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Modal Languages for Topology: Expressivity and Definability
, 2008
"... In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the GoldblattThomason definability theorem in terms of the well established firstorder top ..."
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In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the GoldblattThomason definability theorem in terms of the well established firstorder topological language Lt.
Dynamic Topological Logics Over Spaces with Continuous Functions
 ADVANCES IN MODAL LOGIC
, 2006
"... Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the ωreac ..."
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Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the ωreachability problem for lossy channel systems) that the dynamic topological logic over arbitrary topological spaces as well as those over R^n, for each n ≥ 1, are undecidable. Actually, we prove this result for the natural and expressive fragment of the full dynamic topological language where the topological operators cannot be applied to formulas containing the temporal eventuality. Using Kruskal’s tree theorem we also show that the formulas of this fragment that are valid in arbitrary topological spaces with continuous functions are recursively enumerable, which is not the case for spaces with homeomorphisms.
Propositional logic of continuous transformations in Cantor Space, to appear
 in Symposium on Mathematical Logic ’03 December 1719, Kobe, special issue in Annals of Mathematical Logic
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Nondeterministic Semantics for Dynamic Topological Logic
"... Dynamic Topological Logic (DT L) is a combination of S4, under its topological interpretation, and the temporal logic LT L interpreted over the natural numbers. DT L is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models ..."
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Dynamic Topological Logic (DT L) is a combination of S4, under its topological interpretation, and the temporal logic LT L interpreted over the natural numbers. DT L is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples 〈X, T, f, V 〉, where 〈X, T 〉 is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of DT L over spaces with continuous functions is recursively enumerable. We do this by defining alternative semantics for DT L. Under the standard semantics, DT L is not complete for Kripke frames. However, we introduce the notion of a nondeterministic quasimodel, where the function f is replaced by a binary relation g assigning to each world multiple temporal successors. We place restrictions on the successors so that the logic remains unchanged; under these alternative semantics, DT L becomes Kripkecomplete. We then apply modelsearch techniques to enumerate the set of all valid formulas. Key words: dynamic topological logic, spatial logic, temporal logic 1
A Construction Method for Modal Logics of Space
, 2004
"... I consider myself very fortunate for having the opportunity to work on this thesis under the supervision of Johan van Benthem and Dick de Jongh. Besides shedding a great deal (of very different!) light upon the problems contained in this thesis, they provided the encouragement and support needed to ..."
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I consider myself very fortunate for having the opportunity to work on this thesis under the supervision of Johan van Benthem and Dick de Jongh. Besides shedding a great deal (of very different!) light upon the problems contained in this thesis, they provided the encouragement and support needed to see this volume through to its completion. Thank you. Thanks to Nick Bezhanishvili and Yde Venema, for serving as members on my defense committee and for useful suggestions along the way. To Benedikt Löwe for the same, and for providing inspiring lecture courses, as well as leaving his door open for conversation. To Darko Sarenac for his helpful skepticism and generous hospitality in Palo Alto. And to Niels Molenaar for handling organizational matters right before I was to defend my thesis. In addition, I would like to thank Thomas, Be, Jill, Charles, Alexandra, Chunlai and Café Reibach for making my time in Amsterdam so enjoyable. This thesis is dedicated to my parents Terry and Therese and sister Teena. Without whom.
The modal logic of continuous functions on cantor space
, 2006
"... Let L be a propositional language with standard Boolean connectives plus two modalities: an S4ish topological modality � and a temporal modality �, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L by interpreting L in dynamic topological systems, i.e ..."
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Let L be a propositional language with standard Boolean connectives plus two modalities: an S4ish topological modality � and a temporal modality �, understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language L by interpreting L in dynamic topological systems, i.e. ordered pairs 〈X, f 〉, where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the ZhangMints result.