Dynamic topological logic.(English)Zbl 1067.03028

Three important research areas meet together in the abstract framework of dynamic topological logic (DTL): the topological semantics for S4, topological dynamics, and temporal logic. It is known that S4 can be understood as the logic of topological spaces, and $$\square$$ can be understood as a topological modality (with the meaning of the topological interior). Thus, the topological semantics for S4 is based on topological spaces rather than Kripke frames. On the other hand, topological dynamics studies the asymptotic properties of continuous maps on topological spaces. A dynamic topological system is a topological space $$X$$ together with a continuous function $$f$$ which can be thought of in temporal terms as moving the points of the topological space.
Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces, and are defined for a trimodal language with an S4-ish topological modality, and two temporal modalities (‘next’ and ‘henceforth’) both interpreted using the continuous function $$f$$. In particular, ‘next’ expresses $$f$$’s action on $$X$$ from one moment to the next, and ‘henceforth’ expresses the asymptotic behaviour of $$f$$.
The authors introduce the dynamic topological analogues of Kripke models, the dynamic Alexandrov models, in order to set a precise definition of dynamic topological model; then, a semantic definition of the dynamic topological logic generated by a class $$\mathcal{T}$$ of topological spaces and/or a class $$\mathcal{F}$$ of continuous functions. Later, several specific DTLs are considered, presenting their properties and axiomatizing some of their next-interior fragments. Finally, a sound and complete axiomatization of a DTL is given in a particular trimodal fragment of the language in which the temporal modalities cannot occur in the scope of a topological modality.

MSC:

 03B45 Modal logic (including the logic of norms) 03B44 Temporal logic 54H20 Topological dynamics (MSC2010)
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 [1] Aiello, M.; van Benthem, J.; Bezhanishvili, G., Reasoning about space: the modal way, Journal of logic and computation, 13, 889-920, (2003) · Zbl 1054.03015 [2] Alexandrov, P., Diskrete Räume, Matematicheskii sbornik, 2, 501-518, (1937) · Zbl 0018.09105 [3] Artemov, S.; Davoren, J.; Nerode, A., Modal logics and topological semantics for hybrid systems, technical report MSI 97-05, cornell university, June, 1997, available at [4] N. Bourbaki, General Topology, Part 1, Hermann, Paris, Addison-Wesley, 1966 · Zbl 0145.19302 [5] Brown, J., Ergodic theory and topological dynamics, (1976), Academic Press New York · Zbl 0334.28011 [6] J. Davoren, Modal logics for continuous dynamics, Ph.D. Thesis, Cornell University, 1998 [7] Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, (1981), Princeton University Press Princeton · Zbl 0459.28023 [8] Goldblatt, R., () [9] B. Konev, R. Kontchakov, D. Tishovsky, F. Wolter, M. Zakharyaschev, On dynamic topological and metric logics, 2004 (manuscript) · Zbl 1114.03026 [10] Kozen, D.; Parikh, R., An elementary proof of the completeness of PDL, Theoretical computer science, 14, 113-118, (1981) · Zbl 0451.03006 [11] Kremer, P., Temporal logic over S4: an axiomatizable fragment of dynamic topological logic, Bulletin of symbolic logic, 3, 375-376, (1997) [12] Kremer, P.; Mints, G., Dynamic topological logic, Bulletin of symbolic logic, 3, 371-372, (1997) [13] Kremer, P.; Mints, G.; Rybakov, V., Axiomatizing the next-interior fragment of dynamic topological logic, Bulletin of symbolic logic, 3, 376-377, (1997) [14] Kripke, S., Semantical analysis of modal logic I, normal propositional calculi, Zeitschrift für mathematische logik und grundlagen der Mathematik, 9, 67-96, (1963) · Zbl 0118.01305 [15] Kröger, F., Temporal logics of programs, (1985), Springer-Verlag Berlin · Zbl 0563.68007 [16] McKinsey, J.C.C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, The journal of symbolic logic, 6, 117-134, (1941) · JFM 67.0974.01 [17] McKinsey, J.C.C.; Tarski, A., The algebra of topology, Annals of mathematics, 45, 141-191, (1944) · Zbl 0060.06206 [18] G. Mints, T. Zhang, A proof of topological completeness for S4 in (0, 1) (forthcoming) · Zbl 1133.03009 [19] Prior, A.N., Past, present and future, (1967), Clarendon Press Oxford · Zbl 0169.29802 [20] Rasiowa, H.; Sikorski, R., The mathematics of metamathematics, (1963), Państowowe Wydawnictwo Naukowe Warsaw · Zbl 0122.24311 [21] Segerberg, K., Discrete linear future time without axioms, Studia logica, 35, 273-278, (1976) · Zbl 0343.02017 [22] Segerberg, K., Von wright’s logic of time, (), (in Russian) · Zbl 0599.03029 [23] Segerberg, K., Von wright’s tense logic, (), 603-635 [24] S. Slavnov, Two counterexamples in the logic of dynamic topological systems, Technical Report TR-2003015, Cornell University, 2003 [25] Tang, T.-C., Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication, Bulletin of the American mathematical society, 44, 737-744, (1938) · Zbl 0019.38504 [26] von Wright, G.H., And next, Acta philosophica fennica facs, 18, 293-304, (1965) [27] von Wright, G.H., Always, Theoria, 34, 208-221, (1968) [28] Walters, P., An introduction to ergodic theory, (1982), Springer-Verlag Berlin · Zbl 0475.28009
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