]. We shall then use theseconditions to show that epistemic temporal models that result from sequentially applying aproper subclass of our new Y -arrow–based updates are isomorphic to the generated sequencesof epistemic models from standard Dynamic Epistemic Logic that have been studied bya number of authors [6, 10, 11, 12, 15]. While [11, 12] showed that properties such assynchronicity are necessary of sequences generated in standard Dynamic Epistemic Logic, ourIsomorphism Theorem (Theorem 6.13) demonstrates that the necessity of these propertiesstems from the inherent structure of standard Dynamic Epistemic Logic update modals [U, s]themselves. This provides a new perspective on the results of [11, 12] as well as a method ofcircumventing the restrictions that such properties can impose. In particular, this providesus with a solution to the Problem of Synchronicity in the dynamic-protocol framework ofDynamic Epistemic Temporal Logic.

In the next section, we introduce the language LDETL and the theory TDETL of DynamicEpistemic Temporal Logic. So as to maintain readability of the text, we have moved all

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proofs to an appendix.

2 Syntax

Notation 2.1 (A, Y , Y ). A is a finite nonempty set of symbols not containing the symbolsY and Y . The members of A will be called agents.

To define LDETL, we must first define the internal structure of update modals [U, s]. Thisstructure is built on top of finite Kripke frames. If S is a nonempty set of symbols, then aKripke frame F (for S) is a pair (W F , RF ) consisting of a nonempty set W F whose membersare called worlds and a function RF : S → (W F → 2W

F) mapping each symbol m ∈ S to

a function Rm : W F → 2WF

; to say that F is finite means that W F is finite.1 The internalstructure of update modals [U, s] is given by the structure of the object U , called an updateframe.

Definition 2.2. For a language L, whose formulas we call L-formulas, an L-update frame isa tuple U = (W,R, p) satisfying the following: (W,R) is a finite Kripke frame for A∪{Y, Y }said to be underlying U , and p : W → L is a function mapping each world s ∈ W to anL-formula p(s). A state in U is just a world in the Kripke frame underlying U . Notation:for an L-update frame U , we write WU to denote the first component of the tuple U , wewrite RU to denote the second component of the tuple U , and we write pU to denote thethird component of the tuple U . A pointed L-update frame is a pair (U, s) consisting of anL-update frame U and a state s ∈ WU that will be called the point of (U, s).

Update frames are also called “action models” (or “event models”) in the Dynamic Epis-temic Logic literature [1, 2, 3, 13, 14]. For an update frame U , a state s ∈ WU representsthe communication of the formula pU(s). For an agent a ∈ A, the function RU

a representsagent a’s conditional uncertainty as to which formula is communicated: if s′ ∈ RU

a (s) andthe formula pU(s) was in fact communicated, then agent a will think that the formula pU(s′)is one of the formulas that might have been communicated.

We now define our language LDETL as an extension of the language LSETL of SimpleEpistemic Temporal Logic.

Notation 2.3. N is the set {0, 1, 2, 3, . . . } of natural numbers (this includes 0), and N+ is theset {1, 2, 3, . . . } of positive natural numbers (this excludes 0). Z is the set {. . . ,−3,−2,−1}∪N of integers.

Definition 2.4 (LSETL). LSETL, the Language of Simple Epistemic Temporal Logic, consistsof the formulas formed by the following grammar.

ϕ ::= ⊥ | > | pk | ϕ ? ϕ | ¬ϕ | [a]ϕ

k ∈ N, ? ∈ {→,∨,∧,≡}, a ∈ A ∪ {Y }

Terminology: we call [Y ] the yesterday modal. For each agent a ∈ A, we read the formula[a]ϕ as “agent a believes that ϕ is true.” We read the formula [Y ]ϕ as “ϕ is true in all

1The function RFa gives rise to a binary relation RF

adef= {(x, y) ∈ WF ×WF | y ∈ RF

a (x)} on WF . Wewill conflate RF

a and RFa whenever it is convenient. We will often refer to the members of RF

a as a-arrows.

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Basic Schemes

CL. Schemes for Classical Propositional Logic

Ka. [a](ϕ→ ψ)→ ([a]ϕ→ [a]ψ) for a ∈ AKY . [Y ](ϕ→ ψ)→ ([Y ]ϕ→ [Y ]ψ)

UA. [U, s]q ≡(pU(s)→ q

)for q ∈ {pk,⊥,>}

U?. [U, s](ϕ ? ψ) ≡([U, s]ϕ ? [U, s]ψ

)for ? ∈ {→,∨,∧,≡}

U¬. [U, s]¬ϕ ≡(pU(s)→ ¬[U, s]ϕ

)U[a]. [U, s][a]ϕ ≡

(pU(s)→ ∧

s′∈RUa (s)[a][U, s′]ϕ)

for a ∈ AU[Y ]. [U, s][Y ]ϕ ≡

(pU(s)→ ∧

s′∈RUY (s)[Y ][U, s′]ϕ)∧(

pU(s)→ ∧s′∈RUY (s)[U, s

′]ϕ)

Rules

` ϕ→ ψ ` ϕ` ψ

(MP)a ∈ A ∪ {Y } ` ϕ

` [a]ϕ(MN)

` ϕ` [U, s]ϕ

(UN)

Figure 1: The theory TDETL

possible yesterdays.” Notation: for each a ∈ A∪{Y }, we let 〈a〉 abbreviate ¬[a]¬; we define

for each i ∈ N the formula [a]iϕ by setting [a]0ϕdef= ϕ and [a]i+1ϕ

def= [a]([a]iϕ); for i ∈ N, the

formula 〈a〉iϕ is defined analogously.

Definition 2.5 (LDETL, TDETL). LDETL is the Language of Dynamic Epistemic TemporalLogic. The LDETL-formulas are the formulas that may be formed by the grammar obtainedfrom that in Definition 2.4 by adding the following formula-formation rule: if ϕ is an LDETL-formula and (U, s) is a pointed L-update frame with ∅ 6= L ⊆ LDETL, then [U, s]ϕ is anLDETL-formula. LDETL consists of the LDETL-formulas along with the L-update frames forwhich ∅ 6= L ⊆ LDETL. Terminology: we call [U, s] an update modal. Notation: we let 〈U, s〉abbreviate ¬[U, s]¬. We read the formula [U, s]ϕ as “after update f[U,s], ϕ is true.” Anupdate frame is an LDETL-update frame. A formula is a LDETL-formula. TDETL, the Theoryof Dynamic Epistemic Temporal Logic, is defined in Figure 1.

Since our interest here is in implementing update mechanisms on Kripke models with adesignated Y -relation, we do not impose any of the usual properties on belief or on timethat one might expect [5, 6, 10, 11, 12, 15]. So TDETL should be viewed as the minimaltheory that brings update mechanisms to Simple Epistemic Temporal Logic. Future workwill investigate extensions of this theory that include familiar restrictions on belief and ontime, though we do address the preservation of certain time-related properties in Section 5.

Definition 2.6 (LDETL Complexity; adapted from [14]). To each formula ϕ ∈ LDETL, weassign a positive natural number c(ϕ) ∈ N+, called the complexity of ϕ, as follows: letting|WU | denote the (nonzero, finite) number of states in an update frame U , we make the

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q◦ = q if q ∈ {pk,⊥,>}(ϕ ? ψ)◦ = ϕ◦ ? ψ◦

(¬ϕ)◦ = ¬(ϕ◦)

([a]ϕ)◦ = [a](ϕ◦) if a ∈ A([Y ]ϕ)◦ = [Y ](ϕ◦)

([U, s]q)◦ = (pU(s))◦ → q if q ∈ {pk,⊥,>}([U, s](ϕ ? ψ))◦ = ([U, s]ϕ)◦ ? ([U, s]ψ)◦

([U, s]¬ϕ)◦ = (pU(s))◦ → ¬([U, s]ϕ)◦

([U, s][a]ϕ)◦ = (pU(s))◦ → ∧s′∈RUa (s)[a]([U, s′]ϕ)◦ if a ∈ A

([U, s][Y ]ϕ)◦ =((pU(s))◦ → ∧

s′∈RUY (s)[Y ]([U, s′]ϕ)◦)∧(

(pU(s))◦ → ∧s′∈RUY (s)([U, s

′]ϕ)◦)

([U, s][U ′, s′]ϕ)◦ = ([U, s]([U ′, s′]ϕ)◦)◦

Figure 2: Function ◦ taking LDETL-formulas to LSETL-formulas, Theorem 2.8

following definitions.

c(q)def= 1 for q ∈ {pk,⊥,>}

c(ϕ ? ψ)def= 1 + max{c(ϕ), c(ψ)}

c(¬ϕ)def= 1 + c(ϕ)

c([a]ϕ)def= 1 + c(ϕ) for a ∈ A ∪ {Y }

c([U, s]ϕ)def= (4 + c(U)) · c(ϕ)

c(U)def= (1 + |WU |) ·maxs∈WU c(pU(s))

Notation 2.7 (Ln). For each fragment L of LTDEL and each n ∈ N, we let Ln denote theset of L-formulas ϕ with c(ϕ) ≤ n.

Theorem 2.8 (LDETL Reduction). The equations in Figure 2 define a function that mapseach LDETL-formula ϕ to an LSETL-formula ϕ◦ such that ` ϕ ≡ ϕ◦. Further, the equationsin Figure 2 are complexity-respecting: for each equation in Figure 2, ◦ is applied on theleft-hand side to a formula whose complexity is strictly larger than that of any formula onthe right-hand side to which ◦ is applied.

3 Semantics

Having defined the language LDETL and theory TDETL of Dynamic Epistemic Temporal Logic,we now define the semantics of LDETL. A Kripke model M is a tuple (WM , RM , V M) consistingof a Kripke frame (WM , RM) said to be underlying M and a function V M : {pk | k ∈ N} →2W

Mcalled a (propositional) valuation. A pointed Kripke model is a pair (M,w) consisting

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of a Kripke model M and a world w ∈ WM . The notion of LDETL-truth extends the standardsemantics for Dynamic Epistemic Logic [1, 2, 3, 13, 14] in the following way.

Definition 3.1 (LDETL-Truth, LDETL-Validity). For a pointed Kripke model (M,w) anda formula ϕ, we write M,w |=LDETL

ϕ to mean that ϕ is true at (M,w), and we writeM,w 6|=LDETL

ϕ to mean that ϕ is not true at (or false at) (M,w). The notion of truthof a formula at a pointed Kripke model is defined by the following induction on formulaconstruction.

• M,w 6|=LDETL⊥ and M,w |=LDETL

>.

• M,w |=LDETLpk means that w ∈ V M(pk).

• M,w |=LDETLϕ ? ψ means that M,w |=LDETL

ϕ star M,w |=LDETLψ, where “star” is to

be replaced by the (mathematical) English reading for the binary Boolean connective?.2

• M,w |=LDETL¬ϕ means that M,w 6|=LDETL

ϕ.

• M,w |=LDETL[a]ϕ means that M,x |=LDETL

ϕ for each x ∈ RMa (w).

• M,w |=LDETL[U, s]ϕ means that if M,w |=LDETL

pU(s), then M [U ], (w, s) |=LDETLϕ,

where M [U ] is defined as follows.

WM [U ] def= {(x, t) ∈ WM ×WU : M,x |=LDETL

pU(t)}For a ∈ A:

RM [U ]a (x, t)

def= {(y, u) ∈ WM [U ] : y ∈ RM

a (x) and u ∈ RUa (t)}

RM [U ]Y (x, t)

def= {(y, u) ∈ WM [U ] : y ∈ RM

Y (x) and u ∈ RUY (t)} ∪

{(y, u) ∈ WM [U ] : y = x and u ∈ RUY (t)}

V M [U ](pk)def= {(x, t) ∈ WM [U ] : M,x |=LDETL

pk}

To say that a formula ϕ is valid in a Kripke model M , written M |=LDETLϕ, means that

M,w |=LDETLϕ for each world w ∈ WM . To say that a formula ϕ is valid, written |=LDETL

ϕ,means that M |=LDETL

ϕ for each Kripke model M . When it ought not cause confusion, wemay omit the subscript “LDETL” when writing |=LDETL

.

Given a pointed Kripke model (M,w) representing a multi-agent situation and a pointedupdate frame (U, s) with M,w |= pU(s), the pointed Kripke model

(M [U ], (w, s)

)represents

the situation after the occurrence of the update described by [U, s]. According to Definition3.1, a world (x, t) must satisfy the property that M,x |= pU(t). The set {x ∈ WM |M,x |=pU(t)} of worlds x in M that satisfy pU(t) intuitively represents the set of worlds in M atwhich the formula pU(t) can truthfully be communicated—these are the worlds at which tcan take place.

For each a ∈ A, Definition 3.1 tells us that the relation RM [U ]a is determined by two factors:

agent a’s uncertainty as to which world was the case before the communication (represented

2Read → as “implies,” read ∨ as “or,” read ∧ as “and,” and read ≡ as “if and only if.”

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by RMa ) and agent a’s uncertainty as to which communication has occurred (represented

by RUa ). In particular, suppose (x′, t′) ∈ R

M [U ]a (x, t). Then if the communication pU(t)

corresponding to t actually occurred at world x, then agent a will think it possible that thecommunication pU(t′) corresponding to t′ occurred at world x′.

According to Definition 3.1, the relation RM [U ]Y is determined by two factors. The first

is the interaction between the relations RUY and RM

Y , which adds pairs to RM [U ]Y just as the

interaction between RUa and RM

a did to RM [U ]a for a ∈ A. The second factor is the relation

RUY : if there is a Y -arrow from state t to state t′ in U , then there will be a Y -arrow from

world (x, t) to world (x, t′) in M [U ]. The presence of a Y -arrow from t to t′ in U thus saysthat the communication corresponding to t′ is to be thought of as occurring one time-stepbefore the communication corresponding to t. This addition to the standard definition ofupdates in Dynamic Epistemic Logic [1, 2, 3, 13, 14] allows us to control how an updateaffects the time of worlds in the model M [U ].

Finally, we see that the valuation V M [U ] after the update simply inherits its truth con-ditions from the valuation V M before the update, making our updates purely temporal-epistemic.

Theorem 3.2 (Correctness). For each formula ϕ, we have ` ϕ if and only if |= ϕ.

4 A Simple Example

Suppose Passengers a and b are traveling together by train in China. Further, supposePassenger a understands Mandarin but that Passenger b does not, though Passenger b mis-takenly believes that they are both equally ignorant of the language. Now consider twoscenarios in which an announcement in Mandarin about a delay in arrival is made over theloudspeaker.

1. Passengers a and b are both awake and alert during the announcement.

2. Passenger a is awake and alert, but Passenger b, who is sleepy, dozes off and sleepsthrough the announcement. Waking up a few minutes later without knowing that theannouncement occurred, Passenger b mistakenly thinks that instead of sleeping for afew minutes, he merely blinked.

Taking p to be a propositional letter denoting the statement about late arrival, we rep-resent the first and the second scenarios in our framework using update frames (U1, t1) and(U2, t2), respectively pictured on the left and on the right in Figure 3.

In the first scenario, Passenger b knows that an announcement has taken place, but itprovides him with no new (non-temporal) information, nor does he believe that a gained any(non-temporal) information. In effect, this is a synchronous private announcement to a; afterall, both a and b know that an announcement occurred—so the event is synchronous—butonly a knows the content of the announcement—so the event is private to a. In Figure 3, s1and u are states in which no new (non-temporal) information is conveyed (since > is alwaystrue and thus conveys no new non-temporal information), while t1 is a state in which themessage p is communicated. Since t1 and u are each connected to s1 using a Y -arrow, the

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⊤

s1

p

t1

⊤u

a

a, b

b

Y

Y

a, b, Y

U1

⊤

s2

p

t2

a

Y

b

a, b, Y

U2

Figure 3: Update frames for the synchronous (U1, t1) and asynchronous (U2, t2) privateannouncement of p to a.

communications they represent occur one time-step after the communication represented bys1.

Since s1 is labeled by >, has a reflexive x-arrow for every x ∈ {a, b, Y }, and has noexiting Y -arrows, we see by the definition of truth (Definition 3.1) that any Kripke model Mis embedded into the Kripke model M [U1] by the mapping taking each world y ∈ WM to theworld (y, s1) ∈ WM [U1]. This embedding preserves a copy of the “past situation” M withinthe “current situation” M [U1], which leads us to call s1 a “past state.” So the role of thepast state s1 is to preserve a copy of a given situation M . The states t1 and u then representcommunications that occur one time-step after the situation M . At state t1, Passenger abelieves that t1 represents the only possible communication, while Passenger b believes thatu represents the only possible communication. Since both u and t1 are one time-step afterthe past state s1, the update f[U1,t1] describes the private communication of p to Passengera in which it is common knowledge that one time-step occurs. So we see that

|= (¬〈Y 〉> ∧ ¬[b]p)→ [U1, t1]([a]〈Y 〉> ∧ [a]p ∧ [b]〈Y 〉> ∧ ¬[b]p

).

That is, if no event has yet occurred and Passenger b does not believe p, then, after theoccurrence of f[U1,t1], Passenger a believes that an event occurred and that p is true, whereasPassenger b believes that an event occurred but does not believe that p is true.

In contrast, the second scenario is in effect an asynchronous private announcement toa. After all, while Passenger a knows that an announcement occurred and she knows itscontent, Passenger b has two mistaken beliefs: first, that no announcement occurred, andsecond, that the amount of time between closing and later opening his eyes is essentiallynegligible. b thus does not even think it possible that an event has occurred. Since theannouncement results in b having a mistaken belief about the number of events that haveoccurred, the announcement event is asynchronous. At state t2 in Figure 3, Passenger aknows that p is communicated, but Passenger b mistakenly believes that no event took placebecause the only state he considers possible is the past state s2. Accordingly, we see that

|= (¬〈Y 〉> ∧ ¬[b]p)→ [U2, t2]([a]〈Y 〉> ∧ [a]p ∧ ¬[b]〈Y 〉> ∧ ¬[b]p

).

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That is, if no event has yet occurred and Passenger b does not believe that p is true, then,after the occurrence of f[U2,t2], Passenger a believes that an event occurred and that p is true,whereas Passenger b believes neither that an event occurred nor that p is true.

These scenarios demonstrate the way in which our framework uses Y -arrows to describesynchronous and asynchronous private communications. In particular, we see that Y -arrowscan be used to describe updates that need not preserve synchronicity, as is the case with theasynchronous private announcement.

5 Properties and Preservation

In this section, we define several properties of Kripke models and update frames and thenstudy sufficient conditions for the preservation of these properties after the occurrence of anupdate.

Definition 5.1 (T -Runs, T -Histories, T -Depth). Fix a symbol T ∈ {Y, Y } and let F =(W,R) be a Kripke frame for A ∪ {Y, T}. A T -run (in F ) is a finite nonempty sequence{wi}ni=0 of worlds in F satisfying the property that n ∈ N and for each i ∈ N with i < n,we have that wi+1 ∈ RF

T (wi). We say that a T -run {wi}ni=0 begins at w0 and ends at wn.The length of a T -run {wi}ni=0 is defined as the number n. (Observe that the length of aT -run is one less than the number of worlds that make up the T -run.) To say that a T -runσ′ end-extends a T -run σ means that σ is a (not necessarily proper) prefix of σ′. (Notethat each T -run end-extends itself.) To say that a T -run σ is end-maximal (in F ) meansthat no T -run in F end-extends σ other than σ itself. A T -history (in F ) is a T -run in Fthat is end-maximal. (Note that a nonempty suffix of a T -history is itself a T -history.) Aworld appearing at the end of a T -history in F is said to be T -terminal (in F ). We definea function dFT : W F → N ∪ {∞} as follows: if there is a maximum n ∈ N such that there isa T -history in F of length n that begins at w, then dFT (w) is this maximum n; otherwise, ifno such maximum n ∈ N exists, then dFT (w) is ∞. We will call dFT (w) the T -depth of w.

Definition 5.2. Fix T ∈ {Y, Y } and let F = (W,R) be a Kripke frame for A ∪ {Y, T}.

• T -Depth–Defined (T -DD). To say that F is T -depth–defined (T -DD) means that foreach world w in F , we have that dFT (w) 6=∞.3

• Non–T -Branching. To say that F is non–T -branching means that for each w ∈ W F ,the set RF

T (w) has at most one member.

• T -Synchronous. To say that F is T -synchronous means that F is T -DD and for eacha ∈ A, each w ∈ W F , and each w′ ∈ RF

a (w), we have that dFT (w′) = dFT (w). Thenegation of “T -synchronous” is T -asynchronous.

3We observe that if F is T -depth–defined, then F is T -converse well-founded (that is, for every nonemptyset S of worlds in F , there is a nonempty subset S′ ⊆ S such that each world w ∈ S′ is T -terminal). However,if F is T -converse well-founded, it need not be the case that F is also T -depth–defined. So the notion ofT -depth–definedness is strictly stronger than the notion of T -converse well-foundedness.

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• T–Memory-Preserving. To say that F is T–memory-preserving means that F is T -DDand for each a ∈ A, each w ∈ W F , each v ∈ RF

a (w), and each w′ ∈ RFT (w), there is a

world v′ ∈ RMa (w′) ∩ (RF

T )∗(v), where (RFT )∗ is the reflexive transitive closure of RM

T .4

Diagrammatically:

w

v

w′

v′a

T

T ∗

a

In a T -synchronous Kripke frame, T–memory-preservation is equivalent to the well-known property of Perfect Recall (for T ) [11, 12].5

Convention: for tuples J having a Kripke frame (W J , RJ) underlying J , any use of aproperty or concept from Definition 5.1 or Definition 5.2 in reference to J is meant to bea use of that property or concept in reference to the Kripke frame (W J , RJ) underlying J .Example: for an update frame U , the expression “Y -run in U” is to be identified with theexpression “Y -run in (WU , RU).”

Definition 5.3 (Kripke Model Properties). Let M be a Kripke model.

• Synchronicity. To say that M is synchronous means that M is Y -synchronous. Thenegation of “synchronous” is asynchronous.

• Non–Past-Branching. To say that M is non–past-branching means that M is non–Y -branching.

• Forest-like. To say that M is forest-like means that M is Y -DD and non–past-branching.

• Memory-Preserving. To say thatM is memory-preserving means thatM is Y –memory-preserving.6 In a synchronous Kripke model, memory-preservation is equivalent to thewell-known property of Perfect Recall [11, 12].

Definition 5.4 (Update Frame Properties and Concepts). Let U be an update frame.

• Path-Preserving. A path-preserving run (in U) is a Y -run {si}ni=0 in U satisfying theproperty that for each i ∈ N with i < n, we have |= pU(si)→ pU(si+1). To say that Uis a path-preserving update frame means that each Y -run in U is path-preserving.

4The reflexive transitive closure R∗ : A → (W → 2W ) of R : A → (W → 2W ) is given as follows:v′ ∈ R∗(v) means that there is a finite sequence {vi}ni=0 with n ∈ N such that v0 = v, vn = v′, andvi+1 ∈ R(vi) for each i ∈ N with i < n.

5In words, to say that F satisfies Perfect Recall for T means that for each a ∈ A, each w ∈ WF , eachv ∈ RF

a (w), each w′ ∈ RFT (w), and each v′ ∈ RF

T (v), we have that v′ ∈ RMa (w′). The diagram for Perfect

Recall for T is obtained from the T–memory-preservation diagram by replacing the dashed T ∗-arrow with asolid T -arrow.

6Informally, in a memory-preserving Kripke model, an agent considers a world possible at time t only ifthat world was a future of a world considered possible at time t− 1.

11

• Depth-Respecting. To say that U is depth-respecting means that U is Y -DD and foreach s ∈ WU and each s′ ∈ RU

Y (s), we have that dUY (s′) ≤ dUY (s).

• Past State. To say that a state s ∈ WU is a past state means we have that |= pU(s) ≡ >,that RU

Y (s) = ∅, and that RUa (s) = {s} for each a ∈ A ∪ {Y }.

• Past-Preserving. To say that U is past-preserving means U is Y -DD, path-preserving,and every Y -run in U can be end-extended to a Y -history in U that ends at a paststate.

• Non–Past-Splitting. To say that U is non–past-splitting means that for each s ∈ WU ,we have that RU

Y (s) ∪RUY (s) has at most one member and that RU

Y (s) ∩RUY (s) = ∅.

Having defined these properties, we investigate their preservation under the presence ofupdates in the following two theorems. Theorem 5.5 concerns the behavior of past states inupdate frames, and Theorem 5.6 concerns the preservation of properties in Kripke models.

Theorem 5.5 (Past State). Let U be an update frame and M be a Kripke model.

• If s is a past state in U , then for each ϕ ∈ LDETL and each w ∈ WM , we have thatM [U ], (w, s) |= ϕ if and only if M,w |= ϕ.

• If U is past-preserving and non–past-spliting, s ∈ WU satisfies dUY (s) = n, and w ∈ WM

satisfies M,w |= pU(s), then for each ϕ ∈ LDETL, we have that M [U ], (w, s) |= 〈Y 〉nϕif and only if M,w |= ϕ.

Theorem 5.5 tells us that past states play the role of “maintaining a link to the past”within past-preserving, non–past-splitting update frames. In particular, if s is a past state,then the submodel ofM [U ] consisting of the worlds of the form (w, s) for some world w ∈ WM

is LDETL-indistinguishable from the Kripke model M itself. So the operation (M,w) 7→(M [U ], (w, s)

)retains a copy of the “past” state of affairs (M,w). Furthermore, if U is

past-preserving, then from any world in WM [U ], there is a finite sequence of Y -arrows thatleads back to this “past” state of affairs, thereby “maintaining a link to the past.”

Let us now examine the preservation of properties of the Kripke model M in the presenceof the operation M 7→M [U ].

Theorem 5.6 (Preservation). Let U be an update frame and M be a Kripke model suchthat M,w |= pU(s) for some w ∈ WM and s ∈ WU .

• Y -DD Preservation: if M is Y -DD and U is depth-respecting, then M [U ] is Y -DD.

• Synchronicity Preservation: if M is synchronous and U is depth-respecting, past-preserving, and Y -synchronous, then M [U ] is synchronous.

• Non–Past-Branching Preservation: if M is non–past-branching and U is non–past-splitting, then M [U ] is non–past-branching.

• Forest-likeness Preservation: if M is forest-like and U is depth-respecting and non–past-splitting, then M [U ] is forest-like.

• Memory Preservation: if M is memory-preserving and U is depth-respecting, past-preserving, non–past-splitting, and Y –memory-preserving, thenM [U ] is memory-preserving.

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6 Embedding Standard DEL

In this section, we show that standard (Temporal) Dynamic Epistemic Logic, whose updatemodals contain neither Y - nor Y -arrows, can be embedded in our framework in a naturalway. This provides clear connections between our work and the work in [6, 10, 11, 12, 15]on (Temporal) Dynamic Epistemic Logic, which will be described at the end of this section.

Definition 6.1 (Standard). Choose T ∈ {Y, Y }. To say that a Kripke frame F for A∪{Y, T}is standard means that the set W F of worlds in F does not contain the special symbol [ andfor each s ∈ W F and each m ∈ {Y, T}, we have RF

m(s) = ∅. To say that a Kripke modelor an L-update frame is standard means that the Kripke frame underlying that model or L-update frame is standard. To say that a pointed Kripke model or a pointed L-update frameis standard means that the Kripke model or L-update frame making up the first componentof the pair is standard.

Notation 6.2 (Sequences). Let τ be a finite possibly empty sequence. We write τ · x todenote the sequence obtained from τ by adding x at the end. |τ | denotes the number ofelements in τ . When convenient, we shall use the empty set symbol ∅ to denote the emptysequence. For each integer k ∈ Z, we define the sequence τk as follows: τk = ∅ if k < 0, τk isτ if |τ | ≤ k, and τk is the prefix of τ having k elements if 0 ≤ k < |τ |.

6.1 Temporal Dynamic Epistemic Logic

The language LTDEL of Temporal Dynamic Epistemic Logic [6, 11, 12] adds to the languageLSETL of Simple Epistemic Temporal Logic update modals [U, s] whose update frames U arestandard.

Definition 6.3 (LTDEL; [6]). LTDEL is the Language of Temporal Dynamic Epistemic Logic.The LTDEL-formulas are the formulas that may be formed by the grammar obtained from thatin Definition 2.4 (the definition of LSETL) by adding the following formula-formation rule: ifϕ is an LTDEL-formula and (U, s) is a standard pointed L-update frame with ∅ 6= L ⊆ LTDEL,then [U, s]ϕ is an LTDEL-formula. LTDEL consists of the LTDEL-formulas along with the L-update frames for which ∅ 6= L ⊆ LTDEL.

LTDEL makes explicit the external notion of time that is present in standard DynamicEpistemic Logic. In particular, LTDEL-formulas are evaluated using certain sequences ofKripke models, which allows us to identify the amount of time that has passed with how faralong the current position is in the sequence.

Definition 6.4 (Runs; projection functions π1, π2; adapted from [8, 10, 11, 12, 15]). A runis a nonempty finite sequence {Mi}ni=0 of standard Kripke models satisfying the propertythat for each i ∈ N with 0 < i ≤ n and each w ∈ WMi , we have that w is a pair (v, s) forwhich v ∈ WMi−1 . For an arbitrary pair (x1, x2), we define the projection functions π1 and

π2 by setting π1(x1, x2)def= x1 and π2(x1, x2)

def= x2. Hence for a run {Mi}ni=0 and a world

w ∈ WMi such that i ∈ N and 0 < i ≤ n, we have that π1(w) ∈ WMi−1 . For each k ∈ N andeach i ∈ {1, 2}, we define the iterated projection function πki by

πki (x)def=

{x if k = 0 or x is not a pair,

πi(πk−1i (x)) otherwise.

13

Hence for a run {Mi}ni=0 and a world w ∈ WMi such that i ∈ N and 0 < i ≤ n, for eachk ∈ N with k ≤ n, we have that πk1(w) ∈ WMi−k . A pointed run is a pair (r ·M,w) consistingof a run r ·M and a world w ∈ WM ; the world w is called the point of (r ·M,w).

Definition 6.5 (LTDEL-Truth; [6, 11, 12]). We define a notion of truth for LTDEL-formulasat pointed runs (r ·M,w) by an induction on the construction of LTDEL-formulas.

• r ·M,w 6|=LTDEL⊥ and r ·M,w |=LTDEL

>.

• r ·M,w |=LTDELpk means that w ∈ V M(pk).

• r · M,w |=LTDELϕ ? ψ means that r · M,w |=LTDEL

ϕ star r · M,w |=LTDELψ, where

“star” is to be replaced by the (mathematical) English reading for the binary Booleanconnective ?.

• r ·M,w |=LTDEL¬ϕ means that r ·M,w 6|=LTDEL

ϕ.

• For a ∈ A: r ·M,w |=LTDEL[a]ϕ means that r ·M,x |=LTDEL

ϕ for each x ∈ RMa (w).

• r ·M,w |=LTDEL[Y ]ϕ means that if |r| > 0, then r, π1(w) |=LTDEL

ϕ.

• r · M,w |=LTDEL[U, s]ϕ means that if we have r · M,w |=LTDEL

pU(s), then, letting

r′def= r ·M , it follows that r′ · r′[U ], (w, s) |=LTDEL

ϕ, where r′[U ] is the standard Kripkemodel defined as follows.

W r′[U ] def= {(x, t) ∈ WM ×WU | r′, x |=LTDEL

pU(t)}For a ∈ A :

Rr′[U ]a (x, t)

def= {(y, u) ∈ W r′[U ] | y ∈ RM

a (x) and u ∈ RUa (t)}

Rr′[U ]Y (x, t)

def= ∅

V r′[U ](pk)def= {(x, t) ∈ W r′[U ] | r′, x |=LTDEL

pk}

Validity for LTDEL is defined just as is validity for LDETL (Definition 3.1), except that eachsubscript LDETL is replaced by a subscript LTDEL. When it ought not cause confusion, wemay omit the subscript “LTDEL” in writing |=LTDEL

.

The ↓ operation, which we will define in a moment, provides a connection between LTDEL-truth and LDETL-truth for LSETL-formulas. This connection is described by the CollapseLemma (Lemma 6.7).

Definition 6.6 (↓). Let r = {Mi}ni=0 be a run. We define the Kripke model r↓ as follows.

W r↓ def=

⋃ni=0W

Mi

Rr↓a (v)

def= RMi

a (v), where a ∈ A and v ∈ WMi

Rr↓Y (v)

def=

{{π1(v)} if v ∈ WMi and i > 0,

∅ otherwise.

14

q• = q if q ∈ {pk,⊥,>}(ϕ ? ψ)• = ϕ• ? ψ•

(¬ϕ)• = ¬(ϕ•)

([a]ϕ)• = [a](ϕ•) if a ∈ A([Y ]ϕ)• = [Y ](ϕ•)

([U, s]q)• = (pU(s))• → q if q ∈ {pk,⊥,>}([U, s](ϕ ? ψ))• = ([U, s]ϕ)• ? ([U, s]ψ)•

([U, s]¬ϕ)• = (pU(s))• → ¬([U, s]ϕ)•

([U, s][a]ϕ)• = (pU(s))• → ∧s′∈RUa (s)[a]([U, s′]ϕ)• if a ∈ A

([U, s][Y ]ϕ)• = (pU(s))• → ϕ•

([U, s][U ′, s′]ϕ)• = ([U, s]([U ′, s′]ϕ)•)•

Figure 4: Function • taking LTDEL-formulas to LSETL-formulas, Theorem 6.8

Lemma 6.7 (Collapse). For each pointed run (r, w) and each ϕ ∈ LSETL, we have r, w |=LTDEL

ϕ if and only if r↓, w |=LDETLϕ.

We conclude this subsection with the LTDEL Reduction Theorem, which tells us that everyLTDEL-formula is equivalent to an LSETL-formula.

Theorem 6.8 (LTDEL Reduction). The equations in Figure 4 define a function that mapseach LTDEL-formula ϕ to an LSETL-formula ϕ• such that |=LTDEL

ϕ ≡ ϕ•. Further, the equa-tions in Figure 4 are complexity-respecting: for each equation in Figure 4, • is applied on theleft-hand side to a formula whose complexity is strictly larger than that of any formula onthe right-hand side to which • is applied.

6.2 Embedding Generated Standard DEL Structures

In this subsection, we prove that standard (Temporal) Dynamic Epistemic Logic is closelyconnected with our Dynamic Epistemic Temporal Logic. In particular, we show that theprocess of generating runs in LTDEL from standard Kripke models can be done in an isomor-phic manner within LDETL. This shows that the framework of standard (Temporal) DynamicEpistemic Logic can be embedded within our framework of Dynamic Epistemic TemporalLogic in a natural manner. To show this, we begin with a few preliminary definitions.

Definition 6.9 (Adapted from [8, 10, 11, 12, 15]). An L event-run is a finite possibly emptysequence {(Ui, si)}ni=1 of pointed L-update frames. A standard L event-run is an L event-runwhose constituent pointed L-update frames are all standard.

Definition 6.10 (Generated Structures). Let (M,w) be a standard pointed Kripke model.

• Let σ = {(Ui, si)}ni=1 be an LDETL event-run. We define(m(M,w, σ), m(M,w, σ)

),

15

called the pointed Kripke model generated from (M,w) by σ, to be the pointed Kripkemodel (Mm, wm) appearing at the end of the sequence {(Mi, wi)}mi=0 having the largestinteger m ≤ n subject to the following conditions: (M0, w0) = (M,w) and for eachj ∈ N with j < m, we have

– Mj, wj |=LDETLpUj+1(sj+1) and

– (Mj+1, wj+1) =(Mj[Uj+1], (wj, sj+1)

).

Note: “|=LDETL” and Mj[Uj+1] are given by LDETL-truth (Definition 3.1).

• Let σ = {(Ui, si)}ni=1 be a standard LTDEL event-run. We define(r(M,w, σ), r(M,w, σ)

),

called the pointed run generated from (M,w) by σ, to be the pointed run ({Mi}mi=0, wm)obtained from the sequence {(Mi, wi)}mi=0 of pointed Kripke models having the largestinteger m ≤ n subject to the following conditions: (M0, w0) = (M,w) and for eachj ∈ N with j < m, we have

– {Mi}ji=0, wj |=LTDELpUj+1(sj+1) and

– (Mj+1, wj+1) =({Mi}ji=0[Uj+1], (wj, sj+1)

).

Note: “|=LTDEL” and {Mi}ji=0[Uj+1] are given by LTDEL-truth (Definition 6.5).

In the remainder of this subsection, we will show that a pointed run generated froma standard Kripke model (M,w) by a standard LTDEL event-run σ is isomorphic with apointed Kripke model generated from (M,w) by a LDETL event-run σ] built from σ. This]-translation will also provide mappings between LTDEL-formulas and LDETL-formulas in away that preserves formula truth under the isomorphism. Accordingly, we shall see thatstandard (Temporal) Dynamic Epistemic Logic is naturally embedded within our frameworkof Dynamic Epistemic Temporal Logic. Proceeding, we begin by defining the ]-translation.

Definition 6.11 (]n, ]). For n ∈ N, we define the function ]n : LTDEL → LDETL in Figure 5.

If σ = {(Ui, si)}ni=1 is a standard LTDEL event-run, then we define σ]def= {(U ](i−1)

i , si)}ni=1.

For synchronous, non–past-branching Kripke models, there is a natural connection (de-scribed in the Connection Lemma, Lemma 6.12) between the LDETL-reduced ]-translation ofa LTDEL-formula ϕ and the LTDEL-reduced version of ϕ. This connection plays a crucial rolein the proof of our key result: the Isomorphism Theorem (Theorem 6.13).

Lemma 6.12 (Connection). For each formula ϕ ∈ LTDEL, each synchronous and non–past-branching Kripke model M , and each n ∈ N, we have

M |=LDETL〈Y 〉n[Y ]⊥ →

((ϕ]n)◦ ≡ ϕ•

).

Theorem 6.13 (Isomorphism). Let (M,w) be a standard pointed Kripke model and let σ

be a standard LTDEL event-run. Defining mdef= |r(M,w, σ)|−1, we have each of the following.

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q]ndef= q if q ∈ {pk,⊥,>}

(ϕ ? ψ)]ndef= ϕ]n ? ψ]n

(¬ϕ)]ndef= ¬(ϕ]n)

([a]ϕ)]ndef= [a](ϕ]n) if a ∈ A

([Y ]ϕ)]0def= [Y ]ϕ]0

([Y ]ϕ)]ndef= [Y ]ϕ](n−1) if n > 0

([U, s]ϕ)]ndef= [U]n, s](ϕ](n+1))

WU]n def= WU ] {[} (disjoint union)

for a ∈ A ∪ {Y, Y },

RU]na (s)

def=

RUa (s) if a 6= Y and s 6= [,

{[} if a 6= Y and s = [,

{[} if a = Y and s 6= [,

∅ if a = Y and s = [.

pU]n(s)def=

{(pU(s))]n ∧ 〈Y 〉n[Y ]⊥ if s 6= [,

> if s = [.

Figure 5: Definition of ]n : LTDEL → LDETL for n ∈ N

1. (r(M,w, σ)↓, r(M,w, σ)) and (m(M,w, σ]), m(M,w, σ])) are isomorphic.7

2. For each ϕ ∈ LTDEL, the following are equivalent.

(a) r(M,w, σ), r(M,w, σ) |=LTDELϕ.

(b) m(M,w, σ]), m(M,w, σ]) |=LDETLϕ]m.

The Isomorphism Theorem (Theorem 6.13) provides a strong connection between runsgenerated in a standard (Temporal) Dynamic Epistemic Logic setting and certain Kripkemodels generated in our Dynamic Epistemic Temporal Logic setting. In particular, [11, 12]studies certain structural properties of the forest structure given by a run r(M,w, σ) gener-ated from a standard pointed Kripke model (M,w) by a standard LSETL event-run σ. Forexample, in [11, 12], the authors of define what it means for the run r(M,w, σ) to be syn-chronous (among other properties) and then show that every run generated from a standardpointed Kripke model by a standard LSETL event-run is synchronous.8 Our PreservationTheorem (Theorem 5.6) works together with the Isomorphism Theorem (Theorem 6.13) to

7To say that two (pointed) Kripke models are isomorphic means that there exists an isomorphism betweenthem. An isomorphism between Kripke models M and M ′ is a bijection f : WM → WM ′

satisfying eachof the following: (i) v ∈ VM (pk) if and only if f(v) ∈ VM ′

(pk) for each k ∈ N, and (ii) u ∈ RMa (v) if and

only if f(u) ∈ RM ′

a (f(v)) for each a ∈ A∪ {Y }. An isomorphism between pointed Kripke models (M,w) and(M ′, w′) is an isomorphism f between M and M ′ for which f(w) = w′. See [4] for more information.

8If r(M,w, σ) is a run generated from a standard pointed Kripke model (M,w) by a standard LSETL

event-run σ, then the definition in [11, 12] would have us say that r(M,w, σ) is synchronous if and only ifr(M,w, σ)↓ is synchronous (according to our Definition 5.3).

17

provide a different perspective on this synchronicity result (and on the other results in thosepapers). In particular, our work shows that the results of [11, 12] can be viewed as a conse-quence of the structural properties that are present in the update frame U]n produced froma standard update frame U , thereby pinpointing the source of their result: synchronicity isa necessary consequence of the ]-translation.

7 Conclusion

In its technical essence, this paper is about studying what we can do when we extend thestandard Dynamic Epistemic Logic operation M 7→M [U ] [1, 3] to epistemic temporal modelsusing our Y -arrow mechanism. Essentially, the Y -arrow describes a sufficient condition forthe creation of Y -arrows in the model M [U ]: when there is a Y -arrow in the update frameU from state s to state s′, there will be a Y -arrow in M [U ] from world (x, s) to world (x, s′).Another way to understand this is by way of the hybrid scheme

[U, s][a]ϕ ≡ pU(s)→∧

s′∈WU

∀z.(aUa (s, s′)→ @z(p

U(s′)→ [U, s′]ϕ)),

where aUa is a function mapping each modal label a and each pair (s, s′) of states in U to aformula aUa (s, s′) that may contain the free variable z. This scheme uses the formula aUa (s, s′)to describe the precondition for the creation of a-arrows. For example, the standard DynamicEpistemic Logic precondition for the creation of a-arrows is given by setting

aUa (s, s′)def=

{〈a〉z if s′ ∈ RU

a (s),

⊥ if s′ /∈ RUa (s),

which ensures that there is an a-arrow in M [U ] from world (x, s) to world (x′, s′) if and onlyif there is an a-arrow both in U from state s to state s′ and in M from world x to world x′. InDynamic Epistemic Temporal Logic, the precondition aUY (s, s′) for the creation of Y -arrowsis given by setting

aUY (s, s′)def=

z ∨ 〈Y 〉z if s′ ∈ RU

Y (s) and s′ ∈ RUY (s),

z if s′ ∈ RUY (s) and s′ /∈ RU

Y (s),

〈Y 〉z if s′ /∈ RUY (s) and s′ ∈ RU

Y (s),

⊥ if s′ /∈ RUY (s) and s′ /∈ RU

Y (s),

which takes the standard Dynamic Epistemic Logic precondition 〈Y 〉z as sufficient for Y -arrow creation if there is a Y -arrow in U from s to s′ and takes our new Dynamic EpistemicTemporal Logic precondition z as sufficient for Y -arrow creation if there is a Y -arrow inU from s to s′. This way of understanding our work suggests one natural direction forfuture work: the study of other interesting sufficiency conditions (for example, the conditionaUa (s, s′) = > always creates an a-arrow from world (x, s) to world (x′, s′), making it so thatagent a entertains new possibilities that she may not have entertained before).

But in our view, the main contribution of this paper is our introduction of DynamicEpistemic Temporal Logic as a dynamic-protocol framework that avoids the Problem of Syn-chronicity. Dynamic Epistemic Temporal Logic extends the updates of Dynamic Epistemic

18

Logic so that they operate not just on epistemic models (having an external notion of time)but also on epistemic temporal models (having an internal notion of time) in a way thatallows us to control how an update affects the time of worlds in the model M [U ]. Wepresented two scenarios contrasting synchronous and asynchronous private announcements.This contrast indicates how it is that our framework can be used to reason about discrete-time multi-agent distributed systems that need not be synchronous, thereby illustrating howour framework avoids the Problem of Synchronicity. We then studied sufficient conditionsfor the preservation of various properties of Kripke models, such as synchronicity. Identifyingan isomorphism that connects epistemic temporal models generated in our framework withepistemic temporal models generated by standard updates as in [11, 12], we saw that thenecessity of synchronicity in standardly generated epistemic temporal models stems from thestructure that standard updates possess from the perspective of Dynamic Epistemic Tem-poral Logic. This analysis not only clarifies the way in which these preservation results arisebut it also provides us with a natural and elegant solution to the Problem of Synchronicityin Dynamic Epistemic Logic. For these reasons, we propose the Y -arrow mechanism as astraightforward and simple means of bringing temporal considerations within the reach of theDynamic Epistemic Logic framework. The payoff of our approach is a framework that usesdynamic protocols in harmony with the internal time-keeping mechanism of (Simple) Epis-temic Temporal Logic, thereby avoiding the Problem of Synchronicity in a dynamic-protocolframework.

A Appendix: the Proofs

A.1 Proof of the LDETL Reduction Theorem (Theorem 2.8)

Theorem. The equations in Figure 2 define a function that maps each LDETL-formula ϕ toan LSETL-formula ϕ◦ such that ` ϕ ≡ ϕ◦. Further, the equations in Figure 2 are complexity-respecting: for each equation in Figure 2, ◦ is applied on the left-hand side to a formulawhose complexity is strictly larger than that of any formula on the right-hand side to which◦ is applied.

Proof. To show that equations in Figure 2 define a function that takes each LDETL-formulaϕ to an LSETL-formula ϕ◦, we argue by induction on n ∈ N+ that the equations in Figure 2define a function ◦n mapping LnDETL-formulas to LSETL-formulas.9 We then define a function

◦ mapping LDETL-formulas to LSETL-formulas by setting ϕ◦def= ϕ◦c(ϕ) and argue that ◦ is the

unique function satisfying the equations in Figure 2.10 Along the way, it is shown that the

9In a bit more detail, the equations defining ◦n are obtained from those in Figure 2 as follows: the ◦ onthe left-hand side of an equation is to be replaced by ◦n, each ◦ on the right-hand side of an equation is tobe replaced by ◦n−1, and any equation that then contains ◦n−1 on its right-hand side with n − 1 ≤ 0 is tobe omitted. We then see that ◦1 is well-defined, and we show that ◦n+1 is well-defined if ◦n is well-defined.The latter requires us to prove that our equations in Figure 2 are complexity-respecting. So by induction onn, we conclude that ◦n is well-defined for each n ∈ N+.

10In a bit more detail, making frequent use of the fact that the equations in Figure 2 are complexity-respecting, we proceed in the following way. First, we argue by induction on n ∈ N+ that for each ϕ ∈ Ln

DETL

and each k ∈ N, we have ϕ◦c(ϕ) = ϕ◦c(ϕ)+k . Using this, we argue that the function ◦ defined by ϕ◦def= ϕ◦c(ϕ)

19

equations in Figure 2 are complexity-respecting. We then argue by induction on n ∈ N+

that for each ϕ ∈ LnDETL, we have ` ϕ ≡ ϕ◦. The argument is a straightforward adaptationof the standard argument in Dynamic Epistemic Logic [14].

A.2 Proof of the Correctness Theorem (Theorem 3.2)

Theorem. For each formula ϕ, we have ` ϕ if and only if |= ϕ.

Proof. All but one case of the soundness argument (` ϕ implies |= ϕ) are straightforwardadaptations of the standard Dynamic Epistemic Logic arguments [14], so we shall only provethis one case here. Proceeding, we are to show that

|= [U, s][Y ]ϕ ≡ (pU(s)→ ∧s′∈RUY (s)[Y ][U, s′]ϕ) ∧

(pU(s)→ ∧s′∈RUY (s)[U, s

′]ϕ) .

(2)

Let us first prove the left-to-right direction of this equivalence. So assuming M,w |=[U, s][Y ][ϕ] and M,w |= pU(s) and choosing t ∈ RU

Y (s) and u ∈ RUY (s) arbitrarily, we

prove that M,w |= [Y ][U, t]ϕ and M,w |= [U, u]ϕ. It follows from our assumptions by the

definition of truth that M [U ], (w′, s′) |= ϕ for each (w′, s′) ∈ RM [U ]Y (w, s). If v ∈ RM

Y (w)

and M, v |= pU(t), then (v, t) ∈ RM [U ]Y (w, s) and hence M [U ], (v, t) |= ϕ. We therefore have

that M,w |= [Y ][U, t]ϕ. Further, if M,w |= pU(u), then (w, u) ∈ RM [U ]Y (w, s) and hence

M [U ], (w, u) |= ϕ. We therefore have that M,w |= [U, t]ϕ. This completes the left-to-rightdirection of the equivalence (2).

We now show the right-to-left direction of the equivalence (2). So assuming M,w |=pU(s), M,w |= [Y ][U, t]ϕ for each t ∈ RU

Y (s), and M,w |= [U, u]ϕ for each u ∈ RUY (s), we

prove that M [U ], (w, s) |= [Y ]ϕ. So choose (w′, s′) ∈ RM [U ]Y (w, s) arbitrarily, from which it

follows that M,w′ |= pU(s′). If w′ ∈ RMY (w) and s′ ∈ RM

Y (s), then we have that M,w |=[Y ][U, s′]ϕ and hence that M [U ], (w′, s′) |= ϕ. If it is not the case that w′ ∈ RM

Y (w) ands′ ∈ RM

Y (s), then it must be that w′ = w and s′ ∈ RMY (s), from which it follows that

M [U ], (w′, s′) |= ϕ. So we have shown that we have M [U ], (w′, s′) |= ϕ for each (w′, s′) ∈RM [U ]Y (w, s), from which it follows that M [U ], (w, s) |= [Y ]ϕ. This completes the right-to-left

direction of the equivalence (2). We have therefore shown that Axiom U [Y ] is valid.The completeness argument (|= ϕ implies ` ϕ) follows by the LDETL Reduction Theorem,

the standard modal logic canonical model argument for the language LSETL [4], and thecombination of the LDETL Reduction Theorem with the soundness argument.

A.3 Proof of the Past State Theorem (Theorem 5.5)

Theorem. Let U be an update frame and M be a Kripke model.

• If s is a past state in U , then for each ϕ ∈ LDETL and each w ∈ WM , we have thatM [U ], (w, s) |= ϕ if and only if M,w |= ϕ.

satisfies the equations in Figure 2. Finally, we argue by induction on n ∈ N+ that if † is another functionsatisfying the equations in Figure 2, then ϕ† = ϕ◦ for each ϕ ∈ Ln

DETL.

20

• If U is past-preserving and non–past-spliting, s ∈ WU satisfies dUY (s) = n, and w ∈ WM

satisfies M,w |= pU(s), then for each ϕ ∈ LDETL, we have that M [U ], (w, s) |= 〈Y 〉nϕif and only if M,w |= ϕ.

Proof. We first prove the first item of this theorem. Let s ∈ WU be a past state and letw ∈ WM be a world. Applying the LDETL Reduction Theorem (Theorem 2.8) and theCorrectness Theorem (Theorem 3.2), it sufficies for us to prove that M [U ], (w, s) |= ϕ if andonly if M,w |= ϕ for each LSETL-formula ϕ. But this follows by observing that the definitionsof past state and LDETL-truth imply that the unraveling of the pointed Kripke model (M,w)is isomorphic to the unraveling of the pointed Kripke model

(M [U ], (w, s)

)[4].

We now proceed to the proof of the second item of this theorem. Suppose that U ispast-preserving and non–past-splitting, s ∈ WU satisfies dUY (s) = n, and w ∈ WM satisfies

M,w |= pU(s). Since U is past-preserving and dUY (s) = n, there is a path-preserving Y -

history σdef= {si}ni=0 in U that begins at s and ends at a past state. Since M,w |= pU(s), it

follows from the fact that σ is path-preserving that M,w |= pU(si) for each i ∈ N with i ≤ n.Applying LDETL-truth and the fact that U is past-preserving (and hence path-preserving),

we then have that σ′def= {(w, si)}ni=0 is a Y -history in M [U ]. Since U is non–past-splitting,

it follows that σ′ is the unique Y -run in M [U ] that begins at (w, s) and has length n.We therefore have that M [U ], (w, s) |= 〈Y 〉nϕ if and only if M [U ], (w, sn) |= ϕ for eachϕ ∈ LDETL. Since sn is a past state, we then have by the first item of this theorem thatM [U ], (w, sn) |= ϕ if and only if M,w |= ϕ for each ϕ ∈ LDETL. The result follows.

A.4 Proof of the Preservation Theorem (Theorem 5.6)

Theorem. Let U be an update frame and M be a Kripke model such that M,w |= pU(s)for some w ∈ WM and s ∈ WU .






Proof. We prove each item in turn.


Assume that M is a Y -DD Kripke model and that U is a depth-respecting (and henceY -DD) update frame. We then have that dMY (w) ∈ N for each world w ∈ WM and thatdUY (s) ∈ N for each state s ∈ WU . So to show that M [U ] is Y -DD, it suffices for us to

21

prove that dM [U ]Y (w, s) ≤ dMY (w) + dUY (s) for each (w, s) ∈ WM [U ]. The argument is by

an induction on the sum dMY (w) + dUY (s) ∈ N that breaks up into two cases: w′ = wand w′ 6= w. We omit the details due to space constraints.


Let us first assume the following claim: if M is synchronous (and hence Y -DD) andU is depth-respecting (and hence Y -DD), past-preserving, and Y -synchronous, then

dM [U ]Y (w, s) = dMY (w) + dUY (s) for each (w, s) ∈ WM [U ]. Proceeding under the assump-

tion of this claim, if we have that (w′, s′) ∈ RM [U ]a (w, s) with a ∈ A, it follows that

w′ ∈ RMa (w) and s′ ∈ RU

a (s) by LDETL-truth. But since we have that dMY (w) = dMY (w′)by the synchronicity of M and that dUY (s) = dUY (s′) by the Y -synchronicity of U , it

follows by the claim that dM [U ]Y (w, s) = d

M [U ]Y (w′, s′), which is what we wished to show.

So all that remains is for us to prove the claim. Now we showed in the above argumentfor Y -DD Preservation that having M synchronous (and hence Y -DD) and U depth-

respecting (and hence Y -DD) all together imply dM [U ]Y (w, s) ≤ dMY (w) + dUY (s) for each

(w, s) ∈ WM [U ]. So it suffices for us to show that the additional condition of synchronic-ity on M and the additional conditions of past-preservation and Y -synchronicity on U

all together imply that dMY (w) + dUY (s) ≤ dM [U ]Y (w, s) for each (w, s) ∈ WM [U ]. This is

argued by induction on the sum dMY (w)+dUY (s) ∈ N; in the induction step, we consider

two cases: RUY (s) 6= ∅ and RU

Y (s) = ∅. We omit the details due to space constraints.


Suppose that U is non–past-splitting and that M is non–past-branching. Given anarbitrary (w, s) ∈ WM [U ], we wish to show that R

M [U ]Y (w, s) contains at most one

member. We argue this by showing that under the membership assumptions (w1, s1) ∈RM [U ]Y (w, s) and (w2, s2) ∈ RM [U ]

Y (w, s), we have that (w1, s1) = (w2, s2). Proceedingwith these membership assumptions, it follows by LDETL-truth that we have relationsR1 ∈ {RU

Y , RUY } and R2 ∈ {RU

Y , RUY } such that s1 ∈ R1(s) and s2 ∈ R2(s). But to have

U non–past-splitting means that RUY (s) ∪ RU

Y (s) contains at most one member and

that RUY (s) ∩ RU

Y (s) = ∅. It therefore follows that R1 = R2 and that s1 = s2. Now if

R1 = R2 = RUY , then it follows that w1 = w = w2 by LDETL-truth and our membership

assumptions. Otherwise, if R1 = R2 = RUY , then it follows that w1 ∈ RM

Y (w) andw2 ∈ RM

Y (w) by LDETL-truth and our membership assumptions, and we again concludethat w1 = w2 because M is non–past-branching. We therefore conclude that (w1, s1) =(w2, s2).


A forest-like Kripke model is a Kripke model that is Y -DD and non–past-branching.The proof therefore follows by our arguments on Y -DD Preservation and Non–Past-Branching Preservation.

22


Suppose thatM is memory-preserving (and hence Y -DD) and that U is depth-respecting,past-preserving, non–past-splitting, and Y –memory-preserving. M [U ] is memory-preserving if we have (1) that M [U ] is Y -DD, and (2) that for each a ∈ A, each

(w, s) ∈ WM [U ], each (v, t) ∈ RM [U ]a (w, s), and each (w′, s′) ∈ RM [U ]

Y (w, s), there is a

world (v′, t′) ∈ RM [U ]a (w′, s′) ∩ (R

M [U ]Y )∗(v, t). Item (1) follows immediately from our

assumptions by our argument in the previous item on Y -DD Preservation. So all thatremains is for us to prove Item (2), which we express for convenience using the followingdiagram.

(w, s)

(v, t)

(w′, s′)

(v′, t′)

a

Y

Y ∗

a

Proceeding with our proof of Item (2), suppose we are given (w, s) ∈ WM [U ], (v, t) ∈RM [U ]a (w, s), and (w′, s′) ∈ RM [U ]

Y (w, s). It follows by LDETL-truth that v ∈ RMa (w) and

t ∈ RUa (s). Our argument now breaks up into two cases.

– Case: s′ ∈ RUY (s).

Since U is Y –memory-preserving, there exists a t′ ∈ RUa (s′) ∩ (RU

Y )∗(t):

s

t

s′

t′

a

Y

Y ∗

a

Since U is past-preserving, it follows that (v, t′) ∈ WM [U ]. Applying LDETL-truth,

we then have that (v, t′) ∈ (RM [U ]Y )∗(v, t) and (v, t′) ∈ RM [U ]

a (w, s′):

(w, s)

(v, t)

(w, s′)

(v, t′)

a

Y

Y ∗

a

Conclusion: M [U ] is Y –memory-preserving.

– Case: s′ /∈ RUY (s)

In this case, it follows by LDETL-truth that w′ ∈ RMY (w) and s′ ∈ RU

Y (s). Since Mis Y –memory-preserving there exists a v′ ∈ RM

a (w′) ∩ (RMY )∗(v):

23

w

v

w′

v′a

Y

Y ∗

a

Since U is past-preserving, the singleton Y -run {s} can be end-extended to aY -history σ that ends at a past state. Since s′ ∈ RU

Y (s) and U is non–past-splitting, RU

Y (s) = ∅. We therefore have that σ = {s} and hence that s is itself apast state. Applying LDETL-truth and the definition of past state, it follows thats′ = s = t, that |= pU(s) ≡ >, that s ∈ RU

a (s), and that s ∈ RUY (s). Applying

LDETL-truth, it then follows that (v′, s) ∈ W [U ], that (v′, s) ∈ (RM [U ]Y )∗(v, s), and

that (v′, s) ∈ RM [U ]a (w′, s):

(w, s)

(v, s)

(w′, s)

(v′, s)

a

Y

Y ∗

a

Conclusion: M [U ] is Y –memory-preserving.

A.5 Proof of the Collapse Lemma (Lemma 6.7)

Lemma. For each pointed run (r, w) and each ϕ ∈ LSETL, we have r, w |=LTDELϕ if and only

if r↓, w |=LDETLϕ.

Proof. This can be proved by induction on the construction of ϕ ∈ LSETL. The key case isϕ = [Y ]ψ, which makes use of the fact that r1 = r2·M2, w ∈ WM1 , and π1(w) ∈ WM2 togetherimply that the unraveling of the pointed Kripke model

((r2 ·M2)↓, π1(w)

)is isomorphic to

the unraveling of the pointed Kripke model((r1 ·M1)↓, π1(w)

)[4]. Since ψ ∈ LSETL, it follows

that (r2 ·M2)↓, π1(w) |=LDETLψ is equivalent to (r1 ·M1)↓, π1(w) |=LDETL

ψ if r1 = r2 ·M2.The induction hypothesis is then applied to the left side of this equivalence.

A.6 Proof of the LTDEL Reduction Theorem (Theorem 6.8)

Theorem. The equations in Figure 4 define a function that maps each LTDEL-formula ϕto an LSETL-formula ϕ• such that |=LTDEL

ϕ ≡ ϕ•. Further, the equations in Figure 4 arecomplexity-respecting: for each equation in Figure 4, • is applied on the left-hand side to aformula whose complexity is strictly larger than that of any formula on the right-hand sideto which • is applied.

Proof. This proof is similar to the proof of the LDETL Reduction Theorem, Theorem 2.8,though the last part of the proof uses LTDEL-truth “|=LTDEL

” in place of provability “`”.

24

A.7 Proof of the Connection Lemma (Lemma 6.12)

Lemma. For each formula ϕ ∈ LTDEL, each synchronous and non–past-branching Kripkemodel M , and each n ∈ N, we have

M |=LDETL〈Y 〉n[Y ]⊥ →

((ϕ]n)◦ ≡ ϕ•

).

Proof. We argue by induction on m ∈ N+ that for each ϕ ∈ LmTDEL (Notation 2.7), eachsynchronous and non–past-branching Kripke model M , and each n ∈ N, the validity in thestatement of this theorem holds. Most cases are straightforward, so we shall only addressthe most interesting inductive cases.

• Case: ϕ = [a]ψ with a ∈ A.

To begin, let us argue that M, v |=LDETL〈Y 〉n[Y ]⊥ for each v ∈ RM

a (w). Proceeding,M,w |=LDETL

〈Y 〉n[Y ]⊥ implies that there is a Y -history σ in M that begins at wand has length n. Since M is non–past-branching, σ is the unique Y -history thatbegins at w, and hence it follows that dMY (w) = n. Since M is synchronous, it followsthat dMY (v) = n for each v ∈ RM

a (w) and hence that M, v |=LDETL〈Y 〉n[Y ]⊥ for each

v ∈ RMa (w).

We now prove that M,w |=LDETL(([a]ψ)]n)◦ ≡ ([a]ψ)•. Proceeding, since c(ψ) <

c([a]ψ), it follows by the induction hypothesis, our result from the previous paragraph,and LDETL-truth that M, v |=LDETL

(ψ]n)◦ ≡ ψ• for each v ∈ RMa (w). Applying LDETL-

truth and modal reasoning, it follows that M,w |=LDETL[a](ψ]n)◦ ≡ [a]ψ•. Applying

the definitions of ◦, ]n, and •, we conclude that M,w |=LDETL(([a]ψ)]n)◦ ≡ ([a]ψ)•.

• Case: ϕ = [Y ]ψ and n = 0.

We prove that M,w |=LDETL(([Y ]ψ)]0)◦ ≡ ([Y ]ψ)•. Proceeding, our assumption that

M,w |=LDETL〈Y 〉0[Y ]⊥ means that M,w |=LDETL

[Y ]⊥ and hence that RMY (w) = ∅. It

follows by LDETL-truth that M,w |=LDETL[Y ](ψ]0)◦ and M,w |=LDETL

[Y ]ψ• and hencethat M,w |=LDETL

[Y ](ψ]0)◦ ≡ [Y ]ψ•. Applying the definitions of ◦, ]0, and •, it followsthat M,w |=LDETL

(([Y ]ψ)]0)◦ ≡ ([Y ]ψ)•.

• Case: ϕ = [Y ]ψ and n > 0.

To begin, we note that our assumptions M,w |=LDETL〈Y 〉n[Y ]⊥ and n > 0 together

imply that RMY (w) 6= ∅ and that M, v |=LDETL

〈Y 〉n−1[Y ]⊥ for each v ∈ RMY (w). Since

M is non–past-branching, RMY (w) 6= ∅ implies that RM

Y (w) = {u} for some u ∈ WM .Hence M,u |=LDETL

〈Y 〉n−1[Y ]⊥ and RMY (w) = {u}.

We now prove that M,w |=LDETL(([Y ]ψ)]n)◦ ≡ ([Y ]ψ)•. Since c(ψ) < c([Y ]ψ), it follows

by the induction hypothesis, the results of the previous paragraph, and LDETL-truththat M,u |=LDETL

(ψ](n−1))◦ ≡ ψ• and RMY (w) = {u}. Applying LDETL-truth and modal

reasoning, M,w |=LDETL[Y ](ψ](n−1))◦ ≡ [Y ]ψ•. Applying the definitions of ◦, ]n (for

n > 0), and •, we conclude that M,w |=LDETL(([Y ]ψ)]n)◦ ≡ ([Y ]ψ)•.

25

• Case: ϕ = [U, s][a]χ with a ∈ A.

Choose an arbitrary v ∈ RMa (w). It follows by our argument in the case “[a]ψ” above

that M, v |= 〈Y 〉n[Y ]⊥. (Note that this argument made use of our assumption that Mis synchronous and non–past-branching.) Now choose an arbitrary s′ ∈ RU]n

a (s). Sincewe have that RU]n

a (s) = RUa (s) by the definition of ]n and the fact that [U, s][a]χ ∈

LTDEL (and hence that s 6= [), it follows that c([U, s′]χ) < c([U, s][a]χ) and thus thatM, v |=LDETL

(([U, s′]χ)]n)◦ ≡ ([U, s′]χ)• by the induction hypothesis and LDETL-truth.Since v ∈ RM

a (w) and s′ ∈ RU]na (s) = RU

a (s) were chosen arbitrarily, it follows byLDETL-truth, modal reasoning, and the definition of ]n that

M,w |=LDETL

∧s′∈RU]na (s)[a]([U]n, s′]χ](n+1))◦ ≡∧s′∈RUa (s)[a]([U, s′]χ)• .

(3)

Further, since (pU]n(s))◦ = (pU(s)]n)◦ ∧ 〈Y 〉n[Y ]⊥ by the definitions of ]n and ◦, andsince M,w |=LDETL

〈Y 〉n[Y ]⊥ by LDETL-truth, it follows that M,w |=LDETL(pU(s)]n)◦ ≡

(pU]n(s))◦. In addition, as c(pU(s)) < c([U, s][Y ]χ), it follows by the induction hypoth-esis that M,w |=LDETL

(pU]n(s))◦ ≡ pU(s)•. Applying (3), it follows by LDETL-truth andthe definitions of ◦, •, and ]n that

M,w |=LDETL(([U, s][a]χ)]n)◦ ≡ ([U, s][a]χ)• .

• Case: ϕ = [U, s][Y ]χ.

We prove that M,w |=LDETL(([U, s][Y ]χ)]n)◦ ≡ ([U, s][Y ]χ)•. Proceeding, since [ is a

past state in U]n, it follows by LDETL-truth and the Past State Theorem (Theorem 5.5)that |=LDETL

([U]n, []χ]n) ≡ χ]n. It therefore follows by the LDETL Reduction Theorem(Theorem 2.8), the Correctness Theorem (Theorem 3.2), and the definition of ◦ that|=LDETL

([U]n, []χ]n)◦ ≡ (χ]n)◦. Since we have that c(χ) < c([U, s][Y ]χ) and thatM,w |=LDETL

〈Y 〉n[Y ]⊥, it follows by the induction hypothesis and LDETL-truth thatM,w |=LDETL

(χ]n)◦ ≡ χ•. Hence M,w |=LDETL([U]n, []χ]n)◦ ≡ χ•. Since we have

c(pU(s)) < c([U, s][Y ]χ) and M,w |=LDETL〈Y 〉n[Y ]⊥, the induction hypothesis and

LDETL-truth imply that M,w |=LDETL(pU(s)]n)◦ ≡ pU(s)•. Combining this with our

result from two sentences ago, we have that

M,w |=LDETL

((pU(s)]n)◦ → ([U]n, []χ]n)◦

)≡ (pU(s)• → χ•) .

By the same reasoning as given in the case “[U, s][a]χ”, we have that M,w |=LDETL

(pU(s)]n)◦ ≡ (pU]n(s))◦, whence by LDETL-truth and the definition of •, it follows that

M,w |=LDETL

((pU]n(s))◦ → ([U]n, []χ]n)◦

)≡ ([U, s][Y ]χ)• . (4)

Further, we have that RU]nY (s) = RU

Y (s) = ∅ by the definition of ]n and the fact thatU ∈ LTDEL (and hence that U is standard), and we have that RU]n

Y (s) = {[} by thedefinition of ]n. Accordingly,∧

s′∈RU]nY (s)[Y ]([U]n, s′]χ]n)◦ = > and∧s′∈RU]nY (s)([U]n, s

′]χ]n)◦ = ([U]n, []χ]n)◦ .

26

It therefore follows from (4) by LTDEL-truth and the definition of ◦ that

M,w |=LDETL([U]n, s][Y ]χ]n)◦ ≡ ([U, s][Y ]χ)•

and hence that M,w |=LDETL(([U, s][Y ]χ)]n)◦ ≡ ([U, s][Y ]χ)• by the definition of ]n.

• Case: ϕ = [U, s][U ′, s′]χ.

M,w |= pU]n([) by LDETL-truth and the fact that pU]n([) = >. Further, U]n is depth-respecting, past-preserving, Y -synchronous, and non–past-splitting. So since M issynchronous and non–past-branching, it follows from the Preservation Theorem (The-orem 5.6) that M [U]n] is synchronous and non–past-branching. Since we have thatc([U ′, s′]χ) < c([U, s][U ′, s′]χ), it follows by the induction hypothesis that

M [U]n] |=LDETL〈Y 〉n+1[Y ]⊥ →((([U ′, s′]χ)](n+1))◦ ≡ ([U ′, s′]χ)•

).

(5)

We wish to argue that

M,w |=LDETL[U]n, s]

((([U ′, s′]χ)](n+1))◦ ≡ ([U ′, s′]χ)•

). (6)

If M,w 6|=LDETLpU]n(s), it is immediate that (6) holds. Let us assume that M,w |=LDETL

pU]n(s). Then M,w |=LDETL〈Y 〉n[Y ]⊥ implies that there exists a Y -history {wi}ni=0 in

M with w0 = w. So since M,w |=LDETLpU]n(s), we have that (w, s) · {(wi, [)}ni=0 is a Y -

history in M [U]n] having length n+ 1. It therefore follows that M [U]n], (w, s) |=LDETL

〈Y 〉n+1[Y ]⊥ and hence that

M [U]n], (w, s) |= (([U ′, s′]χ)](n+1))◦ ≡ ([U ′, s′]χ)• (7)

by (5) and LDETL-truth. But (7) implies (6) by LDETL-truth. Applying Axiom U?, theCorrectness Theorem (Theorem 3.2), and LDETL-truth, it follows from (6) that

M,w |=LDETL[U]n, s](([U ′, s′]χ)](n+1))◦ ≡ [U]n, s]([U ′, s′]χ)• .

Since ([U ′, s′]χ)• ∈ LSETL and δ](n+1) = δ for each δ ∈ LSETL, we have

M,w |=LDETL[U]n, s](([U ′, s′]χ)](n+1))◦ ≡[U]n, s](([U ′, s′]χ)•)](n+1) .

Applying the definitions of ]n and ](n+ 1), we have

M,w |=LDETL[U]n, s]([U ′](n+ 1), s′]χ](n+2))◦ ≡([U, s]([U ′, s′]χ)•)]n .

Applying the LDETL Reduction Theorem (Theorem 2.8), the Correctness Theorem, andLDETL-truth, it follows that

M,w |=LDETL([U]n, s]([U ′](n+ 1), s′]χ](n+2))◦)◦ ≡(([U, s]([U ′, s′]χ)•)]n)◦ .

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Since c([U, s]([U ′, s′]χ)•) < c([U, s][U ′, s′]χ), it follows by the induction hypothesis, thefact that M,w |=LDETL

〈Y 〉n[Y ]⊥, and LDETL-truth that

M,w |=LDETL([U]n, s]([U ′](n+ 1), s′]χ](n+2))◦)◦ ≡([U, s]([U ′, s′]χ)•)• .

Applying the definitions of ◦ and •, we have

M,w |=LDETL([U]n, s][U ′](n+ 1), s′]χ](n+2))◦ ≡([U, s][U ′, s′]χ)• .

Applying the definitions of ](n+ 1) and then the definition of ]n, we conclude that

M,w |=LDETL(([U, s][U ′, s′]χ)]n)◦ ≡ ([U, s][U ′, s′]χ)• .

A.8 Proof of the Isomorphism Theorem (Theorem 6.13)

Theorem. Let (M,w) be a standard pointed Kripke model and let σ be a standard LTDEL

event-run. Defining mdef= |r(M,w, σ)| − 1, we have each of the following.

1. (r(M,w, σ)↓, r(M,w, σ)) and (m(M,w, σ]), m(M,w, σ])) are isomorphic.

2. For each ϕ ∈ LTDEL, the following are equivalent.

(a) r(M,w, σ), r(M,w, σ) |=LTDELϕ.

(b) m(M,w, σ]), m(M,w, σ]) |=LDETLϕ]m.

Proof. For each Y -DD Kripke model N and each pair (i, j) ∈ Z2 of integers, we define

N(i..j)def= {w ∈ WN | i ≤ dNY (w) ≤ j}. Notice that N(i..j) = ∅ if j < i or j < 0. We also let

N(..j) abbreviate N(0..j) and we let N(i) abbreviate N(i..i).We fix a standard pointed Kripke model (M,w) for the remainder of the proof. Using

the Preservation Theorem, it is not difficult to see that for each standard LTDEL event-run σ, the Kripke models r(M,w, σ)↓ and m(M,w, σ]) are Y -DD and hence that the setsr(M,w, σ)↓(i..j) and m(M,w, σ])(i..j) are defined for each (i, j) ∈ Z2.

To prove the statement of this theorem, we first argue by induction on n ∈ N that foreach standard LTDEL event-run σ satisfying |σ| ≤ n and |σ| = |r(M,w, σ)| − 1, we have thefollowing items.

1. Let σ = {(Ui, si)}|σ|i=1 and let k range over N.

(a) r(M,w, σk), π|σ|−k1 (x) |=LTDEL

pUk+1(π2(π

|σ|−k−11 (x))

)for each x ∈ r(M,w, σ)↓(|σ|)

and each k < |σ|.(b) r(M,w, σk)[Uk+1] = r(M,w, σk+1) for each k < |σ|.(c) m(M,w, σ]k), π

|σ|−k1 (y) |=LDETL

pUk+1]k(π2(π

|σ|−k−11 (y))

)for each y ∈ m(M,w, σ])(|σ|)

and each k < |σ|.(d) m(M,w, σ]k)[Uk+1]k] = m(M,w, σ]k+1) for each k < |σ|.

28

2. Let τdef= σ|σ|−1. Define the function fσ by

fσ(v)def=

{(fτ (v), [) if v ∈ r(M,w, σ)↓(..|σ| − 1),

v if v ∈ r(M,w, σ)↓(|σ|).

fσ is an isomorphism between

(r(M,w, σ)↓, r(M,w, σ)) and (m(M,w, σ]), m(M,w, σ]))

with πd1(v) = π|σ|1 (fσ(v)) ∈ WM for d ∈ N and v ∈ r(M,w, σ)↓(d).

3. For each ϕ ∈ LTDEL, d ∈ N, and x ∈ r(M,w, σ)↓(d), we have

r(M,w, σ), x |=LTDELϕ iff m(M,w, σ]), fσ(x) |=LDETL

ϕ]d .

In the base case of this induction, σ = ∅. Item 1 is immediate and item 2 follows becauser(M,w, ∅)↓ = M = m(M,w, ∅]) and f∅ : WM → WM is the identity. Choosing x ∈ WM , weprove Item 3 as follows.

• r(M,w, ∅), x |=LTDELϕ iff M,x |=LDETL

ϕ•.

By the LTDEL Reduction Theorem (Theorem 6.8), the fact that r(M,w, ∅)↓ = M andthat ϕ• ∈ LSETL, and the Collapse Lemma (Lemma 6.7).

• M,x |=LDETLϕ• iff M,x |=LDETL

(ϕ]0)◦.

By the Connection Lemma (Lemma 6.12).

• M,x |=LDETL(ϕ]0)◦ iff m(M,w, ∅]), x |=LDETL

ϕ]0

By the LDETL Reduction Theorem (Theorem 2.8), the Correctness Theorem (Theo-rem 3.2), LDETL-truth (Definition 3.1), and the equality m(M,w, ∅]) = M .

For the induction step, we assume the following induction hypothesis: 1, 2, and 3 holdfor every standard LTDEL event-run σ satisfying |σ| ≤ n and |σ| = |r(M,w, σ)| − 1. Wewish to prove that 1, 2, and 3 hold for an arbitrarily chosen standard LTDEL event-run σsatisfying |σ| = n + 1 and |σ| = |r(M,w, σ)| − 1. The proof that item 1 holds for σ followsby the definition of r(M,w, σ), items 2 and 3 of the induction hypothesis for τ = σn, andthe definition of ]n.

We now prove that 2 holds for σ. First, it follows that πd1(v) = π|σ|1 (fσ(v)) ∈ WM

for d ∈ N and v ∈ r(M,w, σ)↓(d) by considering the cases v ∈ r(M,w, σ)↓(..|τ |) and v ∈r(M,w, σ)↓(|σ|). In the first case, we use the induction hypothesis for item 2 to show that

πd1(v) = π|τ |1 (fτ (v)) ∈ WM . In the second case, v = (v′, s) with v′ ∈ r(M,w, σ)↓(..|τ |), so we

apply the argument in the first case to v′. Next, to prove that fσ is the desired isomorphismbetween pointed Kripke models, we must show a number of properties about fσ. The mostdifficult of these are the proofs that fσ is onto the set Wm(M,w,σ]), which comes up in showingthat fσ is a bijection mapping r(M,w, σ) to m(M,w, σ]), and that fσ preserves the epistemicand temporal relations. (Note that the proof that fσ preserves the propositional valuation

follows easily because πd1(v) = π|σ|1 (fσ(v)) ∈ WM for d ∈ N and v ∈ r(M,w, σ)↓(d).) In the

interest of space, we shall sketch the proofs just for these difficult arguments.

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• fσ is onto the set Wm(M,w,σ]).

We prove that v ∈ W r(M,w,σ)↓ implies fσ(v) ∈ Wm(M,w,σ]), considering the cases v ∈r(M,w, σ)↓(..|τ |) and v ∈ r(M,w, σ)↓(|σ|). The first case uses item 2 of the inductionhypothesis to show that fτ (v) ∈ Wm(M,w,τ ]); since pU|σ|](|σ|−1)([) = >, it follows by item1d that fσ(v) = (fτ (v), [) ∈ Wm(M,w,σ]). The second case uses item 2 of the inductionhypothesis to show that fτ (π1(v)) ∈ Wm(M,w,τ ]); since m(M,w, τ ]), fτ (π1(v)) |=LDETL

pU|σ|](|σ|−1)(π2(v)) by item 1c of the induction hypothesis, it follows by item 1d thatfσ(v) = v ∈ Wm(M,w,σ]).

We now prove that for each u ∈ Wm(M,w,σ]), there is a u′ ∈ W r(M,w,σ)↓ such that fσ(u′) =u. As in the previous paragraph, we consider two cases: u ∈ r(M,w, σ)↓(..|τ |) and u ∈r(M,w, σ)↓(|σ|). In the first case, we use item 2 of the induction hypothesis to conclude

that fτ is a bijection; we may then define u′def= f−1τ (π1(u)), and it follows that fσ(u′) =

(fτ (u′), [) = u. In the second case, fσ(u) =

(fτ (π1(u)), π2(u)

)= (π1(u), π2(u)) = u,

with the second equality following by item 2 of the induction hypothesis and the factthat π1(u) ∈ r(M,w, σ)↓(|τ |).

• u ∈ Rr(M,w,σ)↓a (v) iff fσ(u) ∈ Rm(M,w,σ])

a (fσ(v)) for each a ∈ A.

It is not difficult to see that r(M,w, τ)↓ is synchronous. To prove the statement above,we consider two cases.

Case one: v ∈ r(M,w, σ)↓(..|τ |). We have the following.

– u ∈ Rr(M,w,σ)↓a (v) iff u ∈ Rr(M,w,τ)↓

a (v).

Since v ∈ r(M,w, σ)↓(..|τ |) and r(M,w, τ)↓ is synchronous.

– u ∈ Rr(M,w,τ)↓a (v) iff fτ (u) ∈ Rm(M,w,τ ])

a (fτ (v)).

By item 2 of the induction hypothesis.

– fτ (u) ∈ Rm(M,w,τ ])a (fτ (v)) iff fσ(u) ∈ Rm(M,w,σ])

a (fσ(v)).

Since the assumption v ∈ r(M,w, σ)↓(..|τ |) and the synchronicity of r(M,w, τ)↓together imply the equality fσ(x) = (fτ (x), [) for x ∈ {u, v}, the result follows bythe equality pU|σ|](|σ|−1)([) = > and item 1d.

Case two: v ∈ r(M,w, σ)↓(|σ|). We have the following.

– u ∈ Rr(M,w,σ)↓a (v) iff

π1(u) ∈ Rr(M,w,τ)↓a (π1(v)) and π2(u) ∈ RU|σ|

a (π2(v)).

By item 1b, LTDEL-truth, and the fact v ∈ r(M,w, σ)↓(|σ|).– π1(u) ∈ Rr(M,w,τ)↓

a (π1(v)) and π2(u) ∈ RU|σ|a (π2(v)) iff

π1(u) ∈ Rm(M,w,τ ])a (π1(v)) and π2(u) ∈ RU|σ|](|σ|)

a (π2(v)).

By the definition of U|σ|](|σ|), item 2 of the induction hypothesis, and the factthat v ∈ r(M,w, σ)↓(|σ|) and the synchronicity of r(M,w, τ)↓ together imply thatfτ (π1(x)) = π1(x) for x ∈ {u, v}.

– π1(u) ∈ Rm(M,w,τ ])a (π1(v)) and π2(u) ∈ RU|σ|](|σ|)

a (π2(v)) iff

u ∈ Rm(M,w,σ])a (v).

By LDETL-truth and item 1d.

30

• u ∈ Rr(M,w,σ)↓Y (v) iff fσ(u) ∈ Rm(M,w,σ])

Y (fσ(v)).

We consider two cases.

Case one: v ∈ r(M,w, σ)↓(..|τ |). The proof of this case is similar to the proof of theanalogous case of the previous item (note that synchronicity is not needed here).

Case two: v ∈ r(M,w, σ)↓(|σ|). We first show u ∈ Rr(M,w,σ)↓Y (v) implies fσ(u) ∈

Rm(M,w,σ])Y (fσ(v)). Assume u ∈ R

r(M,w,σ)↓Y (v) and hence that u ∈ r(M,w, σ)↓(|τ |).

Then π2(fσ(u)) = [ and hence

fσ(u) = (u, [) ∈ Rm(M,w,σ])Y (u, π2(v))

because [ ∈ RU|σ|](|σ|−1)Y (π2(fσ(v))). The result follows from the fact that fσ(v) = v =

(u, π2(v)).

To prove the converse, we assume fσ(u) ∈ Rm(M,w,σ])Y (fσ(v)). Then π2(fσ(v)) 6= [,

and hence as {[} = RU|σ|](|σ|−1)Y (π2(fσ(v))) and ∅ = R

U|σ|](|σ|−1)Y (π2(fσ(v))), we have the

middle equality offτ (u) = π1(fσ(u)) = π1(fσ(v)) = π1(v) .

Since m(M,w, τ ]), π1(v) |=LDETLpU|σ|](|σ|−1)(π2(fσ(v))) implies that π1(v) ∈ m(M,w, τ ])(|τ |),

and since item 3 of the inductive hypothesis guarantees that fτ preserves the depth ofa world, we have that u ∈ r(M,w, τ)↓(|τ |), and hence fτ (u) = u, whence π1(v) = u.

By the definition of r(M,w, σ)↓, we conclude that u ∈ Rr(M,w,σ)↓Y (v).

This completes the proof of item 2 for σ. To prove item 3 for σ, we choose d ∈ N andx ∈ r(M,w, σ)↓(d). If d < |σ|, then we have the following.

• r(M,w, σ), x |=LTDELϕ iff r(M,w, τ), x |=LTDEL

ϕ.

Since d < |σ|.• r(M,w, τ), x |=LTDEL

ϕ iff m(M,w, τ ]), fτ (x) |=LDETLϕ]d.


• m(M,w, τ ]), fτ (x) |=LDETLϕ]d iff

m(M,w, σ]), fσ(x) |=LDETLϕ]d.

By the Past State Theorem (Theorem 5.5), the equality fσ(x) = (fτ (x), [), and item1d.

If d = |σ|, then we have the following.

• r(M,w, σ), x |=LTDELϕ iff r(M,w, τ), π1(x) |=LTDEL

[U|σ|, s|σ|]ϕ.

By the equality σ = τ · (U|σ|, s|σ|), item 1b for σ, and LTDEL-truth.

• r(M,w, τ), π1(x) |=LTDEL[U|σ|, s|σ|]ϕ iff

m(M,w, τ ]), fτ (π1(x)) |=LDETL([U|σ|, s|σ|]ϕ)](|τ |).


31

• m(M,w, τ ]), fτ (π1(x)) |=LDETL([U|σ|, s|σ|]ϕ)](|τ |) iff

m(M,w, σ]), fσ(x) |=LDETLϕ](|σ|).

By the definition of ](|σ|) (Figure 5), LDETL-truth, item 1d for σ, and the equalityσ = τ · (U|σ|, s|σ|).

This completes the inductive argument. To prove the statement of the theorem, first definem

def= |r(M,w, σ)| − 1. Apply items 1, 2, and 3 to σm. Then observe that r(M,w, σm) =

r(M,w, σ) by the definition of r(M,w, σ) and that this implies that m(M,w, σ]m) = m(M,w, σ])by items 1 and 3. The statement of the theorem follows.

References

[1] Alexandru Baltag and Lawrence S. Moss. Logics for epistemic programs. Synthese,139(2):165–224, 2004.

[2] Alexandru Baltag, Lawrence S. Moss, and S lawomir Solecki. The logic of public an-nouncements, common knowledge, and private suspicions. In Itzhak Gilboa, editor,Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge(TARK VII), pages 43–56, Evanston, Illinois, USA, 1998.

[3] Alexandru Baltag, Hans P. van Ditmarsch, and Lawrence S. Moss. Epistemic logic andinformation update. In Pieter Adriaans and Johan van Benthem, editors, Handbook onthe Philosophy of Information, pages 369–463. Elsevier, 2008.

[4] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic. CambridgeUniversity Press, 2001.

[5] Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. Reasoning aboutKnowledge. The MIT Press, 1995.

[6] Tomohiro Hoshi and Audrey Yap. Dynamic epistemic logic with branching temporalstructures. Synthese, 169(2):259–281, 2009.

[7] Carsten Lutz. Complexity and succinctness of public announcement logic. In Proceed-ings of the 5th International Joint Conference on Autonomous Agents and MultiagentSystems (AAMAS-06), pages 137–144, Hakodate, Japan, May 8–12, 2006. Associationfor Computing Machinery (ACM), New York, New York, USA.

[8] Rohit Parikh and Ramaswamy Ramanujam. A knowledge based semantics of messages.Journal of Logic, Language, and Information, 12:453–467, 2003.

[9] Bryan Renne, Joshua Sack, and Audrey Yap. Dynamic epistemic temporal logic. InX. He, J. Horty, and E. Pacuit, editors, Logic, Rationality, and Interaction, SecondInternational Workshop, LORI 2009, Chongqing, China, October 8–11, 2009, Proceed-ings, volume 5834/2009 of Lecture Notes in Computer Science, pages 263–277. SpringerBerlin/Heidelberg, 2009.

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[10] Joshua Sack. Temporal languages for epistemic programs. Journal of Logic, Language,and Information, 17(2):183–216, 2008.

[11] Johan van Benthem, Jelle Gerbrandy, Tomohiro Hoshi, and Eric Pacuit. Merging frame-works for interaction. Journal of Philosophical Logic, 38(5):491–526, 2009.

[12] Johan van Benthem, Jelle Gerbrandy, and Eric Pacuit. Merging frameworks for inter-action: DEL and ETL. In Proceedings of the 11th Conference on Theoretical Aspects ofRationality and Knowledge (TARK XI), pages 72–81, 2007.

[13] Johan van Benthem, Jan van Eijck, and Barteld Kooi. Logics of communication andchange. Information and Computation, 204(11):1620–1662, 2006.

[14] Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi. Dynamic Epistemic Logic.Synthese Library. Springer, 2007.

[15] Audrey Yap. Dynamic epistemic logic and temporal modality. Forthcoming in Proceed-ings of Dynamic Logic Montreal, 2007.

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