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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- 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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