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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.
DOI 10.1007/s0015301001858 Mathematical Logic
"... The modal logic of continuous functions on the rational ..."
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0 Nonfinite axiomatizability of Dynamic Topological Logic
"... Dynamic topological logic (DT L) is a polymodal logic designed for reasoning about dynamic topological systems. These are pairs 〈X, f〉, where X is a topological space and f: X → X is continuous. DT L uses a language L which combines the topological S4 modality ✷ with temporal operators from linear t ..."
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Dynamic topological logic (DT L) is a polymodal logic designed for reasoning about dynamic topological systems. These are pairs 〈X, f〉, where X is a topological space and f: X → X is continuous. DT L uses a language L which combines the topological S4 modality ✷ with temporal operators from linear temporal logic. Recently, we gave a sound and complete axiomatization DTL ∗ for an extension of the logic to the language L ∗ , where ✸ is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known in the language L, although one proof system, which we shall call KM, was conjectured to be complete by Kremer and Mints. In this paper we show that, given any language L ′ such that L ⊆ L ′ ⊆ L ∗ , the set of valid formulas of L ′ is not finitely axiomatizable. It follows, in particular, that KM is incomplete.
Nonfinite axiomatizability of Dynamic Topological Logic
, 2014
"... Dynamic topological logic (DT L) is a polymodal logic designed for reasoning about dynamic topological systems. These are pairs 〈X, f〉, where X is a topological space and f: X → X is continuous. DT L uses a language L which combines the topological S4 modality with temporal operators from linear t ..."
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Dynamic topological logic (DT L) is a polymodal logic designed for reasoning about dynamic topological systems. These are pairs 〈X, f〉, where X is a topological space and f: X → X is continuous. DT L uses a language L which combines the topological S4 modality with temporal operators from linear temporal logic. Recently, I gave a sound and complete axiomatization DTL ∗ for an extension of the logic to the language L∗, where ♦ is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known over L, although one proof system, which we shall call KM, was conjectured to be complete by Kremer and Mints. In this paper we show that, given any language L ′ such that L ⊆ L ′ ⊆ L∗, the set of valid formulas of L ′ is not finitely axiomatizable. It follows, in particular, that KM is incomplete. 1