An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance

  • Joanna Golińska-Pilarek
  • Angel Mora
  • Emilio Muñoz-Velasco
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5351)

Abstract

We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some heuristics, such as the so called phantom variables, which improve the efficiency of the selection of variables used un the proof.

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References

  1. 1.
    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)CrossRefMATHGoogle Scholar
  2. 2.
    Burrieza, A., Mora, A., Ojeda-Aciego, M., Orłowska, E.: Implementing a relational system for order-of-magnitude reasoning. Technical Report (2008)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Burrieza, A., Muñoz, E., Ojeda-Aciego, M.: Order of magnitude qualitative reasoning with bidirectional negligibility. In: Marín, R., Onaindía, E., Bugarín, A., Santos, J. (eds.) CAEPIA 2005. LNCS (LNAI), vol. 4177, pp. 370–378. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    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)MathSciNetMATHGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Dallien, J., MacCaull, W.: RelDT: A relational dual tableaux automated theorem prover, http://www.logic.stfx.ca/reldt/
  9. 9.
    Formisano, A., Orłowska, E., Omodeo, E.: A PROLOG tool for relational translation of modal logics: A front-end for relational proof systems. In: Beckert, B. (ed.) TABLEAUX 2005 Position Papers and Tutorial Descriptions, Fachberichte Informatik No 12, Universitaet Koblenz-Landau, pp. 1–10 (2005), http://www.di.univaq.it/TARSKI/transIt/
  10. 10.
    Golińska-Pilarek, J., Muñoz-Velasco, E.: Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance. Technical Report (2008)Google Scholar
  11. 11.
    Golińska-Pilarek, J., Orłowska, E.: Relational logics and their applications. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) TARSKI 2006. LNCS (LNAI), vol. 4342, pp. 125–161. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    MacCaull, W., Orłowska, E.: Correspondence results for relational proof systems with application to the Lambek calculus. Studia Logica 71, 279–304 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mavrovouniotis, M.L., Stephanopoulos, G.: Reasoning with orders of magnitude and approximate relations. In: Proc. 6th National Conference on Artificial Intelligence. The AAAI Press/The MIT Press (1987)Google Scholar
  14. 14.
    Mavrovouniotis, M.L.: A belief framework for order-of-magnitude reasoning and other qualitative relations. Artificial Intelligence in Engineering 11(2), 121–134 (1997)CrossRefGoogle Scholar
  15. 15.
    Orłowska, E.: Relational interpretation of modal logics. In: Andréka, H., Monk, D., Nemeti, I. (eds.) Algebraic Logic, Col. Math. Soc. J. Bolyai., vol. 54, pp. 443–471. North Holland, Amsterdam (1988)Google Scholar
  16. 16.
    Raiman, O.: Order of magnitude reasoning. Artificial Intelligence 51, 11–38 (1991)CrossRefGoogle Scholar
  17. 17.
    Randell, D., Cui, Z., Cohn, A.: A spatial logic based on regions and connections. In: Proc. of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR 1992), pp. 165–176 (1992)Google Scholar
  18. 18.
    Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Polish Scientific Publishers, Warsaw (1963)MATHGoogle Scholar
  19. 19.
    Sánchez, M., Prats, F., Piera, N.: Una formalizacin de relaciones de comparabilidad en modelos cualitativos. Boletín de la AEPIA (Bulletin of the Spanish Association for AI) 6, 15–22 (1996)Google Scholar
  20. 20.
    Travé-Massuyès, L., Ironi, L., Dague, P.: Mathematical Foundations of Qualitative Reasoning. AI Magazine, American Asociation for Artificial Intelligence, 91–106 (2003)Google Scholar
  21. 21.
    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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Joanna Golińska-Pilarek
    • 1
  • Angel Mora
    • 2
  • Emilio Muñoz-Velasco
    • 2
  1. 1.Institute of PhilosophyWarsaw University, National Institute of TelecommunicationsPoland
  2. 2.Dept. Matemática AplicadaUniversidad de MálagaSpain

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