## Abstract

We extend the ‘topologic’ framework [13] with dynamic modalities for ‘topological public announcements’ in the style of Bjorndahl [5]. We give a complete axiomatization for this “Dynamic Topo-Logic”, which is in a sense simpler than the standard axioms of topologic. Our completeness proof is also more direct (making use of a standard canonical model construction). Moreover, we study the relations between this extension and other known logical formalisms, showing in particular that it is co-expressive with the simpler (and older) logic of interior and global modality [1, 4, 10, 14]. This immediately provides an easy decidability proof (both for topologic and for our extension).

A. Özgün—Acknowledges financial support from European Research Council grant EPS 313360.

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## Notes

- 1.
Indeed, the original paper [2] on “classical” (non-topological) APAL modality contained a similar attempt of converting an infinitary rule into an finitary rule. That was later shown to be flawed: the finitary rule was not sound for the APAL modality (though it is sound for effort)!

- 2.
For a general introduction to topology we refer to [7]. A

*topological space*\((X, \tau )\) consists of a non-empty set*X*and a “topology” \(\tau \subseteq {\mathcal P}(X)\), i.e. a family of subsets of*X*(called*open sets*) such that \(X, \emptyset \in \tau ,\) and \(\tau \) is closed under finite intersections and arbitrary unions. The complements \(X\setminus U\) of open sets are called*closed*. The collection \(\tau \) is called a*topology*on*X*and elements of \(\tau \) are called*open sets*. An open set containing \(x\in X\) is called an open neighborhood of*x*. The*interior*\( Int ( A)\) of a set \(A\subseteq X\) is the largest open set contained in*A*, i.e., \( Int ( A)=\bigcup \{U\in \tau \ | \ U\subseteq A\}\), while the closure \( cl ( A)\) is the smallest closed set containing*A*. A family \(\mathcal {B}\subseteq \tau \) is called a basis for a topological space \((X,\tau )\) if every non-empty element of \(\tau \) can be written as a union of elements of \(\mathcal {B}\). - 3.
We prefer to talk about “updates”, rather than public announcements, since our setting is single-agent: there is no “publicity” involved. The agent simply learns \(\varphi \) (and implicitly also learns that \(\varphi \) was learnable).

- 4.
In fact, the modality \( int \) can be defined in terms of the public announcement modality as \( int ( \varphi ):=\lnot [\varphi ] \bot \), thus, the language \(\mathcal {L}_{!K int }\) and its fragment \(\mathcal {L}_{!K}\) without the modality \( int \) are also co-expressive.

- 5.
Although Bjorndahl’s formulations of (R\(_!\)) and (R\(_{ int }\)) are unnecessarily complicated: the first is stated as \([\varphi ][\psi ]\chi \leftrightarrow [ int (\varphi ) \wedge [\varphi ] int (\psi ) ]\chi \), while the second as \([\varphi ] int (\psi ) \leftrightarrow \left( int (\varphi ) \rightarrow int ([\varphi ]\psi ) \right) \). It is easy to see that these are equivalent to our simpler formulations, given the other axioms.

- 6.
Indeed, this is because the satisfaction relation for epistemic scenarios in any pseudo-model that happens to be a topo-model agrees with the topo-model satisfaction relation.

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Baltag, A., Özgün, A., Vargas Sandoval, A.L. (2017). Topo-Logic as a Dynamic-Epistemic Logic. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_23

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