‘poisoning’ or colouring that state, i.e., forbidding passage through it, as in [81]. Issues with axiomatizing sabotage modal logic are presented in [22].

Another semantics for the sabotage modality, propagated by Carlos Areces and collaborators in [11, 61], interprets the sabotage modality as removing a pair from the accessibility relation in a model. We then get:

$$M_s \models \langle - \rangle \varphi \quad \text{iff} \quad \text{there is a } (t, u) \in R \text{ such that } M_s^{-tu} \models \varphi$$

where $M^{-tu} = M|R^{-tu}$, the restriction of $M$ to the relation $R^{-tu}$ defined as $R^{-tu} = R \setminus \{(t, u)\}$.

Instead of removing a pair from the relation, these authors also considered other relation-changing updates (and thus moving away from the motivating sabotage game to other applications). They also propose a modality adding a pair to the relation, also known as making a *bridge* between states:

$$M_s \models \langle + \rangle \varphi \quad \text{iff} \quad \text{there is a } (t, u) \notin R \text{ such that } M_s^{+tu} \models \varphi$$

where $M^{+tu}$ is as $M$ except with relation $R^+ = R \cup \{(t, u)\}$. Or we can change the direction of the arrow, known as a *swap* of the states involved in the arrow:

$$M_s \models \langle \pm \rangle \varphi \quad \text{iff} \quad \text{there is a } (t, u) \in R \text{ such that } M_s^{\pm tu} \models \varphi$$

where $M^{\pm tu}$ is as $M$ except with relation $R^{\pm tu} = R \setminus \{(t, u)\} \cup \{(u, t)\}$.

**Example 18** Consider the (anonymous) single-agent model $M$ consisting of states $\{s, t, u\}$ with $V(p) = \{u\}$ and $R = \{(s, t), (s, u)\}$, as visualized below. We also depict a number of updated models according to the sabotage, bridge, and swap semantics.

<figure><img src="image_11.png" /></figure>

For example, $M_s \models \Box p \lor \Box \neg p$, whereas $M_s \models \langle - \rangle \Box p$ because $M_s^{-st} \models \Box p$, but also $M_s \models \langle - \rangle \Box \neg p$ because $M_s^{-su} \models \Box \neg p$. Therefore $M_s \not\models [-] \Box p$. Also, $M_s \not\models \Diamond \Diamond \neg p$ but $M_s \models \langle + \rangle \Diamond \Diamond \neg p$ because $M_s^{+ut} \models \Diamond \Diamond \neg p$, or if you wish because $M_s^{+ts} \models \Diamond \Diamond \neg p$. Finally $M_s \models [\pm](\Box p \lor \Box \neg p)$ because $M_s^{\pm st} \models \Box p$ and (not depicted) $M_s^{\pm su} \models \Box \neg p$.

Sabotage, bridge and swap are all examples of relational change and such modal update logics go by the name of *relation-changing modal logics*. Note that the relational pair that is removed, added, or reversed is not selected according to a logical property, the update is semantic and not syntactic. Also note that it is one pair only, it is particular and not universal. Finally, the sabotage, swap, and bridge updates by Carlos Areces et al. are global updates (there is no relation between the removed pair and the actual state).
However, these authors also proposed local versions of sabotage, swap, and bridge. Johan van Benthem's original sabotage semantics [125] is also a local update: we remove a state different from the actual state. All updates in dynamic epistemic logic that we have seen in prior sections are global.

In such sabotage logics and related logics we can count. If you remove a state from the domain, the resulting domain contains one state less, so if you do this three times it contains three states less. We can therefore count how many states there are in the (finite) domain, just as in for example the graded modal logics of [64]. For example, in a three-state model, $[-]^3 \perp$ is true in the [125] semantics: you can twice remove a state different from the actual state, but not thrice. These logics with counting require more refined notions of bisimulation, because the size of a model is not an invariant for the usual notion of bisimulation. For example, a model is standardly bisimilar to two copies of it. But now no longer. A different notion of bisimulation for sabotage logic is proposed in [22], and for relation-changing logics in general in [12, 62], including for a variation of the $\langle - \rangle$ and $\langle + \rangle$ semantics above, namely where the removal of a pair $(t, u)$ is also defined if the pair was already absent, and the addition of a pair $(t, u)$ is also defined if it was already present.

In such logics we can also give names to the states in domain. If you remove a state from the domain different from the actual state, this allows you to refer to the actual state in the logical language, as in hybrid logics. In [13], relation-changing logics are modelled as hybrid logics.

When we can count worlds and give them names, the logics may lack (finitary) Hilbert-style axiomatizations and may be undecidable, which is indeed the case [22, 15, 14].

## 9.2 Dynamic epistemic logics and relation-changing logics

Let us now relate the domain-changing or relation-changing modal update logics to the dynamic epistemic logics presented in this survey. In relation-changing modal logics we can consider arbitrary iterations or fixpoints of the structural operations wherein a mere pair of the accessibility relation is changed [115, 129]. In this section we investigate to what extent such iterated transformations correspond to dynamic epistemic updates.

First consider refinement modal logic RML. Removing a pair from the accessibility relation of a model is a refinement of that model. Removing any finite number of pairs is also a refinement of that model. As is removing any infinite number of pairs of the relation, or the entire relation, which may be a limit or fixpoint (models need not have countable domains). It may be interesting to investigate on which class of models (all finite models?) we have that

$$ \langle \succeq \rangle \varphi \text{ is equivalent to } \langle - \rangle^* \varphi. $$

Similarly, the quantifier in simulation modal logic can be seen as arbitrary iteration of the bridge modality, such that it is envisageable for some model class that

$$ \langle \preceq \rangle \varphi \text{ is equivalent to } \langle + \rangle^* \varphi. $$
Now consider arrow update logics AUL and AUML, wherein pairs of the accessibility relation come with logical conditions: arrows with source or target formulas. In sabotage logic we delete pairs, whereas in AUL we retain pairs, but this is mere perspective. Retaining all pairs $(s, t)$ with source condition $\varphi$ and target condition $\psi$ in a relation $R_a$ (arrow $\varphi \to_a \psi$) is the same as deleting all pairs that satisfy $\neg\varphi$ at the source or $\neg\psi$ at the target (arrows $\neg\varphi \to_a \top$ and $\top \to_a \neg\psi$).

Assuming a single agent $a$, arbitrarily iterated removal of arrows $\top \to_a \neg p$ appears to correspond to the ‘arrow eliminating’ semantics for public announcement of [79, 94]. Whereas removing all pairs with different values for $p$ at source and target, so preserving all pairs satisfying $p \to_a p$ or $\neg p \to_a \neg p$, corresponds to the standard public announcement semantics, in the model refinement sense of [130]. What works for propositional variables need not work for arbitrary formulas. Public announcement of $\varphi$ is not merely iterated sabotage of arrows $\top \to_a \neg\varphi$, because of Moorean phenomena.

That public announcement can be viewed as iterated removal of states or arrows was suggested by Johan van Benthem. This seems not explicitly addressed in any of the relevant publications [129, 22, 131]. A ‘stepwise public announcement’ (one pair in the accessibility relation only) was proposed in [22] as a variation on the (standard) truthful public announcement semantics (Section 6.2, *Stepwise versions of DEL*, in [22]). In [131], public announcement and ‘stepwise removal’ (sabotage) are combined in a single logic, but there is no suggestion that one can be defined in terms of the other.

## 9.3 Open problems and further research

These are all very open research projects and not specific open problems. We list:

* A decidable and axiomatizable relation-changing modal logic

* A refinement modal logic with refinements as iterated sabotage

* A public announcement logic with announcements as iterated sabotage

## 10 Outlook

We have presented many different dynamic epistemic logics with quantification over information change: arbitrary public announcement logic APAL with quantification over announcements, arbitrary action model logic AAML with quantification over action models, arbitrary arrow update logic AAUL with quantification over arrow updates, and refinement modal logic RML with quantification over refinements. And then the many variations on each of those, with all different logical properties. Some corners of this spectrum are not yet well understood, in particular the various ways to obtain decidable APAL versions, the various simulation modal logics and how they relate to refinement modal logics, and the different complexities of very similar logics quantifying over epistemic actions, whether as action models, arrow updates, or refinements. Promising current developments are
the combination of group epistemic modalities with quantifiers, and the investigation of iterated epistemic actions or quantifiers in relation to epistemic planning. Further developments could involve incorporating features of relation-changing modal logics (sabotage, bridge, swap) with dynamic epistemic logics, and there are subjects beyond the scope of this survey such as succinctness, where we can expect logics with quantifiers to be more succinct than the same logic without. Applications to gossip protocols, security protocols, and epistemic planning are to be developed much further. Let a hundred flowers bloom.

## Acknowledgements

This investigation is a follow-up of a prior survey of logics with quantifiers over information change, entitled 'Quantifying Notes' [137], that also focussed on open problems. Some open problems in [137] have since then been resolved and are now presented as results, for example on GAL and CAL expressivity. Other open problems have come to the fore.

I acknowledge helpful comments and encouragement from: Thomas Ågotnes, Philippe Balbiani, Johan van Benthem, Thomas Bolander, Stéphane Demri, Raul Fervari, Saúl Fernández González, Tim French, Rustam Galimullin, Rob van Glabbeek, Davide Grossi, Wiebe van der Hoek, Sophie Pinchinat, and Igor Sedlár. This survey got under way while I was under lockdown in France in the same apartment complex as Hao Wu and Jinsheng Chen. Their support was invaluable and unforgettable.

Together with Thomas Bolander, Jan van Eijck and Ramanujam, in 2015 I organized a Lorentz Center workshop entitled *To Be Announced*, with, on the request of the organization, a subtitle *Synthesis of Epistemic Protocols* to avoid confusion. See:

https://www.lorentzcenter.nl/to-be-announced-synthesis-of-epistemic-protocols.html.

On various occasions I gave invited seminars entitled *To Be Announced*. “Can we at least make this into ‘*To Be Announced*’, with quotes?” “Please don’t.” The predictable confusion with publishers, supporting staff and events servers has always been hilarious and I hope it will be never-ending.

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