Abstract
In dynamical multiagent systems, agents are controlled by protocols. In choosing a class of formal protocols, an implicit choice is made concerning the types of agents, actions and dynamics representable. This paper investigates one such choice: An intensional protocol class for agent control in dynamic epistemic logic (DEL), called ‘DEL dynamical systems’. After illustrating how such protocols may be used in formalizing and analyzing information dynamics, the types of epistemic temporal models that they may generate are characterized. This facilitates a formal comparison with the only other formal protocol framework in dynamic epistemic logic, namely the extensional ‘DEL protocols’. The paper concludes with a conceptual comparison, highlighting modeling tasks where DEL dynamical systems are natural.
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Notes
 1.
When constructing an action model that produces a pointed Kripke model isomorphic to a particular level in the tobegenerated ETL model, Proposition 3.2 of [21] (which states, roughly, that for almost any two pointed Kripke models, there exists an action model with postconditions that will produce one from the other using product update) is not applicable. The proposition is not applicable as the transforming action model allows only the designated state to survive, from which the desired Kripke model is unfolded. The resulting generated ETL models would therefore (most often) not be ETL isomorphic to the original ETL model, as ETL isomorphisms require that the temporal structure is preserved.
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Acknowledgments
The Center for Information and Bubble Studies is funded by the Carlsberg Foundation. The contribution of R.K. Rendsvig to the research reported in this article was also funded by the Swedish Research Council through the framework project ‘Knowledge in a Digital World’ (Erik J. Olsson, PI): A previous version of this paper occurs in the thesis [57]. We thank Alexandru Baltag, Johan van Benthem, Thomas Bolander, Vincent F. Hendricks and Dominik Klein for valuable comments on elements of this paper, as well as the participants of the 2016 CADILLAC and the 2016 LogiCIC workshops, held in Christiania, Denmark, and Amsterdam, the Netherlands, respectively.
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Appendix: Proofs
Appendix: Proofs
Proposition 1
Let (X, f, x) be a pointed DEL dynamical system given by multipointed action model . Then there exists a singleton uniform DEL protocol that produces the orbit of f from x.
Proof
For each \(k\in \mathbb {N}\), let σ_{k} ∈Γ be such that f^{k}(x)⊧pre(σ_{k}). As is Xdeterministic, for each k there is at most one such σ_{k}. Define a uniform DEL protocol p as the smallest protocol for which p_{k}(s) = Σσ_{k} (for all s ∈ x) whenever σ_{k} exists. Then when p is sequentially applied to x using product update, it produces the sequence \(\langle f^{k}(x)\rangle _{k\in \mathbb {N}}\) of pointed Kripke models (up to the deletion of redundant states not connected to the designated states, cf. Section 2).□
Proposition 2
If saturated ETL model \((\mathcal {H},\underline {H})\) is generated by a pointed DEL dynamical system, then \((\mathcal {H},\underline {H})\) satisfies seven of the eight properties of Section 4.2, namely Synchronicity, Connected TimeSteps, Perfect Recall, Local No Miracles, Precondition Describable, Precondition Describable, and Point Bisimulation Invariance.
Proof
Let (X, f, x) be a pointed DEL dynamical system with orbit \((f^{k}(x))_{k\in \mathbb {N}}\). The length of a state s in f^{k}(x) is len(s) := k + 1.
Let \((\mathcal {H},\underline {H})\) be the saturated ETL model generated by (X, f, x). Given the construction of γ in Def. 6, there exists a family of isomorphisms \(\{g_{k}\}_{k\in \mathbb {N}}\) with each g_{k} mapping [[f^{k}(x)]] to H_{k} := {h ∈ H : len(h) = k} satisfying g_{1}(s) = e_{s} and g_{k+ 1}((s, σ)) = g_{k}(s)e_{σ}. Using this family, it is shown that \((\mathcal {H},\underline {H})\) satisfies the listed properties in order:
 Synchronicity. :

Assume for arbitrary h, h^{′}∈ H that h_{i}h^{′}. Then by the construction of the generated _{i} (Def. 6), \(\exists k\in \mathbb {N}:g_{k}^{1}(h)R_{i}g_{k}^{1}(h^{\prime })\). Hence \(g_{k}^{1}(h),g_{k}^{1}(h^{\prime })\in \left [{\kern 3.5pt}[ f^{k}(x)\right ]{\kern 3.5pt}] \). Thus, \(len(g_{k}^{1}(h))=len(g_{k}^{1}(h^{\prime }))\). Hence, by the construction of g, len(h) = len(h^{′}).
 Perfect Recall. :

Assume for arbitrary he, h^{′}e^{′}∈ H that he_{i}h^{′}e^{′}. Then \(\exists k\in \mathbb {N}:g_{k}^{1}(he)R_{i}g_{k}^{1}(h^{\prime }e^{\prime })\). By construction of f, for the multi pointed action model. As \(g_{k}^{1}(he)R_{i}g_{k}^{1}(h^{\prime }e^{\prime })\), by definition of ⊗ and clean maps, \(g_{k1}^{1}(h)R_{i}g_{k1}^{1}(h^{\prime })\). Hence, by definition of _{i}, it follows that h_{i}h^{′}.
 Local No Miracles. :

Assume that 1) h_{i}h^{′}, 2) h_{1}e_{i}h_{2}e^{′} and 3) h^{∗}h_{1} for arbitrary he, h^{′}e^{′}, h_{1}e, h_{2}e^{′}∈ H. 1) implies that 1*) \(g_{k}^{1}(h)R_{i}g_{k}^{1}(h^{\prime })\) for k = len(h). 3) implies that len(h) = len(h_{1}) by Synchronicity. In conjunction with 2), this implies that 2*) \(g_{k+1}^{1}(h_{1}e)R_{i}g_{k+1}^{1}(h_{2}e^{\prime })\).
By construction of f, . By 2*) and the definition of ⊗ and clean maps, there must be 4) σ_{e}, such that \(\sigma _{e}\mathsf {R}_{i}\sigma _{e^{\prime }}\), for σ_{e} the σ such that g_{k+ 1}((s, σ)) = h_{1}e, for some s ∈ [[f^{k}(x)]], and \(\sigma _{e^{\prime }}\) the σ^{′} such that g_{k+ 1}((t, σ^{′})) = h_{2}e^{′}, for some t ∈ [[f^{k}(x)]].
Now assume that \((g_{k}^{1}(h),\sigma _{e}),(g_{k}^{1}(h^{\prime }),\sigma _{e^{\prime }})\in \left [{\kern 3.5pt}[ f^{k+1}(x)\right ]{\kern 3.5pt}] \). Then 1*), 4) and Def. ⊗ jointly imply that \((g_{k+1}^{1}(h),\sigma _{e})R_{i}(g_{k+1}^{1}(h^{\prime }),\sigma _{e^{\prime }})\). By the definition of the generated _{i}, it thus follows that he_{i}h^{′}e^{′}.
 Connected TimeSteps. :

For arbitrary h, h^{′}∈ H assume len(h) = len(h^{′}). Let \(k\in \mathbb {N}\) such that h, h^{′}∈ H_{k}. Then \(g_{k}^{1}(h),g_{k}^{1}(h^{\prime })\in [{\kern 3.5pt}[ f^{k}(x)]{\kern 3.5pt}]\). By definition of product update ⊗, clean maps and the fact that x is connected, \(g_{k}^{1}(h)R^{*}g_{k}^{1}(h^{\prime })\). By definition of \((\mathcal {H},\underline {H})\) it follows that h^{∗}h^{′}.
 Precondition Describable. :

For arbitrary e ∈ E, let δ_{e} = pre(σ_{e}). Recall that by definition of ⊗, \(\forall k\in \mathbb {N}\), \((g_{k}^{1}(h),\sigma _{e})\in [{\kern 3.5pt}[ f^{k+1}(x)]{\kern 3.5pt}]\) iff \(g_{k}^{1}(h)\models \mathsf {pre}(\sigma _{e})\) and σ_{e} ∈Σ^{k+ 1} (∗). Assume ∃h^{′}∈ H : h^{′}e ∈ H and let h ∈ H such that h^{′}^{∗}h.
⇒: Assume for some \(k\in \mathbb {N}\) that (H_{k}, h)⊧δ_{e}. Then \(g_{k}^{1}(h)\models \mathsf {pre}(\sigma _{e})\). By assumption, σ_{e} ∈Σ_{k+ 1}. By (∗), thus \((g_{k}^{1}(h),\sigma _{e})\in [{\kern 3.5pt}[ f^{k+1}(x)]{\kern 3.5pt}]\). Therefore, he ∈ H.
⇐: Assume he ∈ H. Then for some \(k\in \mathbb {N}\), \(g_{k+1}^{1}(he)=(g_{k}^{1}(h),\sigma _{e})\in [{\kern 3.5pt}[ f^{k+1}(x)]{\kern 3.5pt}]\). By (∗), \(g_{k}^{1}(h)\models {{\mathsf {pre}}}(\sigma _{e})\). And thus h⊧δ_{e}.
 Precondition Describable. :

For arbitrary he ∈ H (in specific for some \(k\in \mathbb {N}\), he ∈ H_{k+ 1}) let \(\delta _{D_{e}}=\mathsf {post}(\sigma _{e})\) where D_{e} = D_{1} ∪ D_{2}, for p, q ∈Φ such that D_{1} = {p : h∉V (p), he ∈ V (p)} and D_{2} = {¬q : h ∈ V (q), he∉V (q)}.
Consider an arbitrary p ∈ D_{1}. Then by definition of the generated ETL model \((\mathcal {H},\underline {H})\), \(g_{k}^{1}(h)\not \in [{\kern 3.5pt}[ p]{\kern 3.5pt}]_{k}\) and \(g_{k+1}^{1}(he)\in [{\kern 3.5pt}[ p]{\kern 3.5pt}]_{k+1}\) (∗). By construction, \(g_{k+1}^{1}(he)=(g_{k}^{1}(h),\sigma )\) for some σ ∈∇_{k+ 1} (∗∗). By definitions of \((\mathcal {H},\underline {H})\) and ⊗, \((g_{k}^{1}(h),\sigma )\in [{\kern 3.5pt}[ p]{\kern 3.5pt}]_{k+1}\) iff post(σ)⊧p. Then, by (∗) and (∗∗), post(σ)⊧p. As p ∈ D_{1} was arbitrary, post(σ)⊧D_{1}.
The argument for post(σ)⊧D_{2} is identical. Conclude that post(σ)⊧D_{e} and thus \(\delta _{D_{e}}\models D_{e}\).
 Point Bisimulation Invariance. :

Let arbitrary \(C(\mathcal {H}h)\underline {h}=(H_{k},\underline {h})\) and \(C(\mathcal {H}h^{\prime })\underline {h^{\prime }}=(H_{l},\underline {h^{\prime }})\) be such that . Hence (∗).
Further, assume for arbitrary h ∈ H_{k} and h^{′}∈ H_{l} that .
⇒: Assume he ∈ H. By construction of g and definition of clean maps, both H_{k} and H_{l} are connected components, i.e, ∀h, h^{′}∈ H_{k} : h^{∗}h^{′} and idem for H_{l}. By the HennessyMilner Theorem (see Section 7), it follows that h and h^{′} satisfy exactly the same modal formulas. Hence, by construction of g and the definition of \(\mathcal {H}\), \(g_{k}^{1}(h)\) and \(g_{l}^{1}(h^{\prime })\) satisfy exactly the same modal formulas as well. Now as he ∈ H, \(g_{k+1}^{1}(he)\in \left [{\kern 3.5pt}[ f^{k+1}(x)\right ]{\kern 3.5pt}] \) and thus \(g_{k}^{1}(h)\models \mathsf {pre}(\sigma _{e})\). Hence \(g_{l}^{1}(h^{\prime })\models \mathsf {pre}(\sigma _{e})\) (∗∗). By (∗) and (∗∗), it follows that \((g_{l}^{1}(h^{\prime }),\sigma _{e})\in \left [{\kern 3.5pt}[ f^{l+1}(x)\right ]{\kern 3.5pt}] \). Hence h^{′}e ∈ H.
⇐: By the same argument, h^{′}e ∈ H implies he ∈ H.
This concludes the proof of Proposition 2. □
Proposition 3
Not all saturated ETL models generated by DEL dynamical systems are Component Collection Describable.
Proof
Let Ms and f be as in Fig. 7 (cf. [56]) and X be the orbit of f from Ms. Then the ETL model generated by the DEL dynamical system (X, f) from initial model Ms is not Component Collection Describable.
Consider A = {f^{n}(Ms) : n is even}. There does not exist a φ such that for all x ∈ X, x⊧φ iff x ∈ A: Assume the modal depth of φ is k. Let m > k. Then f^{m}(Ms)⊧φ iff f^{m+ 1}(Ms)⊧φ as two such models will not differ in the first m + 1 relational steps from the point. Hence the ETL model generated by (X, f) from Ms is not Component Collection Describable. □
Proposition 4
If \((\mathcal {H},\underline {H})\) is a saturated ETL model that satisfies all eight properties of Section 4.2, then there exists a pointed DEL dynamical system that generates \((\mathcal {H},\underline {H})\).
Proof
Proposition 4 is shown by constructing a DEL dynamical system (X, f) with f the clean map of a Xdeterministic multipointed action model and an initial Kripke model x ∈ X such that the saturated ETL model \((\mathcal {H}^{\prime },\underline {H}^{\prime })=(E^{\prime },H^{\prime },\mathtt {R}^{\prime },V^{\prime },\underline {H}^{\prime })\) generated by (X, f) from x is ETL isomorphic to \((\mathcal {H},\underline {H})\). The latter is shown by induction on len(h) of \(h\in \mathcal {H}\) for a map γ^{∗} : E→E^{′}. As in Def. 5, for h = e_{0}...e_{n}, write γ^{∗}(h) := γ^{∗}(e_{0})...γ^{∗}(e_{n}).
As \((\mathcal {H},\underline {H})\) satisfies property Connected TimeSteps, the ETL model is a saturated component branch. To emphasize this, in this proof \((\mathcal {H},\underline {H})\) is written \((\mathcal {H}_{\mathbf {b}},\underline {H})\).
 1.
Initial Kripke model
To obtain a practical and consistent naming of states, the initial Kripke model x = ([[x]], R, [[⋅]], s) is set to be a renaming of the initial component of b: Let \([{\kern 3.5pt}[ x]{\kern 3.5pt}]=\{\sigma _{e}:e\in \mathbf {b}_{0}\underline {h}\}\). For the relations and valuation of the initial model, simply copy over the relations and valuation from the initial component of b: For all \(i\in \mathcal {A}\), let \(\sigma _{e}R_{i}\sigma _{e^{\prime }}\) iff e_{i}e^{′}, and for all p ∈Φ, let σ_{e} ∈ [[p]] iff e ∈ V (p). Finally, let the point of x be the copy of the point of \(\mathbf {b}_{0}\underline {h}\): Let \(s=\sigma _{\underline {h}}\).
 2.
Constructing (X, f)
To define the DEL dynamical system (X, f), first construct a multipointed action model . In words, construct so that for each timestep b_{k} of the component branch b, Γ contains a designated action \(\underline {\sigma }_{k}\) connected to a set of actions [[Σ_{k}]] ⊆ [[Σ]] such that the single pointed action model \({\Sigma }^{\prime }{{~}_{\{\underline {\sigma }_{k}\}}}\) obtained from restricting Σ to [[Σ_{k}]] produces the equivalent of \(\mathbf {b}_{k+1}\underline {h}\) from the Kripke modelequivalent of \(\mathbf {b}_{k}\underline {h}\). In the precondition of \(\underline {\sigma }_{k}\), include a formula \(\delta _{\mathbf {b}_{k}\underline {h}}\) characterizing \(\mathbf {b}_{k}\underline {h}\). As \((\mathcal {H}_{\mathbf {b}},\underline {H})\) is Component Collection Describable by assumption, such a formula exists.
Formally, construct piecewise as follows: Let \(\mathbf {b}_{k}\underline {h}\) and \(\mathbf {b}_{k+1}\underline {h}\) be given. Σ_{k+ 1}σ is constructed such that C(f^{k}(x) ⊗Σ_{k}σ)s^{′} mirrors the structure of \(\mathbf {b}_{k+1}\underline {h}\):
Let the singlepointed action model \({\Sigma }_{k+1}\underline {\sigma }_{k+1}\) be ([[Σ_{k+ 1}]], R_{k+ 1}, pre_{k+ 1}, \(\mathsf {post}_{k+1},\underline {\sigma }_{k+1})\), given by
with δ_{e} and \(\delta _{\mathbf {b}_{k}\underline {h}}\) given by Precondition Describable and Component Collection Describable, respectively.
 \(\mathsf {post}(\sigma _{e})=\delta _{D_{e}}\) :

as given Postcondition Describable.
Let the multipointed action model be given by, for , and \({\Gamma }={\bigcup _{k:\mathbf {b}_{k}\in \mathbf {b}}\{\underline {\sigma }_{k}\}}\). This is welldefined: For pre and post, this follows from Precondition Describable and Postcondition Describable.
Let X be the closure of {x} under the operation . On X, is guaranteed to be deterministic as any two characteristic formulas \(\delta _{\mathbf {b}_{k}\underline {h}}\) and \(\delta ^{\prime }_{\mathbf {b}_{k^{\prime }}\underline {h^{\prime }}}\) are not simultaneously satisfiable. Finally, let f be the clean map of on X. Then (X, f) is a DEL dynamical system with x ∈ X.
 3.
Constructing the Isomorphism
Let \((\mathcal {H}^{\prime },\underline {H}^{\prime })=(E^{\prime },H^{\prime },\mathtt {R}^{\prime },V^{\prime },\underline {H}^{\prime })\) be the saturated ETL model generated by (X, f) from x. Define the two mappings: γ^{‡} : E→[[Σ]] with γ^{‡}(e) = σ_{e} and γ : [[Σ]]→E^{′} for γ(σ) = e_{σ} cf. Def. 6. From these, define γ^{∗} : E→E^{′} as γ^{∗} := γ ∘ γ^{‡}.
Define subsets of E based on history length: For all \(k\in \mathbb {N}\) let E_{0} = {e : e ∈b_{0}} and E_{k+ 1 ≥ 1} = {e : h ∈b_{k}, he ∈b_{k+ 1}}⊆ E. Let \(E^{\prime }_{k},k\in \mathbb {N},\) be given mutatis mutandis. Let \(\gamma _{k}^{\dagger }:E_{k}\longrightarrow [{\kern 3.5pt}[{\Sigma }_{k}]{\kern 3.5pt}]\), \(\gamma _{k}:[{\kern 3.5pt}[{\Sigma }_{k}]{\kern 3.5pt}]\longrightarrow E_{k}^{\prime }\) and \(\gamma _{k}^{*}:E_{k}\longrightarrow E_{k}^{\prime }\) be the restrictions of γ^{‡}, γ and γ^{∗} to E_{k} × [[Σ_{k}]], \([{\kern 3.5pt}[{\Sigma }_{k}]{\kern 3.5pt}]\times E^{\prime }_{k}\) and \(E_{k}\times E^{\prime }_{k}\), respectively. Then, of course, \(\gamma ^{*}=\bigcup _{k\in \mathbb {N}}\gamma _{k}^{*}\), whereby \(\gamma ^{*}(e)=e^{\prime }_{\sigma _{e}}\). By induction on len(h), \(h\in \mathcal {H}\), it is now shown that γ^{∗} is an ETL isomorphism.
Claim: The mapγ^{∗}is a bijection. By the construction of γ^{‡} and γ, each \(\gamma _{k}^{*}\) is an injection: if e ≠ e^{′}, then \(\gamma _{k}^{*}(e)\neq \gamma _{k}^{*}(e^{\prime })\). By construction, it is also guaranteed that \(\gamma _{k}^{*}\) is a surjection: \(\forall e^{\prime }\in E^{\prime }\exists e\in E:\gamma _{k}^{*}(e)=e^{\prime }\). Hence for each \(k\in \mathbb {N}\), \(\gamma _{k}^{*}\) is a bijection.
Furthermore, γ^{∗} is a total map: if e ∈ E_{k} ∩ E_{m}, then \(\gamma _{k}^{\dagger }(e)=\gamma _{m}^{\dagger }(e)\) (i.e., σ_{e} ∈ [[Σ_{k}]] ∩ [[Σ_{m}]]) and γ_{k}(σ_{e}) = γ_{m}(σ_{e}). Thus γ ∘ γ^{‡}(e) is welldefined and in \(E_{k}^{\prime }\cap E_{m}^{\prime }\).
Finally, γ^{∗} inherits injectivity and surjectivity from its restrictions. Hence, the map γ^{∗} is a bijection.
Claim: The mapγ^{∗}is an ETL isomorphism. The claim is shown by 4 inductive subproofs.
 1)
Domain and Temporal Structure.
Base. Let h ∈ H_{0}. This is the case iff γ^{‡}(h) ∈ f^{0}(x) (by construction of initial Kripke model) iff \(\gamma \circ \gamma ^{\dagger }(h)\in H^{\prime }_{0}\) (by Def. 6).
Step. It is shown that he ∈ H_{k+ 1} iff \(\gamma ^{*}(he)\in H^{\prime }_{k+1}\).
⇒: Assume he ∈ H_{k+ 1}. Then h ∈ H_{k}. By the induction hypothesis, γ^{‡}(h) ∈ [[f^{k}(x)]]. By construction of Σ_{k+ 1}, γ^{‡}(e) ∈ [[Σ_{k+ 1}]]. By the same construction and Precondition Describable, γ^{‡}(h)⊧pre(σ_{e}). Hence (γ^{‡}(h), γ^{‡}(e)) ∈ [[f^{k+ 1}(x)]]. By Def. 6, in particular the construction of H_{k+ 1}, \(\gamma ((\gamma ^{\dagger }(h),\gamma ^{\dagger }(e)))\in H^{\prime }_{k+1}\).
⇐: Assume \(\gamma ((\gamma ^{\dagger }(h),\gamma ^{\dagger }(e)))\in H^{\prime }_{k+1}\). Then (γ^{‡}(h), γ^{‡}(e)) ∈ [[f^{k+ 1}(x)]] by Def. 6, so γ^{‡}(h) ∈ [[f^{k}(x)]] and γ^{‡}(e) ∈ [[Σ_{k+ 1}]]. By the induction hypothesis, h ∈ H_{k}. If he∉H_{k+ 1}, a contradiction is reached: pre(γ^{‡}(e)) is satisfied by exactly those γ^{‡}(h) ∈ [[f^{k}(x)]] such that (γ^{‡}(h), γ^{‡}(e)) ∈ [[f^{k+ 1}(x)]] – by the construction of action models in this proof and as \((\mathcal {H}_{\mathbf {b}},\underline {H})\) is Precondition Describable. So he ∈ H_{k+ 1}.
 2)
Epistemic relations.
Base. It follows by construction of initial Kripke model and Def. 6.
Step. It is shown that ∀he, h^{′}e^{′}∈b_{k+ 1}, he_{i}h^{′}e^{′} iff \(\gamma ^{*}(he)\mathtt {R}^{\prime }_{i}\gamma ^{*}(h^{\prime }e^{\prime })\).
⇒: Assume that he_{i}h^{′}e^{′}. By Perfect Recall, h_{i}h^{′}. By the induction hypothesis, γ^{‡}(h)R_{i}γ^{‡}(h^{′}). By construction of Σ_{k+ 1}, γ^{‡}(e)R_{i}γ^{‡}(e^{′}). By definition of ⊗, (γ^{‡}(h), γ^{‡}(e))R_{i}(γ^{‡}(h^{′}), γ^{‡}(e^{′})). By Def. 6, γ((γ^{‡}(h), γ^{‡}(e)))_{i}γ((γ^{‡}(h^{′}), γ^{‡}(e^{′}))).
⇐: Assume γ((γ^{‡}(h), γ^{‡}(e)))_{i}γ((γ^{‡}(h^{′}), γ^{‡}(e^{′}))). By Def. 6, (γ^{‡}(h), γ^{‡}(e))R_{i}(γ^{‡}(h^{′}), γ^{‡}(e^{′})). By definition of ⊗, both γ^{‡}(h)R_{i}γ^{‡}(h^{′}) and γ^{‡}(e)R_{i}γ^{‡}(e^{′}). So by construction of Σ_{k+ 1}, ∃h_{1}e, h_{2}e^{′}∈b_{k+ 1} : h_{1}e_{i}h_{2}e^{′}. By the induction hypothesis, h_{i}h^{′}. Further, note that ∀h, h^{′}∈b_{k} : h^{∗}h^{′}. Hence, by Local No Miracles, he_{i}h^{′}, e^{′}.
 3)
Valuation.
Base. It follows by construction of initial Kripke model and Def. 6.
Step. It is shown that ∀he ∈b_{k+ 1}, he ∈ V (p) iff γ(he) ∈ V^{′}(p).
⇒: Assume he ∈ V (p) . Either i) h ∈ V (p) or ii) h∉V (p). If i), then by the induction hypothesis, γ ∘ γ^{‡}(h) ∈ V^{′}(p). By construction of Σ_{k+ 1}, post(γ^{‡}(e))⊮¬p. Hence (γ^{‡}(h), γ^{‡}(e)) ∈ [[p]]_{k+ 1}. By Def. 6, γ((γ^{‡}(h), γ^{‡}(e))) ∈ V^{′}(p). If ii), then by the induction hypothesis, γ^{‡}(h)∉ [[p]]_{k}. As \((\mathcal {H}_{\mathbf {b}},\underline {H})\) is Postcondition Describable, by construction of Σ_{k+ 1}, post(γ^{‡}(e))⊧p. Thus (γ^{‡}(h), γ^{‡}(e)) ∈ [[p]]_{k+ 1}. By Def. 6, γ((γ^{‡}(h), γ^{‡}(e))) ∈ V^{′}(p).
⇐: Assume γ((γ^{‡}(h), γ^{‡}(e))) ∈ V^{′}(p). Then (γ^{‡}(h), γ^{‡}(e)) ∈ [[p]]_{k+ 1} by Def. 6. Again, either i) γ^{‡}(h) ∈ [[p]]_{k} or ii) γ^{‡}(h)∉ [[p]]_{k}. If i), then by the induction hypothesis, h ∈ V (p). For a contradiction, suppose he∉V (p). Then post(γ^{‡}(e))⊧¬p. But by construction of Σ_{k+ 1}, post(γ^{‡}(e))⊮¬p. This is a contradiction. Hence he ∈ V (p). If ii), then by the definition of ⊗, post(γ^{‡}(e))⊧p. By the induction hypothesis, h∉V (p). If it was the case that he∉V (p), then post(γ^{‡}(e))⊮p. Contradiction. Thus he ∈ V (p).
 4)
Points.
Base. It follows by construction of initial Kripke model and Def. 6.
Step. It is shown that \(\underline {he}\in \underline {H}_{k+1}\) iff \(\gamma ^{*}(\underline {he})\in \underline {H}_{k+1}^{\prime }\). Let f^{k}(x) = Nt and f^{k+ 1}(x) = Ms.
⇒: Assume \(\underline {he}\in \underline {H}_{k+1}\). By the induction hypothesis, \(\gamma ^{\dagger }(\underline {h})=t\). By saturation of b_{k}, \(\exists e\in E:\underline {h}e\in \mathbf {b}_{k+1}\cap \underline {H}\). By construction of Σ_{k+ 1}, γ^{‡}(e) ∈ [[Σ_{k+ 1}]] and, as \((\mathcal {H}_{\mathbf {b}},\underline {H})\) is Precondition Describable, \(\gamma ^{\dagger }(\underline {h})\models \mathsf {pre}(\gamma ^{\dagger }(e))\). By construction of f, f^{k+ 1}(x) = C(Nt ⊗ (Σ_{k+ 1}, γ^{‡}(e))) = C(N ⊗Σ_{k+ 1}, (t, γ^{‡}(e))) = Ms. By Def. 6, \(\gamma (s)=\gamma (\gamma ^{\dagger }(\underline {he}))\in \underline {H}^{\prime }_{k+1}\).
⇐: Assume \(\gamma (\gamma ^{\dagger }(\underline {he}))\in \underline {H}^{\prime }_{k+1}\). By Def. 6, \(f^{k+1}(x)=(C(N\otimes {\Sigma }_{k+1}),(t,\gamma ^{\dagger }(\underline {e})))\). By induction hypothesis, \(\gamma ^{\dagger 1}(t)=h\in \underline {H}_{k}\). By construction of the Action Model, \(\gamma ^{\dagger }(\underline {e})=\gamma ^{1}(e)\) for e such that \(\exists h^{\prime }:h^{\prime }e\in \underline {H}_{k+1}\) and \(h^{\prime }\in \underline {H}_{k}\). As points in H_{m} are unique for all m by Def. 7, h^{′} must be h. Thus \(he\in \underline {H}_{k+1}\).
This concludes the proof of Proposition 4.^{Footnote 1}□
Lemma 1
If there exists an imagefinite and concluding pointed DEL dynamical system that generates \((\mathcal {H},\underline {H})\), then \((\mathcal {H},\underline {H})\) is imagefinite and concluding.
Proof
As the DEL dynamical system is imagefinite, for all \(k\in \mathbb {N}\) with f^{k}(x) = Ms, R_{i} is imagefinite (for all i ∈ I). The construction of the generated ETL model (see Definition 6) ensures that for all H_{k}, _{i} is imagefinite (for all i ∈ I,). Hence \((\mathcal {H},\underline {H})\) is imagefinite.
If the DEL dynamical system terminates, then there is a \(k\in \mathbb {N}\) such that f^{k}(x) is undefined. Let f^{k− 1}(x) = Ms and \(\gamma (s)=\underline {h}\). As f^{k}(x) is undefined, there is no σ such that γ(s)γ(σ) ∈ H. Hence, there is no e such that \(\underline {h}e\in H\). Note that by construction of \(\underline {{\underline {H}}}\), for all \(\underline {h}^{\prime }\in \underline {H}\): \(\underline {h}^{\prime }\sqsubseteq \underline {h}\). Thus, all \(\underline {h}^{\prime }\in \underline {H}\) are finite and hence \((\mathcal {H},\underline {H})\) concludes.
If the DEL dynamical system is periodic, f^{k}(x) = f^{k+m}(x) for some k ≥ 0 and m > 0. Let f^{k}(x) = Ms and f^{k+m}(x) = M^{′}s^{′}, and let \(\gamma (s)=\underline {h}\) and \(\gamma (s^{\prime })=\underline {h}^{\prime }\). By construction, \(\underline {h}\sqsubseteq \underline {h}^{\prime }\). From Ms = M^{′}s^{′} it follows that . Thus, all \(\underline {h}^{\prime \prime }\in \underline {H}\) such that \(\underline {h}^{\prime \prime }\sqsubseteq \underline {h}\) are repeating. Furthermore, note that for all \(n\in \mathbb {N}\), f^{k+n}(x) = f^{k+m+n}(x). Now for arbitrary \(n\in \mathbb {N}\), let f^{k+n}(x) = M^{n}s^{n} and f^{k+m+n}(x) = M^{n′}s^{n′}, and let \(\gamma (s^{n})=\underline {h}^{n}\) and \(\gamma (s^{n\prime })=\underline {h}^{n\prime }\). By same argument as above, \(\underline {h}^{n}\) is repeating. As n is arbitrary, all \(\underline {h}^{\prime \prime }\in \underline {H}\) such that \(\underline {h}^{\prime \prime }\sqsupseteq \underline {h}\) are repeating. Thus, all \(\underline {h}^{\prime \prime }\in \underline {H}\) are repeating and hence \((\mathcal {H},\underline {H})\) concludes. □
Lemma 2
If a saturated ETL model \((\mathcal {H},\underline {H})\) is imagefinite, concluding and satisfies Connected TimeSteps, then \((\mathcal {H},\underline {H})\) is Component Collection Describable.
Proof
Let \((\mathcal {H},\underline {H})\) be an imagefinite and concluding saturated ETL model. Set \(\mathcal {B}:=\{C(\mathcal {H}h):h\in \underline {H}\}\), the set of all connected components in \((\mathcal {H},\underline {H})\). As all \(C(\mathcal {H}h)\in \mathcal {B}\) are imagefinite, by the HennessyMilner Theorem (see Section 7), for each pair \(\underline {h}_{1},\underline {h}_{2}\in \underline {H}\) if , there exists a formula \(\varphi _{\underline {h}_{1},\underline {h}_{2}}\) distinguishing between \(\underline {h}_{1}\) and \(\underline {h}_{2}\): \(\underline {h}_{1}\models \varphi _{\underline {h}_{1},\underline {h}_{2}}\) while \(\underline {h}_{2}\not \models \varphi _{\underline {h}_{1},\underline {h}_{2}}\).
Let be the equivalence class . Then, since \((\mathcal {H},\underline {H})\) is concluding, the set is finite. Therefore, the conjunction is welldefined for any \(\underline {h}_{1}\in \underline {H}\). This conjunction distinguishes \(\underline {h}_{1}\) from any point in \(\underline {H}\) that is not bisimilar to \(\underline {h}_{1}\). Denote this formula \(\varphi _{\underline {h}{}_{1}}\).
Moreover, as is finite, all sets \(A\subseteq \underline {H}\) for which \(\underline {h}\in A\) and implies \(\underline {h}^{\prime }\in A\) are finite. Hence, for any such A, the disjunction \(\bigvee _{\underline {h}_{1}\in A}\varphi _{\underline {h}{}_{1}}\) is welldefined. This disjunction distinguishes the connected components in A from those not in A. □
Lemma 3
Let {(X_{j}, f_{j})}_{j ∈ J} be minimal in generating \(\mathcal {H}\) and let (X_{j}, f_{j}, x_{j}) generate the saturated ETL model \((\mathcal {H}_{j},\underline {H}_{j})\). Then \((\mathcal {H}_{j},\underline {H}_{j})\) is the component branch submodel for some saturated component branch \((\mathbf {b},\underline {H})\) of \(\mathcal {H}\).
Proof
Let \(\mathcal {H}=(E,H,\mathtt {R},V)\) be generated by {(X_{j}, f_{j})}_{j ∈ J}. Let b_{j} be the sequence of connected components \(\mathbf {b}_{j,k}=C(\mathcal {H},h)\) for all \(h_{k}\in \underline {H}_{j}\), ordered on history length k. Suppose for proof by contradiction that b_{j} is both nonmaximal and finite (cf. Def. 7). As b_{j} is finite, there is a \(k\in \mathbb {N}\) such that for all h ∈b_{j, k} there is no e ∈ E such that he ∈b_{j, k+ 1}. However, as b_{j} is nonmaximal, there is a h ∈b_{j, k} and some e ∈ E such that \(\underline {h}e\in H\). This implies that there is an ETL model \((\mathcal {H}_{j^{\prime }},\underline {H}_{j^{\prime }})\) that extends \((\mathcal {H}_{j},\underline {H}_{j})\). Hence there is a DEL dynamical system \((X_{j^{\prime }},f_{j^{\prime }})\), j^{′}∈ J, such that \({f_{j}^{n}}(x_{j})=f_{j^{\prime }}^{n}(x_{j})\) for all n ≤ k − 1. Hence, (X_{k}, f_{k}) is redundant in generating \(\mathcal {H}\), contradicting the assumption that {(X_{j}, f_{j})}_{j ∈ J} is minimal. Thus, \((\mathbf {b}_{j},\underline {H}_{j})\) is a saturated component branch that gives \((\mathcal {H}_{j},\underline {H}_{j})\). □
Theorem 2
Let an imagefinite and concluding ETL model \(\mathcal {H}\) be given. \(\mathcal {H}\) is generatable up to ETL isomorphism by a family of imagefinite and concluding pointed DEL dynamical systems, if, and only if, there exists a saturation of each component branch b of \(\mathcal {H}\) such that \((\mathcal {H}_{\mathbf {b}},\underline {H})\) satisfies all eight properties of Section 4.2.
Proof
Lefttoright: Suppose \(\mathcal {H}\) is generatable by a family of imagefinite and concluding DEL dynamical systems {(X_{j}, f_{j})}_{j ∈ J}. Let \((\mathcal {H}_{j},\underline {H}_{j})\) be the saturated ETL model generated by (X_{j}, f_{j}) (cf. Def. 6). By Lemma 3, \((\mathcal {H}_{j},\underline {H}_{j})\) is a component branch submodel of \(\mathcal {H}\) for some saturated component branch \((\mathbf {b},\underline {H}_{k})\) of \(\mathcal {H}\). As \((\mathcal {H}_{j},\underline {H}_{j})\) was generated by an imagefinite and concluding DEL dynamical system, the saturation \(\underline {H}_{j}\) of \(\mathcal {H}_{j}\) makes \((\mathcal {H}_{j},\underline {H}_{j})\) satisfy all 8 properties of Section 4.2 by Proposition 2 and Lemma 2.
Righttoleft: Let B be the set of all component branches of \(\mathcal {H}\) and assume that for each b ∈ B, there exists a saturation such that \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) satisfies all eight properties of Section 4.2. As \(\mathcal {H}\) is imagefinite and concluding, also each \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) is imagefinite and concluding. By the constructions in the proof of Prop. 4, it follows that each \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) is generatable up to ETL isomorphism by a DEL dynamical system that is imagefinite and concluding.
Let (X_{b}, f_{b}, x_{b}) be the pointed DEL dynamical system that generates \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) up to isomorphism, as given by the construction in the proof of Prop. 4. Let \((\mathcal {H}^{\prime }_{\mathbf {b}},\underline {H}^{\prime }_{\mathbf {b}})\) be the specific ETL model generated by (X_{b}, f_{b}, x_{b}), as given by Def. 6. It will now be shown that the union structure U_{B} = (E_{B}, H_{B}, _{B, i}, V_{B})_{i ∈ I} of \(\{(\mathcal {H}^{\prime }_{\mathbf {b}},\underline {H}^{\prime }_{\mathbf {b}})\}_{\mathbf {b}\in \mathbf {B}}\) is isomorphic to \(\mathcal {H}=(E,H,\mathtt {R}_{i},V)_{i\in I}\). To this end, the existence of a bijection g : H→H_{B} will be shown.
Let g_{b} be the isomorphism between \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) and \((\mathcal {H}^{\prime }_{\mathbf {b}},\underline {H}^{\prime }_{\mathbf {b}})\), guaranteed to exist by Prop. 4 and specifically given by 1) the construction of a DEL dynamical system from a saturated component branch of the proof of Prop. 4 and 2) the construction for generating an ETL model from a DEL dynamical system of Def. 6. Combining the statehistory naming schemes used in these two constructions yield g_{b} given by
The history names from \(\mathcal {H}_{\mathbf {b}}\) are hereby carried over as indices to the histories of \(\mathcal {H}^{\prime }_{\mathbf {b}}\).
Define the mapping g : H→H_{B} by g(h) = g_{b}(h) for \(h\in \mathcal {H}_{\mathbf {b}}\). This mapping is welldefined as either i) for exactly one b ∈B, \(h\in \mathcal {H}_{\mathbf {b}}\) (in which case g(h) is welldefined), or ii) if \(h\in \mathcal {H}_{\mathbf {b}}\) and \(h\in \mathcal {H}_{\mathbf {b^{\prime }}}\), then \(g_{\mathbf {b}}(h)=g_{\mathbf {b}^{\prime }}(h)\). The latter is ensured as history names are carried over in exactly the same way by g_{b} and \(g_{\mathbf {b}^{\prime }}\), by construction. Hence \(g=\bigcup _{\mathbf {b}\in \mathbf {B}}g_{\mathbf {b}}\) is a welldefined map. It is an injection: For h ∈ H and h^{′}∈ H, if h ≠ h^{′}, then g(h)≠g(h^{′}) by the unique naming convention of the maps g_{b}, b ∈B. It is also a surjection: it is welldefined and H = H_{B} as by construction \(H^{\prime }_{\mathbf {b}}=H{}_{\mathbf {b}}\) for all b ∈B, and \(H=\bigcup _{\mathbf {b}\in \mathbf {B}}H{}_{\mathbf {b}}\) and \(H_{\mathbf {B}}=\bigcup _{\mathbf {b}\in \mathbf {B}}H^{\prime }_{\mathbf {b}}\). Hence g is a bijection.
That g is also the sought ETL isomorphism follows as \((\mathcal {H}_{\mathbf {b}},\underline {H}_{\mathbf {b}})\) is isomorphic to \((\mathcal {H}^{\prime }_{\mathbf {b}},\underline {H}^{\prime }_{\mathbf {b}})\), for all b ∈B, cf. Prop. 4. This completes the proof. □
Lemma 4
The saturated ETL model properties Synchronicity, Perfect Recall and Postcondition Describable persist under ETL model union.
I.e.: Let \(\{(\mathcal {H}_{j},\underline {H}_{j})\}_{j\in J}\) be a countable set of saturated ETL models. If all \((\mathcal {H}_{j},\underline {H}_{j})\) satisfy either of the mentioned properties, then the (unsaturated) union structure U_{J} satisfies that property.
Proof
Synchronicity, Perfect Recall and Postcondition Describable persist because they are defined on histories which occur uniquely in the ETL forest, which makes them local by nature. Hence these properties are evaluated locally within a branch to ensure that there will be no conflicts when taking the union of different ETL submodels.□□
Property 5
Local No Miracles, Precondition Describable, Point Bisimulation Invariance and Connected TimeSteps do not persist under union.
Proof
Local No Miracles does not persist under union: Consider a family of two DEL dynamical systems (that each individually satisfy property Local No Miracles.) with multipointed action models f and g with equal initial Kripke model x = {h, h^{′}, h_{1}, h_{2}} where h_{i}h^{′} and h^{∗}h_{1} (by default, as initial models are connected), but disjoint Kripke models at the next level: f^{1}(x) = {he, h^{′}e^{′}} and g^{1}(x) = {h_{1}e, h_{2}e^{′}} where h_{1}e_{i}h_{2}e^{′} while not he_{i}h^{′}e^{′}. In this example, Local No Miracles fails because not he^{∗}h_{1}e.
Precondition Describable does not persist under union: Consider a family of two DEL dynamical systems (that each individually satisfy property Precondition Describable) with multipointed action models f and g with equal initial Kripke model x = {h, h^{′}} with h^{∗}h^{′} (by default as initial models are connected), but disjoint Kripke models at the next level: f^{1}(x) = {he} and g^{1}(x) = {h^{′}e}. Then it is possible that h^{′}⊧δ_{e} while h⊮δ_{e}, which breaks property Precondition Describable. It is left as an open question whether a suitably weakened version of Precondition Describable exists.
That Point Bisimulation Invariance is not preserved under union follows as the property is stated based on a saturation, but the union structure is unpointed, and hence unsaturated. If the union structure would be pointed with the set of points chosen as the union of the sets of points from the united ETL models, then the resulting set of points need not be a saturation, as the united ETL models may overlap, but have distinct points. In that case, the union structure would be “oversaturated”.
Connected TimeSteps does not persist under union as histories within the same timestep are no longer necessarily epistemically connected in a union of disconnected ETL submodels. □
Proposition 5
If an ETL model \(\mathcal {H}\) is generated by a family of pointed DEL dynamical systems (possibly neither imagefinite nor concluding), then \(\mathcal {H}\) satisfies Synchronicity, Perfect Recall, Postcondition Describable and Very Local No Miracles and Local Bisimulation Invariance.
Proof
That \(\mathcal {H}\) satisfies Synchronicity, Perfect Recall and Postcondition Describable follows from Prop. 2 and Lemma 4.
That \(\mathcal {H}\) satisfies Very Local No Miracles follows directly by the additional requirement compared to Local No Miracles that he^{∗}h_{1}e.
That \(\mathcal {H}\) satisfies Local Bisimulation Invariance follows as 1) any ETL model that satisfies Point Bisimulation Invariance also satisfies Local Bisimulation Invariance, 2) Local Bisimulation Invariance is a property local to a connected component, and 3) connected components remain untouched under union.□
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van Lee, H.S., Rendsvig, R.K. & van Wijk, S. Intensional Protocols for Dynamic Epistemic Logic. J Philos Logic 48, 1077–1118 (2019). https://doi.org/10.1007/s1099201909508w
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Keywords
 Dynamic epistemic logic
 Multiagent systems
 Protocols
 Epistemic temporal logic
 Dynamical systems