×

Dynamic topological logic of metric spaces. (English) Zbl 1241.03020

Summary: Dynamic Topological Logic (\(\mathcal {D T L}\)) is a modal framework for reasoning about dynamical systems, that is, pairs \(\langle X,f\rangle \) where \(X\) is a topological space and \(f: X\rightarrow X\) 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 \(\mathbb Q\).
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.

MSC:

03B45 Modal logic (including the logic of norms)
03B80 Other applications of logic
54E35 Metric spaces, metrizability
54E50 Complete metric spaces
54H20 Topological dynamics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] E. Akin The general topology of dynamical systems , Graduate Studies in Mathematics, American Mathematical Society,1993.
[2] S. N. Artemov, J. M. Davoren, and A. Nerode Modal logics and topological semantics for hybrid systems , Technical Report MSI 97-05, Cornell University,1997.
[3] D. Fernández-Duque Dynamic topological completeness for \(\mathbb{R}^2\) , Logic Journal of the IGPL , vol. 15(2007), no. 1, pp. 77-107. · Zbl 1126.03021
[4] —- 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. · Zbl 1168.03010
[5] —- Dynamic topological logic interpreted over minimal systems , Journal of Philosophical Logic , vol. 40(2011), no. 6, pp. 767-804. · Zbl 1243.03027
[6] G. Folland Real analysis: Modern techniques and their applications , Wiley-Interscience,1999. · Zbl 0924.28001
[7] P. Kremer The modal logic of continuous functions on the rational numbers , Archive for Mathematical Logic , vol. 49(2010), no. 4, pp. 519-527. · Zbl 1193.03040
[8] P. Kremer and G. Mints Dynamic topological logic , Annals of Pure and Applied Logic , vol. 131(2005), pp. 133-158. · Zbl 1067.03028
[9] O. Lichtenstein and A. Pnueli Propositional temporal logics: Decidability and completeness , Logic Jounal of the IGPL , vol. 8, no. 1. · Zbl 1033.03009
[10] G. Mints and T. Zhang Propositional logic of continuous transformations in cantor space , Archive for Mathematical Logic , vol. 44(2005), pp. 783-799. · Zbl 1103.03021
[11] M. Nogin and A. Nogin On dynamic topological logic of the real line , Journal of Logic and Computation , vol. 18(2008), no. 6, pp. 1029-1045, doi:10.1093/logcom/exn034. · Zbl 1162.03010
[12] W. Sierpinski Sur une propriété topologique des ensembles dénombrables denses en soi , Fundamenta Mathematicae , vol. 1(1920), pp. 11-16. · JFM 47.0175.03
[13] S. Slavnov Two counterexamples in the logic of dynamic topological systems , Technical Report TR-2003015, Cornell University,2003.
[14] —- On completeness of dynamic topological logic , Moscow Mathematics Journal , vol. 5(2005), no. 2, pp. 477-492. · Zbl 1091.03005
[15] A. Tarski Der Aussagenkalkül und die Topologie , Fundamenta Mathematica , vol. 31(1938), pp. 103-134. · Zbl 0020.33704
[16] J. van Mill The infinite-dimensional topology of function spaces , Elsevier Science, Amsterdam,2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.