Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T20:12:41.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic topological logic of metric spaces

Published online by Cambridge University Press:  12 March 2014

David Fernández-Duque
David Fernández-Duque*
Affiliation:
Group for Logic, Language and Information, Universidad de Sevilla, Calle Camilo José Cela S/N, 41013 Seville, Spain, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Dynamic Topological Logic is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function.

In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.

We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 308 - 328
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Akin, E., The general topology of dynamical systems, Graduate Studies in Mathematics, American Mathematical Society, 1993.Google Scholar
[2]Artemov, S. N., Davoren, J. M., and Nerode, A., Modal logics and topological semantics for hybrid systems, Technical Report MSI 97-05, Cornell University, 1997.CrossRefGoogle Scholar
[3]Fernández-Duque, D., Dynamic topological completeness for ℝ2, Logic Journal of the IGPL, vol. 15 (2007), no. 1, pp. 77–107.CrossRefGoogle Scholar
[4]Fernández-Duque, D., Non-deterministic semantics for dynamic topological logic, Annals of Pure and Applied Logic, vol. 157 (2009), no. 2–3, pp. 110–121, Kurt Gödel Centenary Research Prize Fellowships.Google Scholar
[5]Fernández-Duque, D., Dynamic topological logic interpreted over minimal systems, Journal of Philosophical Logic, vol. 40 (2011), no. 6, pp. 767–804.CrossRefGoogle Scholar
[6]Folland, G., Real analysis: Modern techniques and their applications, Wiley-Interscience, 1999.Google Scholar
[7]Kremer, P., The modal logic of continuous functions on the rational numbers, Archive for Mathematical Logic, vol. 49 (2010), no. 4, pp. 519–527.CrossRefGoogle Scholar
[8]Kremer, P. and Mints, G., Dynamic topological logic, Annals of Pure and Applied Logic, vol. 131 (2005), pp. 133–158.CrossRefGoogle Scholar
[9]Lichtenstein, O. and Pnueli, A., Propositional temporal logics: Decidability and completeness, Logic Jounal of the IGPL, vol. 8, no. 1.Google Scholar
[10]Mints, G. and Zhang, T., Propositional logic of continuous transformations in cantor space, Archive for Mathematical Logic, vol. 44 (2005), pp. 783–799.CrossRefGoogle Scholar
[11]Nogin, M. and Nogin, A., On dynamic topological logic of the real line, Journal of Logic and Computation, vol. 18 (2008), no. 6, pp. 1029–1045, doi:10.1093/togcom/exn034.CrossRefGoogle Scholar
[12]Sierpinski, W., Sur une propriété topologique des ensembles dénombrables denses en soi, Fundamenta Mathematicae, vol. 1 (1920), pp. 11–16.CrossRefGoogle Scholar
[13]Slavnov, S., Two counterexamples in the logic of dynamic topological systems, Technical Report TR-2003015, Cornell University, 2003.Google Scholar
[14]Slavnov, S., On completeness of dynamic topological logic, Moscow Mathematics Journal, vol. 5 (2005), no. 2, pp. 477–492.CrossRefGoogle Scholar
[15]Tarski, A., Der Aussagenkalkül und die Topologie, Fundamenta Mathematica, vol. 31 (1938), pp. 103–134.CrossRefGoogle Scholar
[16]van Mill, J., The infinite-dimensional topology of function spaces, Elsevier Science, Amsterdam, 2001.Google Scholar