Logics of temporalepistemic actions
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Abstract
We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multiagent Kripke models that contain a designated temporal relation. These operations are natural extensions of the wellknown “action models” from Dynamic Epistemic Logic (DEL). Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal structure of a model along with the agents’ basic and higherorder knowledge and beliefs about this structure, including their beliefs about the time. In essence, this approach provides one way to extend the domain of action modelstyle operations from atemporal Kripke models to temporal Kripke models in a manner that allows actions to control the flow of time. We present a number of examples to illustrate the subtleties involved in interpreting the effects of our extended action models on temporal Kripke models. We also study preservation of important epistemictemporal properties of temporal Kripke models under temporal action modelinduced operations, provide complete axiomatizations for two theories of temporal action models, and connect our approach with previous work on time in DEL.
Keywords
Dynamic epistemic logic Epistemic temporal logic Epistemic logic Asynchronicity1 Introduction
Anyone who has been late to an appointment or missed a deadline is aware that it is often difficult to keep track of time. This basic difficulty is the motivation for this paper, which presents a framework called Dynamic Epistemic Temporal Logic that allows us to reason about epistemic agents’ changing beliefs about time, from one point in time to the next. We will develop this framework by combining techniques from the traditions of Epistemic Temporal Logic (ETL) (Parikh and Ramanujam 2003) and Dynamic Epistemic Logic (DEL) (Baltag and Moss 2004; Baltag et al. 1998, 2008; van Benthem et al. 2006; van Ditmarsch et al. 2007).
Our main contribution in this paper is to extend the modal “action model” operators from DEL by incorporating explicit temporal information as to the relative time at which various modelchanging events occur. This change to the standard DEL setup allows for a nuanced study of the relationship between dynamic actions, agent belief, and time. A number of authors have already looked at these issues (van Benthem et al. 2009; Dégremont et al. 2011; Hoshi 2009; Hoshi and Yap 2009; Hoshi 2010). Like these (and other) authors, we are interested in the doxastic/epistemic states of reasoning agents and the representation of these states by possibly asynchronous systems. These states can be considered from two perspectives: the “static” perspective of the underlying model (i.e., from the point of view of a “snapshot in time,” which includes a snapshot of agents’ knowledge/beliefs about time in that moment) and the “dynamic” perspective of action modelinduced changes to the underlying model (i.e., from the point of view of a “progression of snapshots,” which can be used to understand agents’ evolving beliefs about time from one temporalepistemic snapshot to the next). The difference between this and previous work is that our action models incorporate a binary relation representing the relative time at which the events of the action model occur. Standard action models were designed to include only doxastic/epistemic information explicitly, but our “temporal action models” explicitly include both doxastic/epistemic and temporal information. Our aim, like that of standard DEL, is to represent all key information “dynamically” using flexible relations defined in the action models themselves. By including a relation for time, our framework extends the domain of applicability of the “action model approach” to a wider class of models in which time plays an explicit role in the action model specification language.
As we will see, our framework is sufficiently flexible to accommodate asynchronicity. However, we do not yet have a precise characterization of the exact class of possibly asynchronous systems our approach can capture. The determination of independent criteria that capture the models of a certain framework (e.g., Epistemic Temporal Logic) that are representable within our setting remains an open problem. Some such criteria are known for DEL itself (van Benthem et al. 2009) and we speculate that many features of this work might be put to use for the analysis of our own framework. However, this is a complex task that we must leave for future work.
With respect to the outline of this paper, Sect. 2 introduces the syntax and semantics of Dynamic Epistemic Temporal Logic (DETL). Section 3 then highlights several features of this system by presenting a number of examples that illustrate different ways of measuring the time at a world in a model. The proof system and completeness results for DETL appear in Sect. 4. In Sect. 5, we study the preservation under updates of several modeltheoretic properties that one might wish to enforce so as to ensure models have sensible temporal structure. Finally, we conclude in Sect. 6 by connecting DETL to other work concerned with adding time to DEL.
2 Dynamic epistemic temporal logic
We begin with a nonempty finite set \(\fancyscript{A}\) of agents and a disjoint nonempty set \(\fancyscript{P}\) of (propositional) letters. Our semantics is based on Kripke models (with yesterday). These are structures Open image in new window consisting of a nonempty set \(W^M\) of (possible) worlds, a binary epistemic accessibility relation \(\rightarrow ^M_a\) for each \(a\in \fancyscript{A}\) indicating the worlds \(w'\leftarrow ^M_a w\) agent \(a\in \fancyscript{A}\) considers possible at w, a binary temporal accessibility relation Open image in new window indicating the worlds \(w' \leadsto ^M w\) to be thought of as a “yesterday” of (i.e., fall one clocktick before) world w,^{1} and a (propositional) valuation \(V^M:\fancyscript{P}\rightarrow \wp (W^M)\) indicating the set \(V^M(p)\) of worlds at which propositional letter \(p\in \fancyscript{P}\) is true. For now, we do not place any restrictions on the behavior of these relations, but later (in Definition 7) we will introduce several desirable properties that they will typically have in concrete examples. For a binary relation R, a pair \((w,v)\in R\) is often called an “R arrow.” A pointed Kripke model (with yesterday), sometimes called a situation, is a pair (M, w) consisting of a Kripke model and a world \(w\in W^M\) called the point. To say that a Kripke model (pointed or not) is atemporal means that it contains no Open image in new window arrows.
Pointed Kripke models (M, w) describe fixed (i.e., “static”) epistemictemporal situations in which agents have certain beliefs about time, propositional truth, and the beliefs of other agents. We now define (epistemictemporal) action models, which transform a situation (M, w) into a new situation (M[U], (w, s)) according to a certain “product operation” \(M\mapsto M[U]\) defined in a moment (in Definition 5).
Definition 1

\(W^U\) is a nonempty finite set of informational events the agents may experience.

For each \(a\in \fancyscript{A}\), the object \(\rightarrow ^U_a\) is a binary (epistemic) accessibility relation. The relation \(\rightarrow _a^U\) designates the events \(s'\leftarrow ^U_a s\) that agent a thinks are consistent with her experience of event s.

Open image in new window is a binary temporal relation indicating the events \(s' \leadsto ^U s\) that occur as a “yesterday” of (i.e., fall one timestep before) event s.

\(\mathsf {pre}^U:W^U\rightarrow F\) is a precondition function assigning a precondition (formula) \(\mathsf {pre}^U(s)\in F\) to each event s. The precondition \(\mathsf {pre}^U(s)\) of event s is the condition that must hold in order for event s to occur.

\(\mathfrak {A}(F)\) is the set of action models over F,

\(\mathfrak {A}^a(F)\) is the set of atemporal action models over F,

\(\mathfrak {A}_*(F)\) is the set of pointed action models over F, and

\(\mathfrak {A}^a_*(F)\) is the set of pointed atemporal action models over F.
Atemporal action models were developed by Baltag and Moss (2004), Baltag et al. (1998) and have been adapted or extended in various ways in the Dynamic Epistemic Logic literature in order to reason about knowledge and belief change; see the textbook (van Ditmarsch et al. 2007) for details and references. Our contribution here is the inclusion of temporal arrows Open image in new window within action models. To say more about this, we first introduce some additional terminology.
Definition 2
(Progressions, Histories, Depth \({{\mathrm{d}}}(w)\)) We shall use the word state to refer either to a world of a Kripke model or an event of an action model. A progression is a finite nonempty sequence \(\langle {w_i}\rangle _{i=0}^n\) of states having \(w_i\leadsto w_{i+1}\) for each \(i<n\). We say that a progression \(\langle {w_i}\rangle _{i=0}^n\) begins at \(w_0\) and ends at \(w_n\). The length of a progression \(\langle {w_i}\rangle _{i=0}^n\) is the number n, which is equal to the number of \(\leadsto \) arrows it takes to link up the states making up the progression (i.e., one less than the number of states in the progression). A pastextension of a progression \(\sigma \) is another progression obtained from \(\sigma \) by adding zero or more extra states at the beginning of the sequence (i.e., in the “pastlooking direction” from x to y in the arrow Open image in new window ). A pastextension is proper if more than zero states were added. A history is a progression that has no proper pastextension. For each state w, we define \({{\mathrm{d}}}(w)\) as follows: if there is a maximum \(n\in \mathbb {N}\) such that there is a history of length n that ends at w, then \({{\mathrm{d}}}(w)\) is this maximum n; otherwise, if no such maximum \(n\in \mathbb {N}\) exists, then \({{\mathrm{d}}}(w) = \infty \). We call \({{\mathrm{d}}}(w)\) the depth of w. A state w satisfying \({{\mathrm{d}}}(w)=0\) is said to be initial.
We will present a number of examples shortly showing that the inclusion of temporal Open image in new window arrows in both Kripke models and action models allow us to reason about time in a Dynamic Epistemic Logicstyle framework. The basic idea is this: if the depth \({{\mathrm{d}}}(w)\) of a world w is finite, then the depth \({{\mathrm{d}}}(w)\) of w indicates the time at w; likewise, if the depth \({{\mathrm{d}}}(s)\) of an event s is finite, then the depth \({{\mathrm{d}}}(s)\) of s indicates the relative time at which event s takes place. Notice that this notion of “time” can be a little strange: if \(t_1\leadsto s\) and \(t_2\leadsto s\) with \(t_1\ne t_2\), then \({{\mathrm{d}}}(s)\ge \max \{{{\mathrm{d}}}(t_1),{{\mathrm{d}}}(t_2)\}\) but we might think it odd to say that “the time is \({{\mathrm{d}}}(s)\)” given that there is “branching” in the past direction. Therefore, in order to make our notion of time coherent and useful, there are a number of things we do. First, we introduce our multimodal language \({L_{\mathsf {DETL}}}\) having doxastic modalities \(\Box _a\varphi \) (“agent a believes \(\varphi \)”) for each \(a\in \fancyscript{A}\), the temporal modality \([Y]\varphi \) (“\(\varphi \) was true ‘yesterday’ (i.e., one timestep ago)”), and action model modalities \([U,s]\varphi \) (“after action (U, s) occurs, \(\varphi \) is true”).
Definition 3
Definition 4
(Past State) Let U be an action model. A past state is an event s in U that has no yesterday: there is no \(s'\leadsto ^U s\).
Every history \(s_0\leadsto s_1\leadsto s_2\leadsto \cdots \leadsto s_n\) begins with a past state (see Definition 2). The past state \(s_0\) plays a special role in the semantics by copying part or all of the input Kripke model. The next definition shows how this is done. The sequential execution of successive events \(s_1,\ldots ,s_n\) then transforms this copy.
Definition 5

\(M,w\models \Box _a\varphi \) means \(M,v\models \varphi \) for each \(v\leftarrow ^M_aw\).

\(M,w\models [Y]\varphi \) means \(M,v\models \varphi \) for each \(v\leadsto ^M w\).
 \(M,w\models [U,s]\varphi \) means \(M,w\models \mathsf {pre}^U(s)\) implies \(M[U],(w,s)\models \varphi \), where

\(W^{M[U]} \mathop {=}\limits ^{ \text{ def }}\{ (v,t) \in W^M \times W^U \mid M,v\models \mathsf {pre}^U(t)\}\).

We have \((v,t)\rightarrow ^{M[U]}_a(v',t')\) if and only if both \(v\rightarrow ^M_av'\) and \(t\rightarrow ^U_a t'\).
 We have \((v',t') \leadsto ^{M[U]} (v,t)\) if and only if we have one of the following:

\(v' \leadsto ^M v, t'=t\), and t is a past state; or

\(v'=v\) and \(t' \leadsto ^U t\).


\(V^{M[U]}(p) \mathop {=}\limits ^{ \text{ def }}\{(v,t)\in W^{M[U]} \mid v \in V^M(p)\}\).

After taking the update product \(M\mapsto M[U]\), the epistemic relation \(\rightarrow _a\) behaves as it does in DEL (van Ditmarsch et al. 2007); that is, one pair is epistemically related to another iff they are related componentwise. This is analogous to the notion of synchronous composition in process algebra (van Glabbeek 2001). However, our relations \(\rightarrow _a\) are epistemic, whereas the relations in process algebra are temporal.
The behavior of our temporal relation \(\leadsto \) after the update product \(M\mapsto M[U]\) is analogous to the notion of asynchronous composition from process algebra (Aceto et al. 2001): one component of the pair makes the transition while the other component remains fixed. However, in our case, if \((v',t') \leadsto ^{M[U]} (v,t)\), then the component that makes the transition depends on whether t is a past state. If t is indeed a past state, then the first component makes the transition (\(v'\leadsto v\)) and the second component remains fixed (\(t'=t\)). Otherwise, if t is not a past state, then it is instead the first component that remains fixed (\(v'=v\)) and the second component that makes the transition (\(t'\leadsto t\)).
If M and U are atemporal, then our operation \(M\mapsto M[U]\) is equivalent to the standard “product update” from DEL and M[U] is atemporal.
Definition 6
(Epistemic Past State) Let U be an action model. An epistemic past state is a past state s in U whose precondition is a validity (i.e., \(\models \mathsf {pre}^U(s)\)) and whose only epistemic arrows are the reflexive arrows \(s\rightarrow ^U_a s\) for each agent \(a\in \fancyscript{A}\).
Like a past state (Definition 4), an epistemic past state \(s_0\) in a history \(s_0\leadsto \cdots \leadsto s_n\) plays the special role of copying the initial state of affairs before the remaining events in the history take place. However, there is a key difference: a past state may copy only part of the initial state of affairs, whereas an epistemic past state will always make a complete copy. A later result (Theorem 4) will explain this further. Therefore, a history \(s_0\leadsto \cdots \leadsto s_n\) beginning with an epistemic past state \(s_0\) may be thought of as describing the following construction: the epistemic past state \(s_0\) makes a complete copy of the initial state of affairs (thereby remembering the “past” just as it was) and then the remaining events \(s_1,\dots ,s_n\) transform this copy (appending “future” states of affairs one by one). A series of examples in Sect. 3 will explain this in further detail.
Definition 7

Persistence of Facts (for Kripke models only): \(w\leadsto w' \Rightarrow (w\in V(p)\Leftrightarrow w'\in V(p))\) for all w and \(w'\).
This says that propositional letters retain their values across temporal \(\leadsto \) arrows.

DepthDefinedness: \({{\mathrm{d}}}(w)\ne \infty \) for all w.
This says that every world/event has a finite depth.

Knowledge of the Past: \((w'\leadsto w\rightarrow _a v)\Rightarrow \exists v'(v'\leadsto v)\) for all \(a\in \fancyscript{A}, w', w\), and v.
This says that agents know if there is a past (i.e., a state reachable via a backward step along a \(\leadsto \) arrow).

Knowledge of the Initial Time: \(w\rightarrow _av\wedge \lnot \exists w'(w'\leadsto w)\Rightarrow \lnot \exists v'(v'\leadsto v)\) for all \(a\in \fancyscript{A}, w\), and v.
This says that agents know if there is no past.

Uniqueness of the Past: \((w'\leadsto w\wedge w''\leadsto w)\Rightarrow (w'=w'')\) for all \(w', w\), and \(w''\).
This says that there is only one possible past.

Perfect Recall: \((w\leadsto v\rightarrow _a v')\Rightarrow \exists w'(w\rightarrow _aw'\leadsto v')\) for all \(a\in \fancyscript{A}, w, v\), and \(v'\).
This says that agents do not forget what they knew in the past.^{2}

Synchronicity: The structure is depthdefined and \(w\rightarrow _a w'\) implies \({{\mathrm{d}}}(w)={{\mathrm{d}}}(w')\) for all \(a\in \fancyscript{A}, w\), and \(w'\).
This says that there is no uncertainty, disagreement, or mistakenness among the agents with regard to the depth of a world/event.

History Preservation (for action models U only): \(s'\leadsto ^U s\) implies \(\models \mathsf {pre}^U(s)\rightarrow \mathsf {pre}^U(s')\) for all \(s'\) and s; further, every past state in U is an epistemic past state.
This says that a noninitial event s can take place only if its predecessor \(s'\) can as well, and initial events can always take place (Definition 6). Whenever an event s in a history preserving action model U is executable at a world w of a Kripke model M (i.e., \(M,w\models \mathsf {pre}^U(s)\)), the temporal structure of any partial or full history \(s_n\leadsto \cdots \leadsto s_0=s\) in U is preserved as the partial or full history \((w,s_n)\leadsto \cdots \leadsto (w,s_0)=(w,s)\) in the updated model M[U]. In this way, history preserving action models preserve the executable parts of their own histories.
 Past Preservation (for pointed action models (U, s) Epistemic Past Stateonly): The action model U is history preserving; further, every progression \(s_n\leadsto ^U\cdots \leadsto ^Us_0=s\) that ends at s can be pastextended to a historythat begins at an epistemic past state \(s_{n+m}\). To say an event \(t\in W^U\) is past preserving means the action (U, t) with point t is past preserving. This says that there is a “link to the past” (i.e., an epistemic past state) via a sequence of Open image in new window arrows. Since past preserving action models are history preserving, it follows that past preserving action models preserve executable parts of their own histories and maintain a “link to the past.” (This is explained in detail in and around the forthcoming Theorem 4.)$$\begin{aligned} s_{n+m}\leadsto ^U \cdots \leadsto ^U s_{n+1} \leadsto ^U s_n\leadsto ^U\cdots \leadsto ^Us_0=s \end{aligned}$$

Timeadvancing (for pointed action models (U, s) only): The action (U, s) is past preserving and the point s is not a past state.
This says that the “past” is at least one timestep away.
For the moment, we do not require that our Kripke models or action models satisfy any of the above properties. This will change in Sect. 5, where we study the preservation of Kripke model properties under action models satisfying appropriate properties, and in Sect. 6, where we impose a number of these properties in order to study connections between our framework and other approaches to the study of time in Dynamic Epistemic Logic.
A note on the depth of worlds (Definition 2) in the updated model M[U]: if world w and event t both have finite depth, then the maximum depth of the resulting world \((w,t)\in W^{M[U]}\) is \({{\mathrm{d}}}(w)+{{\mathrm{d}}}(t)\). The reason: we can take at most \({{\mathrm{d}}}(t)\) backward steps in the second coordinate (fix the first coordinate and proceed backward in the second until a past state is reached), and we can take at most \({{\mathrm{d}}}(w)\) backward steps in the first coordinate (fix the second coordinate and proceed backward in the first until an initial world is reached). The actual depth of (w, t) does not need to obtain its maximum: when stepping backward in either coordinate (with the other fixed), we may reach a pair \((w',s')\) whose world \(w'\) violates the precondition of the event \(s'\) (i.e., \(M,w'\not \models \mathsf {pre}^U(s')\)) and therefore the pair \((w',s')\) will not be a member of \(W^{M[U]}\). However, if the action model U is history preserving, then this problem is avoided and hence \({{\mathrm{d}}}(w,t)={{\mathrm{d}}}(w)+{{\mathrm{d}}}(t)\).
Finally, we note that we can express finite depth explicitly in our language.
Theorem 1
In the next section, we will discuss how the depth of a world w can be used as an explicit measure of the time at world w. If one adopts this measure of time, then it follows from Theorem 1 that we can express the time of a world within our language \({L_{\mathsf {DETL}}}\): we say that “the time at world w is n” to mean that w satisfies \(D_n\).
3 Examples
In this section, we will illustrate several features of our system. This will demonstrate the way in which our system can represent interesting epistemic situations as well as shed some light on the ways in which time is treated in dynamic systems. First, we will define explicit and implicit measures of time. An explicit measure of time is one in which the time of a world w in model M can be determined solely by inspection of M. As we have seen, the depth of a world in a Kripke model (see Definition 2) can provide an explicit measure of the time at that world, and this can be expressed explicitly in our language (Theorem 1). In contrast, an implicit measure of time is one in which the time of a world w in model M cannot be determined solely be inspection of M. For example, if we measure the time at a world in M by the number of updates that have led up to M, this can provide an implicit measure of the time at that world. These are not the only possible implicit and explicit measures, but they are certainly natural ones within our \(\mathsf {DETL}\) framework.
Now we will consider ways in which explicit and implicit representations of time might differ. More specifically, we will consider cases in which there is only a single update (implicitly increasing the time by 1) but at which the explicit time at the actual world changes by a number other than 1. Second, we will add in the epistemic aspect, by demonstrating the ways in which agents can hold differing beliefs about temporal and epistemic features of their situation.
3.1 Explicit and implicit representations of time
Example 1
The action \((U_2,s)\) (Fig. 2) represents the public announcement of p. Applied to our initial situation (M, w) (Fig. 1), the result is \((M[U_2],(w,s))\).
The language of ordinary DEL is the atemporal fragment of \({L_{\mathsf {DETL}}}\) without the [Y] modality. In this language, the only way to refer to the agents’ knowledge before the announcement is with reference to the original situation (M, w). This is because ordinary DEL lacks \(\leadsto \) arrows, both in action models and in the underlying Kripke models.
Example 1 showed “standard” temporal behavior: a single update increases the time of the actual world by 1. Two obvious ways in which updates could be “nonstandard” are by not increasing the time when an update takes place or by increasing the time by a number greater than 1.
Example 2
Here we consider the effect of the action \((U_3,t)\) (Fig. 3) on our initial situation (M, w) (Fig. 1). This action has a structure that is nearly identical to that of action \((U_2,s)\) in Example 1 (Fig. 2); in fact, these actions are based on the same underlying action model (i.e., \(U_2=U_3\)). However, the actual events of \((U_2,s)\) and \((U_3,t)\) are different. The result of the update with the latter action is the pointed model \((M[U_3],(w,t))\). Note that the resultant actual world (w, t) is among the worlds (u, t), (w, t), and (v, t) that make up the “copy” of the initial model M.
We have designed our system so that the initial world w and its copy (w, t) satisfy the same formulas. In this way, we may identify each initial world with its copy, so that the collection of copied worlds (and the arrows interconnecting them) may be identified with the initial model itself. This allows us to reason about what was the case in the initial model by evaluating formulas only within the resultant model. In effect, we can “forget” the initial model because all of its information is copied over to the resultant model.
To make this work, both w and its copy (w, t) must satisfy the same formulas. We guarantee this by designing our system so that it ignores all “future” worlds, by which we mean the worlds accessible from the point only via a link \(x\leadsto y\) from a “past” world x to a “future” world y.^{3} So from the point of view of our theory, the time1 worlds (u, s) and (w, s) in Fig. 3 are ignored in the resultant time0 situation \((M[U_3],(w,t))\) because the time1 worlds can only be reached via a forward \(\leadsto \) arrow. This leaves only the “copy” of the initial model M consisting of the worlds (u, t), (w, t), and (v, t). The resultant situation \((M[U_3],(w,t))\) is therefore equivalent to the initial situation (M, w) from the point of view of our theory. In other words, the action \((U_3,t)\) does not change the state of affairs at all.^{4}
Remark 1
The previous two examples illustrate some motivation behind our choice not to include a [T] operator, as defined in (1). If our language had included such an operator, two worlds that are intended to represent the exact same state of affairs could disagree about the truth of formulas. In Example 2, a nontimeadvancing update transforms (M, w) into \((M[U_3],(w,t))\), but the worlds w and (w, t) are meant to represent the same situation and so should satisfy the same formulas. However, (M, w) and \((M[U_3],(w,t))\) disagree on the truth of the formula \(\langle T\rangle p\). But as we have defined \({L_{\mathsf {DETL}}}\) without the [T]operator, we can easily show that for any \({L_{\mathsf {DETL}}}\)formula \(\varphi \), we have \(M, w \models \varphi \) iff \(M[U], (w,t) \models \varphi \) because t is an epistemic past state (see Theorem 4). This avoids the problem illustrated here where two worlds that are supposed to represent the same situation disagree on the truth of formulas.
Example 1 also illustrates the semantic difference between an action model operator [U, s] and the operator [T]. The truth of \([U,s]\varphi \) is determined by evaluating \(\varphi \) in a new model, while the truth of \([T]\varphi \) is determined by evaluating \(\varphi \) within the model as it currently stands. In considering our initial model (M, w) (Fig. 1), note that \(M, w \not \models \langle {T}\rangle p\). However, as we can see, \(M, w \models \langle {U_2,s}\rangle p\). So while we informally read the formula \(\langle {T}\rangle \varphi \) as claiming that \(\varphi \) will hold “tomorrow” (and that there is at least one possible “tomorrow” world), this is only from the perspective of a static model—it does not consider all possible ways in which that model might evolve given different updates.
The [U, s] and [T] operators are not the only options for “tomorrow” operators. Section 2 mentions the way in which the update operators [U, s] serve as dynamic parameterized operators. And it is also possible to define dynamic unparameterized operators that work by quantifying over the parameterized ones, and these also are different from [T] operators. Call our unparameterized operator [N], and we can define: \(M,w\models [N]\varphi \) if and only if \(M,w\models [U,s]\varphi \) for all action models (U, s) with only epistemic preconditions (this constraint is imposed on the action models, so as to ensure the wellfoundedness of the semantics) (Balbiani et al. 2008). Since all public announcement action models fall into this category, we have that \(M,w\models \langle N\rangle p\) (since \(M,w\models \langle U_2,s\rangle p\)) and yet \(M,w\not \models \langle T\rangle p\).
We believe that these interpretive issues involving [T] reflect its complex relationship with the update modalities. Indeed, [T] may not even have a clear interpretation in the context of our framework, which is part of the rationale for leaving it out of our language. But such a situation is not unusual in the epistemic logic tradition. For instance, a common system for modeling agents’ beliefs is \(\mathsf {KD45}\), whose epistemic accessibility relation does not have to be symmetric. In such systems, it is not clear that the converse of the accessibility relation has a clear semantic interpretation, but this is not viewed as problematic. So for us, the relation Open image in new window is one example of many from modal logic of a relation that has a corresponding modality but whose converse does not. (Also, the semantic asymmetry of having a [Y] operator but no corresponding [T] operator is analogous to an asymmetry in ordinary DEL, which has [U, s] modalities but no converse \([U,s]^{1}\) modalities.) As a result, our system \(\mathsf {DETL}\) is one that has a dynamic parameterized future (accessed via the update modals [U, s]) and a static unparameterized past (accessed via the yesterday modal [Y]).
Example 3
In this example, the time at the actual world increases by 2 even though only a single update \((U_4,r)\) (Fig. 4) takes place. With simple modifications of \((U_4,r)\), we could increase the time by any finite number.
These three examples demonstrate the differences between explicit and implicit measures of time; in particular, the number of updates (the implicit time) need not equal the depth of the actual world (the explicit time). In this paper, we will adopt the convention that the time at the actual world is measured by its depth (explicit time). While this is by no means a necessary choice, it has the advantage of allowing us to determine the time at a world solely by inspection of the model to which it belongs.
3.2 Agents mistaken about time
Given that we are measuring the time at a world by its depth, we can represent situations in which agents are unable to distinguish between worlds that have different times. These situations can be brought about by \(\mathsf {DETL}\) actions, as the following example illustrates.
Example 4
In this example, \((U_5,r)\) (Fig. 5) represents the sequenced public announcement of p followed by the asynchronous semiprivate announcement of q to agent b. This increases the time at the actual world by 2, as in Example 3. However, in the present example, agent a is uncertain whether the time has increased by 1 or by 2.
We contrast Example 4 with the following.
Example 5
\((U_6,r)\) (Fig. 6) couples a public announcement of p with the simultaneous semiprivate announcement of q to agent b. When we compare this with Example 4 (pictured in Fig. 5), we note that the agents’ respective knowledge gain is identical with respect to the propositional facts: a learns that p is true but not whether q is true, while b learns both p and q. However, in the current example (pictured in Fig. 6), b learns p and q simultaneously instead of successively, and a’s knowledge differs accordingly.
4 Proof system and completeness
Definition 8
The theory of Dynamic Epistemic Temporal Logic, \(\mathsf {DETL}\), is defined in Fig. 7. Axioms whose name begins with “U” are called reduction axioms.^{5}
Many axioms of \(\mathsf {DETL}\) are the same as in Dynamic Epistemic Logic.^{6} In defining \(\mathsf {DETL}\), we have not imposed any of the properties from Definition 7 on action models, nor have we designed the theory to be sound for Kripke models having properties from Definition 7 that one might expect. So \(\mathsf {DETL}\) should be viewed as the minimal theory that brings update mechanisms to a basic Epistemic Temporal Logic. However, we will study the preservation of these properties in Sect. 5, and we study a \(\mathsf {DETL}\)based theory satisfying a number of these properties in Sect. 6.
Theorem 2
(\({L_{\mathsf {DETL}}}\) Reduction) For every \({L_{\mathsf {DETL}}}\)formula \(\varphi \), there is an action modelfree \({L_{\mathsf {SETL}}}\)formula \(\varphi ^\circ \) such that \(\mathsf {DETL}\vdash \varphi \leftrightarrow \varphi ^\circ \).
Proof
The proof is a straightforward adaptation of the standard argument from Dynamic Epistemic Logic (van Ditmarsch et al. 2007). \(\square \)
Theorem 3
(Soundness and Completeness) \(\mathsf {DETL}\vdash \varphi \) if and only if \(\models \varphi \).
Proof

Case: s is a past state.
Assume \(M,w\models [U,s][Y]\varphi \). By the definition of truth, \(M[U],(w',s)\models \varphi \) for each \((w',s)\leadsto ^{M[U]}(w,s)\). Therefore, if \(v\leadsto ^M w\) satisfies \(M,v\models \mathsf {pre}^U(s)\), then \(M[U],(v,s)\models \varphi \). Conclusion: \(M,w\models [Y][U,s]\varphi \). Conversely, assume \(M,w\models [Y][U,s]\varphi \) and \((w',s)\leadsto ^{M[U]}(w,s)\). The second assumption implies both that \(w'\leadsto ^M w\)—and hence \(M,w'\models [U,s]\varphi \) by the first assumption—and that \(M,w'\models \mathsf {pre}^U(s)\). But then \(M[U],(w',s)\models \varphi \). Conclusion: \(M,w\models [U,s][Y]\varphi \).

Case: s is not a past state. Assume \(M,w\models [U,s][Y]\varphi \). By the definition of truth, \(M[U],(w,t)\models \varphi \) for each \((w,t)\leadsto ^{M[U]}(w,s)\). If \(s'\leadsto ^Us\) satisfies \(M,w\models \mathsf {pre}^U(s')\), then \((w,s')\leadsto ^{M[U]}(w,s)\) and hence \(M[U],(w,s')\models \varphi \). Conclusion: \(M,w\models \bigwedge _{s'\leadsto ^U s}[U,s']\varphi \). Conversely, assume \(M,w\models \bigwedge _{s'\leadsto ^U s}[U,s']\varphi \) and \((w,t)\leadsto ^{M[U]}(w,s)\). The second assumption implies both that \(t\leadsto ^U s\)—and hence \(M,w\models [U,t]\varphi \) by the first assumption—and that \(M,w\models \mathsf {pre}^U(t)\). But then \(M[U],(w,t)\models \varphi \). Conclusion: \(M,w\models [U,s][Y]\varphi \).
5 Preservation results
In this section, we study the preservation of properties of Kripke models defined previously in Definition 7. These properties have been of interest in the study of time in Dynamic Epistemic Logic (van Benthem et al. 2009, 2007; Dégremont et al. 2011; Sack 2010, 2008; Yap 2011) and so it will be useful for our purposes to understand the conditions under which they are preserved within our \(\mathsf {DETL}\) setting. Theorem 4 concerns the behavior of past states in action models, and Theorem 5 concerns the preservation of Kripke model properties.
Theorem 4
 (a)
If s is an epistemic past state, then (M[U], (w, s)) and (M, w) satisfy the same \({L_{\mathsf {DETL}}}\)formulas.
 (b)If (U, s) is past preserving, then there is a historysuch that \((M[U],(w,s_0))\) and (M, w) satisfy the same \({L_{\mathsf {DETL}}}\)formulas.$$\begin{aligned} (w,s_0) \leadsto ^{M[U]} (w,s_1) \leadsto ^{M[U]} \cdots \leadsto ^{M[U]} (w,s_n) = (w,s) \end{aligned}$$
Proof
 (a)
Since the language of \({L_{\mathsf {DETL}}}\) does not include forwardlooking tomorrow operators [T], it follows by the standard argument in modal logic (Blackburn et al. 2001) that bisimilar worlds satisfy the same action modelfree \({L_{\mathsf {DETL}}}\)formulas (i.e., the same \({L_{\mathsf {SETL}}}\)formulas).^{8} By \({L_{\mathsf {DETL}}}\) Reduction (Theorem 2) and soundness (Theorem 3), bisimilar worlds also satisfy the same \({L_{\mathsf {DETL}}}\)formulas. Finally, one can show that there is a bisimulation between (w, s) and w. The result follows.
 (b)
Past preservation of (U, s) implies there exists a history \(s_0\leadsto ^U\cdots \leadsto ^Us_n=s\) that begins at an epistemic past state \(s_0\). The result therefore follows by part (a).\(\square \)
We now examine the relationship between our conditions on action models (Definition 7) and the preservation of certain Kripke models properties (also Definition 7) under the update operation \(M\mapsto M[U]\).
Theorem 5
 (a)
If M satisfies persistence of facts, then so does M[U].
 (b)
If M and U are depthdefined, then so is M[U].
 (c)
If M and U satisfy knowledge of the past and U is history preserving, then M[U] satisfies knowledge of the past.
 (d)
If M satisfies knowledge of the initial time and U is history preserving, then M[U] satisfies knowledge of the initial time.
 (e)
If M and U satisfy uniqueness of the past, then so does M[U].
 (f)
If M and U satisfy perfect recall and U is history preserving, then M[U] satisfies perfect recall.
 (g)
If M and U are synchronous and U is history preserving, then M[U] is synchronous.
Proof
We prove each item in turn.
(a) If M satisfies persistence of facts, then so does M[U]. Suppose that M satisfies persistence of facts and \((w,s) \leadsto ^{M[U]} (w',s')\). It follows that \(w=w'\) or \(w\leadsto ^M w'\). Now M satisfies persistence of facts, so \(w\in V^M(p)\) iff \(w'\in V^M(p)\). Applying the fact that \((v,t)\in V^{M[U]}(p)\) iff \(v\in V^M(p)\), we have \((w,s)\in V^{M[U]}(p)\) iff \((w',s')\in V^{M[U]}(p)\).

Base case: \({{\mathrm{d}}}(s)=0\).
It follows that s is a past state. Therefore \((w',s')\leadsto ^{M[U]}(w,s)\) implies \(w'\leadsto ^M w\) and \(s'=s\). We now prove (4) by a subinduction on \({{\mathrm{d}}}(w)\). In the subinduction base case, \({{\mathrm{d}}}(w)=0\) and therefore there is no \(w'\leadsto ^M w\), which implies there is no \((w',s)\leadsto ^{M[U]}(w,s)\). But then \({{\mathrm{d}}}(w,s)=0={{\mathrm{d}}}(w)+{{\mathrm{d}}}(s)\), which completes the subinduction base case. For the subinduction step, we assume that (4) holds for all worlds v having \(0\le {{\mathrm{d}}}(v)<{{\mathrm{d}}}(w)\) (the “subinduction hypothesis”) and we prove (4) holds for world w itself. If \({{\mathrm{d}}}(w,s)=0\), then (4) follows immediately because depths are nonnegative integers. So let us assume that \({{\mathrm{d}}}(w,s)>0\). Then we may choose an arbitrary \((w',s)\leadsto ^{M[U]}(w,s)\), from which it follows that \(w'\leadsto ^M w\). Since M is depthdefined, \({{\mathrm{d}}}(w')\le {{\mathrm{d}}}(w)1\) and so we may apply the subinduction hypothesis:Hence$$\begin{aligned} {{\mathrm{d}}}(w',s) \le {{\mathrm{d}}}(w')+{{\mathrm{d}}}(s) \le {{\mathrm{d}}}(w)1+{{\mathrm{d}}}(s). \end{aligned}$$$$\begin{aligned} {{\mathrm{d}}}(w,s)= & {} 1+\max \{{{\mathrm{d}}}(w',s)\mid (w',s)\leadsto ^{M[U]}(w,s)\}\\\le & {} 1+\max \{{{\mathrm{d}}}(w)1+{{\mathrm{d}}}(s) \mid (w',s)\leadsto ^{M[U]}(w,s)\}\\= & {} {{\mathrm{d}}}(w)+{{\mathrm{d}}}(s) \end{aligned}$$ 
Induction step: we suppose (4) holds for all events t having \(0\le {{\mathrm{d}}}(t)<{{\mathrm{d}}}(s)\) (the “induction hypothesis”) and prove that (4) holds for event s itself.
s is not a past state because \({{\mathrm{d}}}(s)>0\). Therefore \((w',s')\leadsto ^{M[U]}(w,s)\) implies \(w'=w\) and \(s'\leadsto ^U s\). If \({{\mathrm{d}}}(w,s)=0\), then (4) follows immediately because depths are nonnegative integers. So let us assume \({{\mathrm{d}}}(w,s)>0\). Then we may choose an arbitrary \((w,s')\leadsto ^{M[U]}(w,s)\), from which it follows that \(s'\leadsto ^U s\). Since U is depthdefined, we have \({{\mathrm{d}}}(s')\le {{\mathrm{d}}}(s)1\) and so we may apply the induction hypothesis:Hence$$\begin{aligned} {{\mathrm{d}}}(w,s') \le {{\mathrm{d}}}(w)+{{\mathrm{d}}}(s') \le {{\mathrm{d}}}(w)+{{\mathrm{d}}}(s)1. \end{aligned}$$$$\begin{aligned} {{\mathrm{d}}}(w,s)= & {} 1+\max \{{{\mathrm{d}}}(w,s')\mid (w,s')\leadsto ^{M[U]}(w,s)\}\\\le & {} 1+\max \{{{\mathrm{d}}}(w)+{{\mathrm{d}}}(s)1 \mid (w,s')\leadsto ^{M[U]}(w,s)\}\\= & {} {{\mathrm{d}}}(w)+{{\mathrm{d}}}(s) \end{aligned}$$

Case: \(w' \leadsto ^M w\) and \(s'=s\) is a past state.
Since \(w'\leadsto ^M w \rightarrow ^M_a v\) and M satisfies knowledge of the past, there exists \(v' \leadsto ^M v\). Since U is history preserving and s is a past state, s is an epistemic past state. From this we obtain two things. First, \((v',s)\in W^{M[U]}\) because epistemic past states have valid preconditions. Second, applying the fact that \(s \rightarrow ^U_a t\), it follows that \(s=t\) because \(\rightarrow _a\) arrows leaving epistemic past states are all reflexive. Since \(v' \leadsto ^M v\) and s is a past state, we conclude that \((v',s)\leadsto ^{M[U]} (v,s)=(v,t)\).

Case: \(w'=w\) and \(s'\leadsto ^U s\).
Since \(s'\leadsto ^U s \rightarrow ^U_a t\) and U satisfies knowledge of the past, there exists \(t' \leadsto ^U t\). Applying this to the assumption that U is history preserving and the fact that \((v,t)\in W^{M[U]}\), it follows that \((v,t') \in W^{M[U]}\). But then \((v,t') \leadsto ^{M[U]} (v,t)\).

Case: s is not a past state.
Since s is not a past state, there exists \(s' \leadsto ^U s\). Since U is history preserving and \((w,s)\in W^{M[U]}\), it follows that \((w,s')\in W^{M[U]}\). But then \((w,s') \leadsto ^{M[U]} (w,s)\), which contradicts our assumption that \({{\mathrm{d}}}(w,s)=0\).

Case: s is a past state.
Since U is history preserving and s is a past state, s is in fact an epistemic past state. Applying the fact that \(s \rightarrow ^U_a t\), it follows that \(s=t\) because \(\rightarrow _a\) arrows leaving epistemic past states are all reflexive. Since \((v',t')\leadsto ^{M[U]} (v,t)\) and \(t=s\) is a past state, it follows that \(t'=t=s\) and \(v\leadsto ^M v'\). Also, it follows from the fact that s is a past state and \({{\mathrm{d}}}(w,s)=0\) that we have \({{\mathrm{d}}}(w)=0\). Since M satisfies knowledge of the initial time, it follows from \({{\mathrm{d}}}(w)=0\) and \(w\rightarrow ^M_a v\) that \({{\mathrm{d}}}(v)=0\), but this contradicts \(v\leadsto ^M v'\).

Case: s is a past state.
Since s is a past state, if follows from \((v_1,t_1)\leadsto ^{M[U]}(w,s)\) and \((v_2,t_2)\leadsto ^{M[U]}(w,s)\) that we have \(s=t_1=t_2, v_1\leadsto ^M w\), and \(v_2\leadsto ^M w\). Since M satisfies uniqueness of the past, it follows that \(v_1=v_2\). Hence \((v_1,t_1)=(v_2,t_2)\).

Case: s is not a past state.
Since s is not a past state, if follows from \((v_1,t_1)\leadsto ^{M[U]}(w,s)\) and \((v_2,t_2)\leadsto ^{M[U]}(w,s)\) that we have \(w=v_1=v_2, t_1\leadsto ^U s\), and \(t_2\leadsto ^U s\). Since U satisfies uniqueness of the past, it follows that \(t_1=t_2\). Hence \((v_1,t_1)=(v_2,t_2)\).

Case: \(w'=w\) and \(s'\leadsto ^U s\).
Since \(s' \leadsto ^U s \rightarrow ^U_a t\) and U satisfies perfect recall, there exists \(t'\) satisfying \(s' \rightarrow ^U_a t' \leadsto ^U t\). Since U is history preserving and \((v,t)\in W^{M[U]}\), it follows that \((v,t')\in W^{M[U]}\). But then \((v,t')\leadsto ^{M[U]}(v,t)\). Since \(w\rightarrow ^M_av\) and \(s'\rightarrow ^U_at'\), we have \((w',s')=(w,s')\rightarrow ^{M[U]}_a(v,t')\).

Case: \(w'\leadsto ^M w\) and \(s'=s\) is a past state.
Since \(w' \leadsto ^M w \rightarrow ^M_a v\) and M satisfies perfect recall, there exists \(v'\) satisfying \(w' \rightarrow ^M_a v' \leadsto ^M v\). Further, since s is a past state and U is history preserving, s is an epistemic past state. From this two things follow. First, \((v',s)\in W^{M[U]}\) because epistemic past states have valid preconditions. Second, applying the fact that \(s\rightarrow ^U_a t\), it follows that \(s=t\) because \(\rightarrow _a\) arrows leaving epistemic past states are all reflexive. Since \(v'\leadsto ^M v\) and s is a past state, we have \((v',s)\leadsto ^{M[U]}(v,s)=(v,t)\). Further, since \(w'\rightarrow ^M_av'\) and \(s\rightarrow ^U_at=s\), we have \((w',s')=(w',s)\rightarrow ^{M[U]}_a(v',s)\).

Base case: \({{\mathrm{d}}}(t)=0\).
It follows that t is a past state. Since U is history preserving, t is an epistemic past state. Therefore, \(u\in W^M\) implies \((u,t)\in W^{M[U]}\) because epistemic past states have valid preconditions, and hence \(u\leadsto ^M u'\) implies \((u,t)\leadsto ^{M[U]}(u',t)\). It follows by an easy subinduction on \({{\mathrm{d}}}(v)\) that \({{\mathrm{d}}}(v,t)={{\mathrm{d}}}(v)\). Since \({{\mathrm{d}}}(t)=0\), this proves (5).

Induction step: assume the result holds for events s with \({{\mathrm{d}}}(s)<{{\mathrm{d}}}(t)\) (the “induction hypothesis”) and prove the result holds for event t with \({{\mathrm{d}}}(t)>0\).
Theorem 5 describes conditions under which properties of epistemic temporal models are preserved under updates. We will use this theorem later to show that a wellstudied atemporal Dynamic Epistemic Logic approach to reasoning about time is limited to the class of Kripke models that necessarily satisfy all of the properties we have defined. This highlights one of the key advantages of our \(\mathsf {DETL}\) framework: it may be used to reason about situations that do or do not satisfy these (or other) properties. The choice is left to the user.
6 Connections with previous work
6.1 \(\mathsf {RDETL}\)
Theorem 5 studied the preservation of Kripke model properties under certain actions. We chose these properties because they have been of interest in many studies of time in Dynamic Epistemic Logic (van Benthem et al. 2009, 2007; Dégremont et al. 2011; Sack 2010, 2008; Yap 2011). We now focus our attention on the class of Kripke models that satisfy these properties. This provides a paradigmatic example demonstrating how our \(\mathsf {DETL}\) framework can be used to reason about a wellstudied account of time in Dynamic Epistemic Logic.
Definition 9
(Restricted (forestlike) models) A Kripke model M is restricted (or forestlike) if it satisfies persistence of facts, depthdefinedness, knowledge of the past, knowledge of the initial time, uniqueness of the past, and perfect recall (Definition 7). Let \(\mathfrak {R}\) be the class of all the restricted Kripke models and \(\mathfrak {R}_{*}\) the class of all pointed restricted Kripke models.
The restricted models satisfy all the constraints on Kripke models given in Definition 7. Although synchronicity was not explicitly named as one of the properties of a restricted model, it is not hard to show that synchronicity does follow from the other properties (argue by induction on the depth of worlds, making use of perfect recall, knowledge of the past, and knowledge of the initial time).
We now define a fragment of \({L_{\mathsf {DETL}}}\) whose update modals preserve these restricted models.
Definition 10
(Language \(L_{\mathsf {RDETL}}\)) The language \(L_{\mathsf {RDETL}}\) of restricted \(\mathsf {DETL}\) is the sublanguage of \({L_{\mathsf {DETL}}}\) obtained by removing all actions [U, s] that are based on an action model U that fails to satisfy one or more of persistence of facts, depthdefinedness, knowledge of the past, history preservation, knowledge of the initial time, uniqueness of the past, or perfect recall. This removal applies recursively to preconditions as well.
The restrictions on the action models in \(L_{\mathsf {RDETL}}\) are those that appear in the Preservation Theorem (Theorem 5). Hence updating a restricted model by an action model in \(\mathfrak {A}(L_{\mathsf {RDETL}})\) yields another restricted model.
Definition 11
( \(\mathsf {RDETL}\) Semantics) We write \(M,w\models _{\mathsf {RDETL}}\varphi \) to mean that \((M,w)\in \mathfrak {R}_*\) and \(M,w\models \varphi \). We write \(\models _{\mathsf {RDETL}}\varphi \) to mean that \(M,w\models \varphi \) for every \((M,w)\in \mathfrak {R}_*\).
6.1.1 Proof system for \(\mathsf {RDETL}\)
Definition 12
( \(\mathsf {RDETL}\) Theory) The theory of Restricted Dynamic Epistemic Temporal Logic, \(\mathsf {RDETL}\), is defined in Fig. 8. We write \(\vdash _{\mathsf {RDETL}}\varphi \) to mean that \(\varphi \) is a theorem of \(\mathsf {RDETL}\).
Theorem 6
(Soundness and Completeness for \(\mathsf {RDETL}\)) \(\vdash _{\mathsf {RDETL}}\varphi \) iff \(\models _{\mathsf {RDETL}}\varphi \) for each \(\varphi \in L_{\mathsf {RDETL}}\).
Proof
Theorem 3 already establishes the soundness of the \(\mathsf {DETL}\) schemes and rules. Soundness for the remaining schemes is straightforward to prove.
The completeness proof can be divided into two stages. First, prove the Reduction Theorem: every \(L_\mathsf {RDETL}\)formula is \(\mathsf {RDETL}\)provably equivalent to an action modelfree formula in \({L_{\mathsf {SETL}}}\). This follows by the proof of Theorem 2. Second, prove completeness of action modelfree formulas with respect to the class of restricted Kripke models: \(\nvdash _\mathsf {RDETL}\psi \) for a given \(\psi \in {L_{\mathsf {SETL}}}\) implies there is a restricted situation (M, w) for which \(M,w\not \models \psi \). We outline a proof of the second stage.

\(W^{\varOmega \times \mathbb {Z}}\mathop {=}\limits ^{ \text{ def }}W^\varOmega \times \mathbb {Z}\).

For each \(a\in \fancyscript{A}\): \((w,k)\rightarrow ^{\varOmega \times \mathbb {Z}}_a (w',k')\) if and only if \(w\rightarrow _a^\varOmega w'\) and \(k=k'\).

Open image in new window if and only if Open image in new window and \(k'=k1\).

\(V^{\varOmega \times \mathbb {Z}}(p) \mathop {=}\limits ^{ \text{ def }}V^\varOmega (p) \times \mathbb {Z}\).

Open image in new window , where \(R^*\) denotes the reflexivetransitive closure of a binary relation R.

For each \(a\in \fancyscript{A}\): \({\rightarrow _a^M}\mathop {=}\limits ^{ \text{ def }}{\rightarrow _a^{\varOmega \times \mathbb {Z}}}\cap (W^M\times W^M)\).

\(V^M(p)\mathop {=}\limits ^{ \text{ def }}V^{\varOmega \times \mathbb {Z}}(p)\cap W^M\).

\(W^{M'}\mathop {=}\limits ^{ \text{ def }}\{(w,k)\in W^M\mid 0\le k\le m\}\).

For each \(a\in \fancyscript{A}\): \({\rightarrow _a^{M'}}\mathop {=}\limits ^{ \text{ def }}{\rightarrow _a^M}\cap (W^{M'}\times W^{M'})\).

\(V^{M'}(p)\mathop {=}\limits ^{ \text{ def }}V^M(p)\cap W^{M'}\).

\(W^F\mathop {=}\limits ^{ \text{ def }}\{[w,k]\mid (w,k)\in W^{M'}\}\).

For each \(a\in \fancyscript{A}\): \(A\rightarrow ^F_a B\) iff \(\exists (w,k)\in A\) and \(\exists (v,j)\in B\) with \((w,k)\rightarrow ^{M'}_a(v,j)\).

Open image in new window iff \(\exists (w,k)\in A\) and \(\exists (v,j)\in B\) with Open image in new window .

\(V^F(p)\mathop {=}\limits ^{ \text{ def }}\{A\in W^F\mid p\in \mathsf {Cl}_m(\lnot \psi ) \text { and } \forall (w,k)\in A:M',(w,k)\models p\}\).
Truth preservation: By what we have shown above, it follows that for each \((w,k)\in W^\varOmega \times \{0,\dots ,m\}\) and each \(\varphi \in \mathsf {Cl}_k(\lnot \psi )\), we have \(\varOmega ,w\models \varphi \) if and only if \(F,[w,k]\models \varphi \). In particular, we have \(F,[\varGamma _{\lnot \psi },m]\not \models \psi \). So to complete the proof, it suffices for us to show that \(F\in \mathfrak {R}\) (i.e., F is a restricted model).
 F satisfies uniqueness of the past:Suppose not. From \([w',k']\leadsto ^F[w,k]\) and \([w'',k'']\leadsto ^F[w,k]\), we have \(k'=k''=k1\). From \([w',k1]\ne [w'',k1]\), it follows that there exists \(\varphi \in \mathsf {Cl}_{k1}(\lnot \psi )\) such that, without loss of generality, \(F,[w',k1]\models \varphi \) and \(F,[w'',k1]\not \models \varphi \). Defining$$\begin{aligned} ([w',k']\leadsto ^F[w,k] \wedge [w'',k'']\leadsto ^F[w,k])\Rightarrow ([w',k']=[w'',k'']). \end{aligned}$$we have \(\chi \in \mathsf {Cl}_k(\lnot \psi )\) and \(F,[w,k]\not \models \chi \) and hence that \(\varOmega ,w\not \models \chi \) by Truth preservation. But then it follows by the Truth Lemma that the maximal consistent set w fails to contain an instance \(\chi \) of the Uniqueness of the Past axiom, a contradiction.$$\begin{aligned} \chi \mathop {=}\limits ^{ \text{ def }}\lnot [Y]\varphi \rightarrow [Y]\lnot \varphi , \end{aligned}$$
 F satisfies persistence of facts:Suppose not. Then we have$$\begin{aligned}{}[w,k]\leadsto ^F[v,j] \Rightarrow ([w,k]\in V^F(p)\Leftrightarrow [v,j]\in V^F(p)). \end{aligned}$$Let \(\chi \mathop {=}\limits ^{ \text{ def }}[Y]p\leftrightarrow (\lnot [Y]\bot \rightarrow p)\). Since F satisfies uniqueness of the past, it follows from (6) that \(F,[v,j]\not \models \chi \). Since \([w,k]\leadsto ^F[v,j]\), we have \(j\ge 1\). Further, it follows by (6) that \([w,k]\in V^F(p)\) or \([v,j]\in V^F(p)\). Applying the definition of \(V^F(p)\), we have that \(p\in \mathsf {Cl}_m(\lnot \psi )\) and therefore that \(p\in \mathsf {Cl}_0(\lnot \psi )\). So since \(j\ge 1\), we have \(\chi \in \mathsf {Cl}_j(\lnot \psi )\). But then it follows from \(F,[v,j]\not \models \chi \) by Truth preservation that \(\varOmega ,v\not \models \chi \). Applying the Truth Lemma, the maximal consistent set v fails to contain an instance \(\chi \) of the Persistence of Facts axiom, a contradiction.$$\begin{aligned} \lnot ([w,k]\in V^F(p)\Leftrightarrow [v,j]\in V^F(p)). \end{aligned}$$(6)

F is depthdefined: \({{\mathrm{d}}}([w,k])\ne \infty \).
For each \([v,j]\in W^F\), we have \(0\le j\le m\). Further, Open image in new window implies \(j'=k'1\). It follows that \({{\mathrm{d}}}([w,k])\ne \infty \).
 F satisfies knowledge of the past:Suppose not. Letting \(\chi \mathop {=}\limits ^{ \text{ def }}\lnot [Y]\bot \rightarrow \Box _a\lnot [Y]\bot \), it follows that \(F,[w,k]\not \models \chi \). Further, from \([w',k']\leadsto ^F[w,k]\), we have \(k\ge 1\) and therefore that \(\chi \in \mathsf {Cl}_k(\lnot \psi )\). But then it follows by Truth preservation that \(\varOmega ,w\not \models \chi \). Applying the Truth Lemma, the maximal consistent set w fails to contain an instance \(\chi \) of the Uniqueness of the Past axiom, a contradiction.$$\begin{aligned} ([w',k']\leadsto ^F[w,k]\rightarrow ^F_a[v,j])\Rightarrow \exists [v',j']([v',j']\leadsto ^F[v,j]). \end{aligned}$$
 F satisfies knowledge of the initial time:Suppose \([w,k]\rightarrow ^F_a[v,j]\) and \(\lnot \exists [w',k']([w',k']\leadsto ^F[w,k])\). If \(k=0\), then it follows by \([w,k]\rightarrow ^F_a[v,j]\) that \(j=0\) and therefore that \(\lnot \exists [v',j']([v',j']\leadsto ^F[v,j])\) by Trimming. So let us assume that \(k>0\). Further, toward a contradiction, we assume that \(\exists [v',j']([v',j']\leadsto ^F[v,j])\). Letting \(\chi \mathop {=}\limits ^{ \text{ def }}[Y]\bot \rightarrow \Box _a[Y]\bot \), it follows that \(F,[w,k]\not \models \chi \). But since \(k>0\), we have \(\chi \in \mathsf {Cl}_k(\lnot \psi )\) and hence it follows by Truth preservation that \(\varOmega ,w\not \models \chi \). Applying the Truth Lemma, the maximal consistent set w fails to contain an instance \(\chi \) of the Knowledge of the Initial Time axiom, a contradiction.$$\begin{aligned}&[w,k]\rightarrow ^F_a[v,j]\wedge \lnot \exists [w',k']([w',k']\leadsto ^F[w,k])\\&\quad \Rightarrow \lnot \exists [v',j']([v',j']\leadsto ^F[v,j]). \end{aligned}$$
 F satisfies perfect recall:Suppose \([w,k]\leadsto ^F[v,j]\rightarrow ^F_a[v',j']\). Then \(k+1=j=j'\). Now \([v,k+1]=[v,j]\rightarrow ^F_a[v',j']=[v',k+1]\) implies \(\exists (v_*,k+1)\in [v,k+1]\) and \(\exists (v'_*,k+1)\in [v',k+1]\) with \((v_*,k+1)\rightarrow ^{M'}_a(v'_*,k+1)\). Hence \(v_*\rightarrow ^\varOmega _av'_*\). Now from \([w,k]\leadsto ^F[v,j]=[v,k+1]\), we have that \(F,[v,k+1]\models \lnot [Y]\bot \). Since \(k+1\ge 1\), it follows that \(\lnot [Y]\bot \in \mathsf {Cl}_{k+1}(\lnot \psi )\) and therefore we have by Truth preservation that \(\varOmega ,v_*\models \lnot [Y]\bot \). By the definition of truth, it follows that there is a \(w_*\in W^\varOmega \) with \(w_*\leadsto ^\varOmega v_*\). But then \([w_*,k]\leadsto ^F[v,j]=[v,k+1]\), from which it follows by uniqueness of the past for F that \([w_*,k]=[w,k]\) and therefore that \((w_*,k)\in [w,k]\). Now \(\varOmega \) satisfies perfect recall (for if it did not, we could find a violation of an instance of the Perfect Recall axiom at a maximal consistent set, a contradiction). Therefore, since we have \(w_*\leadsto ^\varOmega v_*\rightarrow ^\varOmega _a v'_*\), it follows by perfect recall for \(\varOmega \) that there is a \(w'_*\in W^\varOmega \) satisfying \(w_*\rightarrow ^\varOmega _a w'_*\leadsto ^\varOmega v'_*\). But then \([w,k]=[w_*,k]\rightarrow ^\varOmega _a[w'_*,k]\leadsto ^\varOmega [v'_*,k+1]=[v',k+1]=[v',j]\). \(\square \)$$\begin{aligned} ([w,k]\leadsto ^F[v,j]\rightarrow ^F_a[v',j'])\Rightarrow \exists [w',k']([w,k]\rightarrow ^F_a[w',k']\leadsto ^F[v',j']). \end{aligned}$$
6.2 \(\mathsf {YDEL}\)
In this section, we relate \(\mathsf {DETL}\) to a more conservative approach to adding time to \(\mathsf {DEL}\), which, as is generally studied in the literature (Baltag and Moss 2004; Baltag et al. 1998, 2008; van Benthem et al. 2006; van Ditmarsch et al. 2007), does not use \(\leadsto \) arrows in its action models. Further, the semantics of \(\mathsf {DEL}\) does not use Kripke models with designated timekeeping arrows \(\leadsto \). In order to draw this comparison, we will define an extension of \(\mathsf {DEL}\) called “Yesterday Dynamic Epistemic Logic,” or \(\mathsf {YDEL}\) (see Sack 2008, 2010; Yap 2011), that records a history of the updates made. We will then show that \(\mathsf {YDEL}\) reasoning can be done within the \(\mathsf {DETL}\) setting, since (modulo translation) \(\mathsf {YDEL}\) is sound and complete with respect to a particular class of \(\mathsf {DETL}\) models.
Definition 13
\({L_{\mathsf {YDEL}}}\) is the atemporal fragment of \({L_{\mathsf {DETL}}}\). For reasons explained in a moment, we assume that the special symbol \(\flat \) is used neither as a world nor as an event in \({L_{\mathsf {YDEL}}}\).
We will evaluate \(\mathsf {YDEL}\) formulas on restricted Kripke models (Definition 9).
Definition 14
The forthcoming Corollary 1 shows that \(M\oplus U\in \mathfrak {R}\) whenever \(M\in \mathfrak {R}\) and U is atemporal. The function of the symbol \(\flat \) is to serve as an epistemic past state, preserving a copy of M in \(M \oplus U\) (Lemma 1). Since \(\mathsf {YDEL}\) uses atemporal action models, which contain no \(\leadsto \) arrows, the mechanism for preserving the previous model M is “hardcoded” in the semantics. However, by defining a translation from \({L_{\mathsf {YDEL}}}\) to \({L_{\mathsf {DETL}}}\) (Definition 15), we will be able to show that “atemporal” \(\mathsf {YDEL}\) reasoning can be captured in \(\mathsf {DETL}\).
Lemma 1
Let (M, w) be a situation and (U, s) be an atemporal action satisfying \(M,w\models \mathsf {pre}^U(s)\). The function \(f:W^M\rightarrow W^{M\oplus U}\) defined by \(f(w)\mathop {=}\limits ^{ \text{ def }}(w,\flat )\) is a bisimulation.
Proof
w and \((w,\flat )\) have the same valuation. If \(f(w)=(w,\flat ) \rightarrow ^{M \oplus U}_a (v,t)\), then \(t=\flat \) and \(w \rightarrow ^M_a v\). If \(w \rightarrow ^M_a v\), then \((w, \flat ) \rightarrow ^{M \oplus U}_a (v,\flat )\).\(\square \)
Before defining the translation from \(\mathsf {YDEL}\) to \(\mathsf {DETL}\), we will first show how \(\mathsf {YDEL}\) works by illustrating the way in which \(M \oplus U\) is constructed.
Example 6
Figure 9 pictures an initial situation and a \(\mathsf {YDEL}\) action. In the initial situation, neither agent knows whether p is true. The action informs a that p is true but tells b only that a was either informed of p or provided with trivial information. After applying the action to the situation, we obtain the resultant situation in Fig. 10.
We now show how \(\mathsf {YDEL}\) reasoning is captured in \(\mathsf {DETL}\).
6.2.1 Translation of \(L_\mathsf {YDEL}\) into \(L_\mathsf {DETL}\)
We define a translation from \(\mathsf {YDEL}\) formulas and action models to \(\mathsf {DETL}\) formulas and action models. This translation acts on action models by adding a new epistemic past state \(\flat \) along with an arrow \(\flat \leadsto s\) to each action s. See Fig. 11 for an example.
Definition 15
The function \(\sharp \) transforms the atemporal action models used by \(\mathsf {YDEL}\) into \(\mathsf {DETL}\) action models having epistemic past states. As it turns out, such action models are in fact \(\mathsf {RDETL}\) action models (Definition 10).
Lemma 2
If U is an atemporal action model, then \(U^\sharp \in \mathfrak {A}(L_{\mathsf {RDETL}})\).
The proof of this lemma is straightforward. It follows that the image of \(\sharp \) is contained in \(L_{\mathsf {RDETL}}\cup \mathfrak {A}(L_{\mathsf {RDETL}})\). This containment is strict: every history in \(U^\sharp \) has length 1, while the length of histories in \(\mathsf {RDETL}\) action models is unbounded.
Theorem 7
 (a)
\(M\oplus U=M[U^\sharp ]\), and
 (b)
\(M,w\models _\mathsf {YDEL}\varphi \) iff \(M,w\models _\mathsf {RDETL}\varphi ^\sharp \).
Proof
 1.
For each \(M\in \mathfrak {R}\) and each \(U\in \mathfrak {A}^a(L_i)\): \(M\oplus U=M[U^\sharp ]\).
 2.For each \((M,w)\in \mathfrak {R}_*\) and each \(\varphi \in L_i\):$$\begin{aligned} M,w\models _\mathsf {YDEL}\varphi \ \textit{iff}\ M,w\models _\mathsf {RDETL}\varphi ^\sharp . \end{aligned}$$

\(t,t'\ne \flat \) and \(t\rightarrow ^U_at'\); or

\(t=t'=\flat \).

\(t=\flat , t'\ne \flat \), and \(v=v'\); or

\(t=t'=\flat \) and \(v\leadsto ^M v'\).

\(v\leadsto ^Mv', t=t'\), and t is a past state; or

\(v=v'\) and \(t\leadsto ^{U^\sharp } t'\).
Finally, we have \((v,t)\in V^{M\oplus U}(p)\) if and only if \(v\in V^M(p)\) if and only if \((v,t)\in V^{M[U^\sharp ]}(p)\). Here we made tacit use of the fact that \(W^{M\oplus U}=W^{M[U^\sharp ]}\). Conclusion: \(V^{M\oplus U}=V^{M[U^\sharp ]}\).
This completes the proof of Statement 1. The proof of Statement 2 then proceeds by a subinduction on the construction of \(L_{i+1}\)formulas. Most cases are obvious, so we only address the case for \(L_{i+1}\)formulas \([U,s]\varphi \). Proceeding, we have \(M,w\models _\mathsf {YDEL}[U,s]\varphi \) if and only if \(M,w\not \models _\mathsf {YDEL}\mathsf {pre}^U(s)\) or \(M\oplus U,(w,s)\models _\mathsf {YDEL}\varphi \). By Statement 2 of the induction hypothesis, the latter is equivalent to “\(M,w\not \models _\mathsf {RDETL}\mathsf {pre}^U(s)^\sharp \) or \(M\oplus U,(w,s)\models _\mathsf {RDETL}\varphi ^\sharp \),” which is itself equivalent to “\(M,w\not \models _\mathsf {RDETL}\mathsf {pre}^{U^\sharp }(s)\) or \(M[U^\sharp ],(w,s)\models _\mathsf {RDETL}\varphi ^\sharp \)” (by the definition of \(U^\sharp \) for the left disjunct and Statement 1 of the induction hypothesis for the right). But this is equivalent to \(M,w\models _\mathsf {RDETL}[U^\sharp ,s]\varphi ^\sharp \) by the \(\mathsf {RDETL}\) semantics. Since \([U^\sharp ,s]\varphi ^\sharp =([U,s]\varphi )^\sharp \), the result follows. \(\square \)
A corollary of Theorem 7 is that restricted models are closed under the \(\mathsf {YDEL}\) update operation \(M\mapsto M\oplus U\).
Corollary 1
If \(M\in \mathfrak {R}, U\in \mathfrak {A}({L_{\mathsf {YDEL}}})\), and \(W^{M\oplus U}\ne \emptyset \), then \(M\oplus U\in \mathfrak {R}\).
Proof
Fix \(M\in \mathfrak {R}\) and \(U\in \mathfrak {A}({L_{\mathsf {YDEL}}})\) with \(W^{M\oplus U}\ne \emptyset \). Since \(U\in \mathfrak {A}({L_{\mathsf {YDEL}}})\) if and only if U is atemporal (Definition 13), it follows by Lemma 2 that \(U^\sharp \in \mathfrak {A}(L_{\mathsf {RDETL}})\). Applying Preservation (Theorem 5) and the definition of \(L_{\mathsf {RDETL}}\) (Definition 10), we have \(M[U^\sharp ]\in \mathfrak {R}\). By Theorem 7, \(M\oplus U=M[U^\sharp ]\in \mathfrak {R}\). \(\square \)
6.2.2 Connecting the theories of \(\mathsf {YDEL}\) and \(\mathsf {RDETL}\)
Definition 16
The theory \(\mathsf {YDEL}\) is defined in Fig. 12. Note that all axioms and rules refer to formulas in \({L_{\mathsf {YDEL}}}\) and hence to action models in \(\mathfrak {A}({L_{\mathsf {YDEL}}})\).
Theorem 8
The theory of \(\mathsf {YDEL}\) is sound and complete with respect to \(\mathfrak {R}_{*}\).
Proof

Soundness for \([U,s][Y]\varphi \leftrightarrow (\mathsf {pre}^U(s)\rightarrow \varphi )\).
We first prove the lefttoright direction of the equivalence. Suppose \(M,w \models [U,s][Y] \varphi \) and \(M,w \models \mathsf {pre}^U(s)\). This implies that for every \((v,t) \leadsto ^{M \oplus U} (w,s)\), we have \(M \oplus U, (v,t) \models \varphi \). By definition of \(M\oplus U\), we have \((w, \flat ) \leadsto ^{M \oplus U} (w,s)\), and so \(M \oplus U (w,\flat ) \models \varphi \). By Lemma 1, \(M,w \models \varphi \). Now we prove the righttoleft direction. Suppose \(M,w \models \mathsf {pre}^U(s)\) and \(M,w \models \varphi \) (the case where \(M,w\not \models \mathsf {pre}^U(s)\) is immediate). By definition of \(M\oplus U\) and uniqueness of the past, \((w,t) \leadsto ^{M \oplus U} (w,s)\) implies \(t=\flat \). By the fact that \(M,w \models \varphi \) and Lemma 1, we have \(M \oplus U, (w,\flat ) \models \varphi \). Hence \(M,w \models [U,s][Y] \varphi \).

Soundness for (UN) follows from the fact that restricted models are closed under the operation \(M\mapsto M\oplus U\) (Corollary 1).
Corollary 2
7 Conclusion
We have presented Dynamic Epistemic Temporal Logic (\(\mathsf {DETL}\)), a general framework for reasoning about transformations on Kripke models with a designated timekeeping relation Open image in new window . Our “temporal” action models are a generalization of the atemporal action models familiar from Dynamic Epistemic Logic. We showed by way of a number of examples how temporal action models can be used to reason about and control the flow of time. We also highlighted some key design choices that allow this framework to avoid conceptual complications relating to time and that enable us to define actions that preserve a complete copy of the past state of affairs. This leads to one natural choice for understanding time: the time of a world is the depth of that world (i.e., the maximum number of “backward” temporal steps one can take from that world, whenever this maximum exists).
Kripke models with a designated timekeeping relation are essentially the models of Epistemic Temporal Logic. Therefore, one way of understanding our work is as follows: we extend the domain of action model operations from those on (atemporal) Kripke models to (what are essentially) the models of Epistemic Temporal Logic. We showed that a number of properties that may arise in the latter models—such as Persistence of Facts, Perfect Recall, and Synchronicity—are preserved under the application of temporal action models that themselves satisfy certain related properties. This makes it possible to use our \(\mathsf {DETL}\) framework to develop Dynamic Epistemic Logicstyle theories of temporal Kripke models. Such logics can be used to reason about objective changes in time along with the agents’ basic and higherorder knowledge and beliefs about changes in this structure. As an example, we showed how the \(\mathsf {DETL}\) approach can be used to define the logic \(\mathsf {RDETL}\) of “restricted” Dynamic Epistemic Temporal Logic, which is essentially the Dynamic Epistemic Logic of synchronous actions with the “yesterday” temporal operator [Y]. We proved that \(\mathsf {RDETL}\) reasoning captures the reasoning of \(\mathsf {YDEL}\), the first Dynamic Epistemic Logic of synchronous time with the yesterday modal.
The \(\mathsf {DETL}\) approach is not, however, limited to synchronous systems. We presented one example where a synchronous model is transformed into an asynchronous one, leaving one agent sure that two clock ticks occurred, and the other uncertain as to whether it was one or two. We contrasted this with a synchronous variant in which the agents’ knowledge about atemporal information is the same, but the knowledge change is compressed into a single clock tick that is common knowledge. Here we see that the difference is easily discernible by a simple examination of the temporal action models involved. In essence, our theory extends the types of knowledge change describable by atemporal action models to the temporal setting, which gives us a great deal of control as to the relationship between how much time passes and what the agents perceive of this passage. And we of course also inherit many features (and drawbacks) of the atemporal action model approach.
One direction for future work is to extend our temporal language to include more than just the onestep “yesterday” operator Y. For example, it would also be interesting (and challenging) to consider backwardlooking “since” operator and other operators familiar from temporal logic (Goldblatt 2006).^{10}
In closing, we mention a recent study of time in Dynamic Epistemic Logic that looks at asynchronous systems (Dégremont et al. 2011). The basic idea is that an atemporal action model operates on a temporal Kripke model in such a way that an agent experiences a single clock tick if her knowledge of atemporal information changes, but will otherwise be uncertain as to whether the clock ticked. So, for example, if agent a does not know p, a public announcement of p will transform her knowledge in a synchronous manner: the clock will tick once, she will learn p, and she will know that the clock ticked once. But if the public announcement of p then occurs again, the clock will tick but, since she already knows p and hence her atemporal knowledge will not change, she will be uncertain as to whether the clock ticked once or not at all. The result is an asynchronous situation.
Though we have shown (by way of an example) that \(\mathsf {DETL}\) can reason about some asynchronous updates, we have not proved that it can reason about every such update. Nor have we shown that it can reason about a certain class of asynchronous updates that can be independently identified according to some desirable properties it satisfies. In particular, it is not clear if there is a \(\mathsf {DETL}\) action model for all the updates that can be produced by the framework of Dégremont et al. (2011). Moreover, the latter approach is based on “protocols” constraining the sequences of actions that can occur, something we have left out of the present study for simplicity. Another complication is that the asynchronous updates of Dégremont et al. (2011) essentially insert agent arrows \(\rightarrow _a\) based on whether a certain knowledge condition is satisfied, whereas our temporal action models do not allow us to conditionally insert arrows. This suggests that there may be connections with “arrow update logics” (Kooi and Renne 2011a, b) that allow such conditions on arrows. In particular, it has been shown that generalized arrow updates are equivalent to atemporal action models in terms of update expressivity (Kooi and Renne 2011b). Therefore, an arrow update version of \(\mathsf {DETL}\) might suggest a natural way to represent asynchronous updates like those of Dégremont et al. (2011), and this may turn out to be equivalent in update expressivity to our present approach, just as in the atemporal case. If this is so, then it may be the case that “conditional” arrow changes are already within the scope of our current approach, albeit indirectly.
In conclusion, we believe that \(\mathsf {DETL}\) presents a viable option for developing Dynamic Epistemic Logicstyle theories of Epistemic Temporal Logic. While this paper [and its early predecessor (Renne et al. 2009)] present the first steps of this study, there is clearly still much more work to be done.
Footnotes
 1.
For all structures X, let \(\leadsto ^X\) denote the converse of Open image in new window and let \(\leftarrow ^X_a\) denote the converse of \(\rightarrow ^X_a\). Our discussion of temporal issues will typically use \(\leadsto \) rather than Open image in new window because the former follows the natural direction of time’s flow. Here and elsewhere, we will omit superscripts on relations when doing so ought not cause confusion.
 2.
We note that because synchronicity requires the structure to be depthdefined and perfect recall does not, perfect recall does not imply synchronicity. However, the conjunction of perfect recall, depthdefinedness, and uniqueness of the past are together equivalent to the conjunction of synchronicity and uniqueness of the past. See Goranko and Pacuit (2014) for a discussion of a weaker version of perfect recall that does not stand in a similar relationship to synchronicity. In the interest of simplicity, we do not consider this more complicated weaker version here.
 3.
Formally, a \({L_{\mathsf {DETL}}}\)formula \(\varphi \) is true at a world x if and only if \(\varphi \) is true at x even after we delete all worlds y satisfying the property that every path from x to y contains at least one \(\leadsto \) arrow (followed in the “forward” direction \(z\leadsto z'\) from “past” z to “future” \(z'\)). This is so because \({L_{\mathsf {DETL}}}\) has no [T] operator, as defined in (1).
 4.
This is similar to the way in which the “do nothing” Propositional Dynamic Logic (PDL) program \( skip \) does not change the state of the system. However, there is a difference: the PDL program does not change the structure of the model, though the action \((U_3,t)\) does. Nevertheless, from the point of view of language equivalence, this change is inconsequential: the “before” situation and the “after” situation satisfy the same \({L_{\mathsf {DETL}}}\)formulas, and so our intention is that these situations are to be identified.
 5.
While these may not have the typical “look” of reduction axioms as they are commonly found in Dynamic Epistemic Logics, they can nevertheless be considered reduction axioms in the sense that they allow us to prove the reduction to \({L_{\mathsf {SETL}}}\) as per Theorem 2.
 6.
The primary difference is with those concerning the Ymodality. There are two cases: s is a past state, and s is not a past state. In both cases, U[Y] is a simplification of U\(\Box _a\), reflecting the involvement of asynchronous composition rather than synchronous (see the discussion after Definition 5). In U\(\Box _a\), the conjunction reflects the transitions made in the action model, while the modality \(\Box _a\) that follows reflects the transitions made in original model. Note that if s is a past state, then there is no \(s'\leadsto ^U s\), so we can remove the conjunction. If s not a past state, then it is the first coordinate rather than the second coordinate that must be fixed in a \(\leadsto \) transition in the updated model. Hence we remove the modality [Y] that would otherwise follow the conjunction.
 7.
The axiomatization of the validities of the action modelfree sublanguage \({L_{\mathsf {SETL}}}\) are obtained from the axiomatization of \(\mathsf {DETL}\) by deleting each of the reduction axioms, deleting the rule UN, and restricting the language to that of \({L_{\mathsf {SETL}}}\). What results is just multimodal \(\mathsf {K}\), with one \(\mathsf {K}\) modality for each agent modal \(\Box _a\) and one \(\mathsf {K}\) modality for the yesterday modal [Y]. Completeness thereby follows by the standard construction (e.g., in (Blackburn et al. 2001, Chap. 4) using the “modal similarity type” of \({L_{\mathsf {SETL}}}\)).
 8.
In detail: a bisimulation is a nonempty binary relation B between the worlds of Kripke models (with yesterday) Open image in new window and Open image in new window such that \(wBw'\) implies w and \(w'\) satisfy the same propositional letters; and, for each binary relation symbol Open image in new window specified by the structures, \(wBw'\) and \(wR^Mv\) implies there is a \(v'\in W^{M'}\) such that \(w'R^{M'}v'\) and \(vBv'\), and \(wBw'\) and \(w'R^{M'}v'\) implies there is a \(v\in W^M\) such that \(wR^Mv\) and \(vBv'\). Note that the definition of bisimulation only considers temporally reachable worlds “in the past” (i.e., in the direction from x to y in the arrow Open image in new window ).
 9.
 10.
Thanks to an anonymous reviewer for this suggestion.
Notes
Acknowledgments
The authors thank the reviewers for their feedback and suggestions. Bryan Renne was funded by Veni grant 27520030 from the Netherlands Organisation for Scientific Research (NWO). Joshua Sack was funded by Vidi grant 639072904 from the NWO.
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