Abstract
The syntax, semantics and an axiom system for an extension of Propositional Dynamic Logic (PDL) for order of magnitude qualitative reasoning which formalizes the concepts of closeness and distance is introduced in this paper. In doing this, we use some of the advantages of PDL: firstly, we exploit the possibility of constructing complex relations from simpler ones for defining the concept of closeness and other programming commands such as while ... do and repeat ... until; secondly, we employ its theoretical support in order to show that the satisfiability problem is decidable and the completeness of our system. Moreover, the specific axioms of our logic have been obtained from the minimal set of formulas needed in our definition of qualitative sum of small, medium and large numbers. We also present some of the advantages of our approach on the basis of an example.
Partially supported by projects TIN2006-15455-C03-01 and P6-FQM-02049.
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
Areces, C., ten Cate, B.: Hybrid Logics. In: Blackburn, P., Van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3, pp. 821–868. Elsevier, Amsterdam (2007)
Bennett, B., Cohn, A.G., Wolter, F., Zakharyaschev, M.: Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning. Applied Intelligence 17(3), 239–251 (2002)
Benthem, J., Eijck, J., Kooi, B.: Logics of communication and change. Information and Computation 204(11), 1620–1662 (2006)
Blackburn, P., Van Benthem, J.: Modal Logic: A semantic perspective. In: Blackburn, P., Van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3, pp. 58–61. Elsevier, Amsterdam (2007)
Bollig, B., Kuske, D., Meinecke, I.: Propositional dynamic logic for message-passing systems. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 303–315. Springer, Heidelberg (2007)
Bugaychenko, D., Soloviev, I.: MASL: A logic for the specification of multiagent real-time systems. In: Burkhard, H.-D., Lindemann, G., Verbrugge, R., Varga, L.Z. (eds.) CEEMAS 2007. LNCS (LNAI), vol. 4696, pp. 183–192. Springer, Heidelberg (2007)
Burrieza, A., Muñoz-Velasco, E., Ojeda-Aciego, M.: A Propositional Dynamic Logic Approach for Order of Magnitude Reasoning. In: Geffner, H., Prada, R., Machado Alexandre, I., David, N. (eds.) IBERAMIA 2008. LNCS (LNAI), vol. 5290, pp. 11–20. Springer, Heidelberg (2008)
Burrieza, A., Muñoz-Velasco, E., Ojeda-Aciego, M.: A logic for order of magnitude reasoning with negligibility, non-closeness and distance. In: Borrajo, D., Castillo, L., Corchado, J.M. (eds.) CAEPIA 2007. LNCS (LNAI), vol. 4788, pp. 210–219. Springer, Heidelberg (2007)
Burrieza, A., Ojeda-Aciego, M.: A multimodal logic approach to order of magnitude qualitative reasoning with comparability and negligibility relations. Fundamenta Informaticae 68, 21–46 (2005)
Burrieza, A., Ojeda-Aciego, M., Orłowska, E.: Relational approach to order-of-magnitude reasoning. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) TARSKI 2006. LNCS (LNAI), vol. 4342, pp. 105–124. Springer, Heidelberg (2006)
Dague, P.: Symbolic reasoning with relative orders of magnitude. In: Proc. 13th Intl. Joint Conference on Artificial Intelligence, pp. 1509–1515. Morgan Kaufmann, San Francisco (1993)
Duckham, M., Lingham, J., Mason, K., Worboys, M.: Qualitative reasoning about consistency in geographic information. Information Sciences 176(6, 22), 601–627 (2006)
Golińska-Pilarek, J., Muñoz-Velasco, E.: Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance. International Journal of Computer Mathematics (to appear, 2009)
Harel, D., Kozen, D.: Dynamic Logic. MIT Press, Cambridge (2000)
Heinemann, B.: A PDL-like logic of knowledge acquisition. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 146–157. Springer, Heidelberg (2007)
Mirkowska, C., Salwicki, A.: Algorithmic Logic. Kluwer Academic Publishers, Norwell (1987)
Nayak, P.: Causal Approximations. Artificial Intelligence 70, 277–334 (1994)
Passy, S., Tinchev, T.: An essay in combinatory dynamic logic. Information and Computation 93(2), 263–332 (1991)
Raiman, O.: Order of magnitude reasoning. Artificial Intelligence 51, 11–38 (1991)
Sanchez, M., Prats, F., Piera, N.: Una formalización de relaciones de comparabilidad en modelos cualitativos. Boletín de la AEPIA (Bulletin of the Spanish Association for AI) 6, 15–22 (1996)
Travé-Massuyès, L., Ironi, L., Dague, P.: Mathematical Foundations of Qualitative Reasoning. AI Magazine, American Asociation for Artificial Intelligence, 91–106 (2003)
Travé-Massuyès, L., Prats, F., Sánchez, M., Agell, N.: Relative and absolute order-of-magnitude models unified. Annals of Mathematics and Artificial Intelligence 45, 323–341 (2005)
Wolter, F., Zakharyaschev, M.: Qualitative spatio-temporal representation and reasoning: a computational perspective. In: Lakemeyer, G., Nebel, B. (eds.) Exploring Artificial Intelligence in the New Millenium. Morgan Kaufmann, San Francisco (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Burrieza, A., Muñoz-Velasco, E., Ojeda-Aciego, M. (2010). Closeness and Distance Relations in Order of Magnitude Qualitative Reasoning via PDL . In: Meseguer, P., Mandow, L., Gasca, R.M. (eds) Current Topics in Artificial Intelligence. CAEPIA 2009. Lecture Notes in Computer Science(), vol 5988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14264-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-14264-2_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14263-5
Online ISBN: 978-3-642-14264-2
eBook Packages: Computer ScienceComputer Science (R0)