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Journal of Logic and Computation Advance Access originally published online on August 6, 2008
Journal of Logic and Computation 2008 18(6):1029-1045; doi:10.1093/logcom/exn034
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© The Author, 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

On Dynamic Topological Logic of the Real Line

Maria Nogin

Department of Mathematics, California State University, Fresno 5245 N Backer Ave, M/S PB 108 Fresno, CA 93740, USA.
E-mail: mnogin{at}csufresno.edu

Aleksey Nogin

HRL Laboratories, LLC, 3011 Malibu Canyon Rd, Malibu, CA 90265, USA.
E-mail: anogin{at}hrl.com

Received 20 May 2008.


  Abstract

This article explores the topological interpretations of the modal language with two modalities—{square}, which is interpreted as the interior operation and {bigcirc} (‘next’) which is interpreted as the pre-image operation for a continuous function. It is known that the {square}{bigcirc} logic S4C is complete with respect to topological interpretations in Rn for n≥2, yet it is incomplete with respect to topological interpretations in R. We focus on the logic Formula{square}{bigcirc}(R) of all the {square}{bigcirc} formulas that are sound with respect to topological interpretations in R. In this article we present two formulas in Formula{square}{bigcirc}(R)–S4C, and prove that they are sound in R and independent. We also establish that the previously known examples of formulas in Formula{square}{bigcirc}(R)–S4C are instances of a particular consequence of one of the two formulas presented.

Keywords: Dynamic topological logic; dynamic modal logic; S4C


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