Tableau Development for a Bi-intuitionistic Tense Logic

  • John G. Stell
  • Renate A. Schmidt
  • David Rydeheard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8428)


The paper introduces a bi-intuitionistic logic with two modal operators and their tense versions. The semantics is defined by Kripke models in which the set of worlds carries a pre-order relation as well as an accessibility relation, and the two relations are linked by a stability condition. A special case of these models arises from graphs in which the worlds are interpreted as nodes and edges of graphs, and formulae represent subgraphs. The pre-order is the incidence structure of the graphs. These examples provide an account of time including both time points and intervals, with the accessibility relation providing the order on the time structure. The logic we present is decidable and has the effective finite model property. We present a tableau calculus for the logic which is sound, complete and terminating. The MetTel system has been used to generate a prover from this tableau calculus.


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  1. [Met]
  2. [Av02]
    Aiello, M., van Benthem, J.: A Modal Walk Through Space. Journal of Applied Non-Classical Logics 12, 319–363 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  3. [AO07]
    Aiello, M., Ottens, B.: The Mathematical Morpho-Logical View on Reasoning about Space. In: Veloso, M.M. (ed.) IJCAI 2007, pp. 205–211. AAAI Press (2007)Google Scholar
  4. [ANvB98]
    Andréka, H., Németi, I., van Benthem, J.: Modal Languages and Bounded Fragments of Predicate Logic. Journal of Philosophical Logic 27, 217–274 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. [Bén00]
    Bénabou, J.: Distributors at Work (2000),
  6. [Blo02]
    Bloch, I.: Modal Logics Based on Mathematical Morphology for Qualitative Spatial Reasoning. Journal of Applied Non-Classical Logics 12, 399–423 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. [BHR07]
    Bloch, I., Heijmans, H.J.A.M., Ronse, C.: Mathematical Morphology. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 857–944. Springer (2007)Google Scholar
  8. [GPT10]
    Goré, R., Postniece, L., Tiu, A.: Cut-elimination and Proof Search for Bi-Intuitionistic Tense Logic. arXiv e-Print 1006.4793v2 (2010)Google Scholar
  9. [Grä99]
    Grädel, E.: On the restraining power of guards. Journal of Symbolic Logic 64, 1719–1742 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. [HV93]
    Heijmans, H., Vincent, L.: Graph Morphology in Image Analysis. In: Dougherty, E.R. (ed.) Mathematical Morphology in Image Processing, pp. 171–203. Marcel Dekker (1993)Google Scholar
  11. [Rau74]
    Rauszer, C.: A formalization of the propositional calculus of H-B logic. Studia Logica 33, 23–34 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  12. [RZ96]
    Reyes, G.E., Zolfaghari, H.: Bi-Heyting Algebras, Toposes and Modalities. Journal of Philosophical Logic 25, 25–43 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  13. [SH07]
    Schmidt, R.A., Hustadt, U.: The Axiomatic Translation Principle for Modal Logic. ACM Transactions on Computational Logic 8, 1–55 (2007)CrossRefMathSciNetGoogle Scholar
  14. [ST08]
    Schmidt, R.A., Tishkovsky, D.: A General Tableau Method for Deciding Description Logics, Modal Logics and Related First-Order Fragments. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS, vol. 5195, pp. 194–209. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. [ST11]
    Schmidt, R.A., Tishkovsky, D.: Automated Synthesis of Tableau Calculi. Logical Methods in Computer Science 7, 1–32 (2011)CrossRefMathSciNetGoogle Scholar
  16. [ST13]
    Schmidt, R.A., Tishkovsky, D.: Using Tableau to Decide Description Logics with Full Role Negation and Identity. To appear in ACM Transactions on Computational Logic (2013)Google Scholar
  17. [Ser82]
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press (1982)Google Scholar
  18. [Ste12]
    Stell, J.G.: Relations on Hypergraphs. In: Kahl, W., Griffin, T.G. (eds.) RAMICS 2012. LNCS, vol. 7560, pp. 326–341. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. [TSK12]
    Tishkovsky, D., Schmidt, R.A., Khodadadi, M.: The Tableau Prover Generator MetTeL2. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 492–495. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  20. [TS13]
    Tishkovsky, D., Schmidt, R.A.: Refinement in the Tableau Synthesis Framework. arXiv e-Print 1305.3131v1 (2013)Google Scholar
  21. [Wan08]
    Wansing, H.: Constructive negation, implication, and co-implication. Journal of Applied Non-Classical Logics 18, 341–364 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. [ZWC01]
    Zakharyaschev, M., Wolter, F., Chagrov, A.: Advanced Modal Logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 3, pp. 83–266. Kluwer (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • John G. Stell
    • 1
  • Renate A. Schmidt
    • 2
  • David Rydeheard
    • 2
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.School of Computer ScienceUniversity of ManchesterManchesterUK

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