## Journal of Symbolic Logic

### Dynamic topological logic of metric spaces

David Fernández-Duque

#### Abstract

Dynamic Topological Logic (𝒟𝒯ℒ) 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.

#### Article information

Source
J. Symbolic Logic, Volume 77, Issue 1 (2012), 308-328.

Dates
First available in Project Euclid: 20 January 2012

https://projecteuclid.org/euclid.jsl/1327068705

Digital Object Identifier
doi:10.2178/jsl/1327068705

Mathematical Reviews number (MathSciNet)
MR2951643

Zentralblatt MATH identifier
1241.03020

#### Citation

Fernández-Duque, David. Dynamic topological logic of metric spaces. J. Symbolic Logic 77 (2012), no. 1, 308--328. doi:10.2178/jsl/1327068705. https://projecteuclid.org/euclid.jsl/1327068705

#### References

• E. Akin The general topology of dynamical systems, Graduate Studies in Mathematics, American Mathematical Society,1993.
• 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.
• D. Fernández-Duque Dynamic topological completeness for $\mathbb{R}^2$, Logic Journal of the IGPL, vol. 15(2007), no. 1, pp. 77–107.
• –––– 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.
• –––– Dynamic topological logic interpreted over minimal systems, Journal of Philosophical Logic, vol. 40(2011), no. 6, pp. 767–804.
• G. Folland Real analysis: Modern techniques and their applications, Wiley-Interscience,1999.
• P. Kremer The modal logic of continuous functions on the rational numbers, Archive for Mathematical Logic, vol. 49(2010), no. 4, pp. 519–527.
• P. Kremer and G. Mints Dynamic topological logic, Annals of Pure and Applied Logic, vol. 131(2005), pp. 133–158.
• O. Lichtenstein and A. Pnueli Propositional temporal logics: Decidability and completeness, Logic Jounal of the IGPL, vol. 8, no. 1.
• G. Mints and T. Zhang Propositional logic of continuous transformations in cantor space, Archive for Mathematical Logic, vol. 44(2005), pp. 783–799.
• 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.
• W. Sierpinski Sur une propriété topologique des ensembles dénombrables denses en soi, Fundamenta Mathematicae, vol. 1(1920), pp. 11–16.
• S. Slavnov Two counterexamples in the logic of dynamic topological systems, Technical Report TR-2003015, Cornell University,2003.
• –––– On completeness of dynamic topological logic, Moscow Mathematics Journal, vol. 5(2005), no. 2, pp. 477–492.
• A. Tarski Der Aussagenkalkül und die Topologie, Fundamenta Mathematica, vol. 31(1938), pp. 103–134.
• J. van Mill The infinite-dimensional topology of function spaces, Elsevier Science, Amsterdam,2001.