Document Open Access Logo

Dynamic Cantor Derivative Logic

Authors David Fernández-Duque, Yoàv Montacute

PDF
Thumbnail PDF

File

LIPIcs.CSL.2022.19.pdf
  • Filesize: 0.72 MB
  • 17 pages

Document Identifiers

Author Details

David Fernández-Duque
  • Department of Mathematics, Ghent University, Belgium
  • Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic
Yoàv Montacute
  • Computer Laboratory, University of Cambridge, UK

Cite AsGet BibTex

David Fernández-Duque and Yoàv Montacute. Dynamic Cantor Derivative Logic. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CSL.2022.19

Abstract

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f: X → X. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all T_D dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation - something known to be impossible over the class of all spaces.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
Keywords
  • dynamic topological logic
  • Cantor derivative
  • temporal logic
  • modal logic

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Sergei Artemov, Jennifer Davoren, and Anil Nerode. Modal logics and topological semantics for hybrid systems. Technical Report MSI 97-05, 1997. Google Scholar
  2. Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic, volume 53 of Cambridge Tracts in Theoretical Computer Scie. Cambridge University Press, Cambridge, 2001. Google Scholar
  3. George Boolos. The logic of provability. The American Mathematical Monthly, 91(8):470-480, 1984. URL: https://doi.org/10.1080/00029890.1984.11971467.
  4. Steven L. Brunton and J. Nathan Kutz. Data-driven science and engineering: Machine learning, dynamical systems, and control. Cambridge University Press, 2019. Google Scholar
  5. Moody T. Chu. Linear algebra algorithms as dynamical systems. Acta Numerica, 17:1-86, 2008. Google Scholar
  6. Weinan E. A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics, 5(1):1-11, 2017. Google Scholar
  7. Leo Esakia. Diagonal constructions, Löb’s formula and Cantor’s scattered spaces. Studies in logic and semantics, 132(3):128-143, 1981. Google Scholar
  8. Leo Esakia. Weak transitivity-restitution. In Nauka, editor, Study in Logic, volume 8, pages 244-254, 2001. In Russian. Google Scholar
  9. Leo Esakia. Intuitionistic logic and modality via topology. Ann. Pure Appl. Log., 127(1-3):155-170, 2004. URL: https://doi.org/10.1016/j.apal.2003.11.013.
  10. David Fernández-Duque. Tangled modal logic for spatial reasoning. In Toby Walsh, editor, IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pages 857-862. IJCAI/AAAI, 2011. URL: https://doi.org/10.5591/978-1-57735-516-8/IJCAI11-149.
  11. David Fernández-Duque. Non-finite axiomatizability of dynamic topological logic. ACM Transactions on Computational Logic, 15(1):4:1-4:18, 2014. URL: https://doi.org/10.1145/2489334.
  12. David Fernández-Duque and Petar Iliev. Succinctness in subsystems of the spatial μ-calculus. Journal of Applied Logics, 5(4):827-874, 2018. URL: https://www.collegepublications.co.uk/downloads/ifcolog00024.pdf.
  13. David Fernández-Duque. A sound and complete axiomatization for dynamic topological logic. The Journal of Symbolic Logic, 77(3):947-969, 2012. URL: https://doi.org/10.2178/jsl/1344862169.
  14. Robert Goldblatt and Ian M. Hodkinson. Spatial logic of tangled closure operators and modal mu-calculus. Annals of Pure and Applied Logic, 168(5):1032-1090, 2017. Google Scholar
  15. Guillaume Hanrot, Xavier Pujol, and Damien Stehlé. Analyzing blockwise lattice algorithms using dynamical systems. In Annual Cryptology Conference, pages 447-464. Springer, 2011. Google Scholar
  16. Boris Konev, Roman Kontchakov, Frank Wolter, and Michael Zakharyaschev. Dynamic topological logics over spaces with continuous functions. In Guido Governatori, Ian M. Hodkinson, and Yde Venema, editors, Advances in Modal Logic, volume 6, pages 299-318, London, 2006. College Publications. Google Scholar
  17. Boris Konev, Roman Kontchakov, Frank Wolter, and Michael Zakharyaschev. On dynamic topological and metric logics. Studia Logica, 84(1):129-160, 2006. URL: https://doi.org/10.1007/s11225-006-9005-x.
  18. Philip Kremer and Grigori Mints. Dynamic topological logic. Annals of Pure and Applied Logic, 131:133-158, 2005. Google Scholar
  19. Andrey Kudinov. Topological modal logics with difference modality. In Guido Governatori, Ian M. Hodkinson, and Yde Venema, editors, Advances in Modal Logic 6, pages 319-332. College Publications, 2006. URL: http://www.aiml.net/volumes/volume6/Kudinov.ps.
  20. Hai Lin and Panos J. Antsaklis. Hybrid dynamical systems: An introduction to control and verification. Found. Trends Syst. Control, 1(1):1-172, March 2014. URL: https://doi.org/10.1561/2600000001.
  21. John C. C. McKinsey and Alfred Tarski. The algebra of topology. Annals of Mathematics, 2:141-191, 1944. Google Scholar
  22. Henning S. Mortveit and Christian M. Reidys. An Introduction to Sequential Dynamical Systems. Springer-Verlag, Berlin, Heidelberg, 2007. Google Scholar
  23. Amir Pnueli. The temporal logic of programs. In Proceedings 18th IEEE Symposium on the Foundations of CS, pages 46-57, 1977. Google Scholar
  24. Harold Simmons. Topological aspects of suitable theories. Proceedings of the Edinburgh Mathematical Society, 19(4):383-391, 1975. URL: https://doi.org/10.1017/S001309150001049X.
  25. Craig Smorynski. Self-reference and modal logic. Springer, 1985. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact