Abstract
We consider a certain, in a way hybrid extension of a system called topologic, which has been designed for reasoning about the spatial content of the idea of knowledge. What we add to the language of topologic are names of both points and neighbourhoods of points. Due to the special semantics of topologic these names do not quite behave like nominals in hybrid logic. Nevertheless, corresponding satisfaction operators can be simulated with the aid of the global modality, which becomes, therefore, another means of expression of our system. In this paper we put forward an axiomatization of the set of formulas valid in all such hybrid scenarios, and we prove the decidability of this logic. Moreover, we argue that the present approach to hybridizing modal concepts for knowledge and topology is not only much more powerful but also much more natural than a previous one.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)
Meyer, J.J.C., van der Hoek, W.: Epistemic Logic for AI and Computer Science. In: Cambridge Tracts in Theoretical Computer Science, vol. 41, Cambridge University Press, Cambridge (1995)
Moss, L.S., Parikh, R.: Topological reasoning and the logic of knowledge. In: Moses, Y. (ed.) Theoretical Aspects of Reasoning about Knowledge (TARK 1992), pp. 95–105. Morgan Kaufmann, San Francisco (1992)
Dabrowski, A., Moss, L.S., Parikh, R.: Topological reasoning and the logic of knowledge. Annals of Pure and Applied Logic 78, 73–110 (1996)
Blackburn, P.: Representation, reasoning, and relational structures: a hybrid logic manifesto. Logic Journal of the IGPL 8, 339–365 (2000)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. In: Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, Cambridge (2001)
Areces, C., Blackburn, P., Marx, M.: The computational complexity of hybrid temporal logics. Logic Journal of the IGPL 8, 653–679 (2000)
Heinemann, B.: Knowledge over dense flows of time (from a hybrid point of view). In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 194–205. Springer, Heidelberg (2002)
Heinemann, B.: Separating sets by modal formulas. In: Haeberer, A.M. (ed.) AMAST 1998. LNCS, vol. 1548, pp. 140–153. Springer, Heidelberg (1998)
Heinemann, B.: Generalizing the modal and temporal logic of linear time. In: Rus, T. (ed.) AMAST 2000. LNCS, vol. 1816, pp. 41–56. Springer, Heidelberg (2000)
Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)
Bourbaki, N.: General Topology, Part 1. Hermann, Paris (1966)
Heinemann, B.: Extended canonicity of certain topological properties of set spaces. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 135–149. Springer, Heidelberg (2003)
Krommes, G.: A new proof of decidability for the modal logic of subset spaces. In: ten Cate, B. (ed.) Proceedings of the Eighth ESSLLI Student Session, Vienna, Austria, pp. 137–147 (2003)
Blackburn, P., Spaan, E.: A modal perspective on the computational complexity of attribute value grammar. Journal of Logic, Language and Information 2, 129–169 (1993)
Georgatos, K.: Knowledge theoretic properties of topological spaces. In: Masuch, M., Polos, L. (eds.) Logic at Work 1992. LNCS, vol. 808, pp. 147–159. Springer, Heidelberg (1994)
Georgatos, K.: Knowledge on treelike spaces. Studia Logica 59, 271–301 (1997)
Heinemann, B.: A hybrid treatment of evolutionary sets. In: Coello Coello, C.A., de Albornoz, Á., Sucar, L.E., Battistutti, O.C. (eds.) MICAI 2002. LNCS (LNAI), vol. 2313, pp. 204–213. Springer, Heidelberg (2002)
Heinemann, B.: An application of monodic first-order temporal logic to reasoning about knowledge. In: Reynolds, M., Sattar, A. (eds.) Proceedings TIME-ICTL 2003, pp. 10–16. IEEE Computer Society Press, Los Alamitos (2003)
McKinsey, J.C.C.: A solution to the decision problem for the Lewis systems S2 and S4, with an application to topology. Journal of Symbolic Logic 6, 117–141 (1941)
Gabelaia, D.: Modal definability in topology. Master’s thesis, ILLC, Universiteit van Amsterdam (2001)
Weiss, M.A., Parikh, R.: Completeness of certain bimodal logics for subset spaces. Studia Logica 71, 1–30 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Heinemann, B. (2004). A Hybrid Logic of Knowledge Supporting Topological Reasoning. In: Rattray, C., Maharaj, S., Shankland, C. (eds) Algebraic Methodology and Software Technology. AMAST 2004. Lecture Notes in Computer Science, vol 3116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27815-3_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-27815-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22381-8
Online ISBN: 978-3-540-27815-3
eBook Packages: Springer Book Archive