Dual Tableau-Based Decision Procedures for Fragments of the Logic of Binary Relations

  • Domenico CantoneEmail author
  • Marianna Nicolosi-Asmundo
Part of the Outstanding Contributions to Logic book series (OCTR, volume 17)


In this paper, written to honor the career of Ewa Orłowska, we survey the main results on dual tableau-based decision procedures for fragments of the logic of binary relations. Specifically, we shall review relational fragments representing well known classes of first-order logic, of modal and multi-modal logics, and of description logics. We shall also examine a relational fragment admitting the use of a simple form of entailment within dual tableau decision procedures.


Logic of binary relations Non-classical logics Dual tableau systems Decision procedures Relational composition operation Relational entailment 



The authors would like to thank the anonymous reviewers for their very helpful comments. This work was supported by the Polish National Science Centre research project DEC-2011/02/A/HS1/00395.


  1. Baader, F., Calvanese, D., McGuinness, D. L., Nardi, D., & Patel-Schneider, P. F. (Eds.). (2003). The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge: Cambridge University Press.Google Scholar
  2. Baader, F. (2009). Description logics. In S. Tessaris et al. (Eds.), Reasoning Web. Semantic Technologies for Information Systems, 5th International Summer School 2009, Tutorial Lectures (Vol. 5689, pp. 1–39). Lecture Notes in Computer Science. Brixen-Bressanone: Springer.Google Scholar
  3. Beth, W. E. (1955). Semantic entailment and formal derivability. Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde,18(13), 309–342 (reprinted in Hintikka, 1969).Google Scholar
  4. Cantone, D., Nicolosi Asmundo, M., & Orłowska, E. (2010). Dual tableau-based decision procedures for some relational logics. In W. Faber & N. Leone (Eds.), Proceedings of the 25th Italian Conference on Computational Logic (Vol. 598). CEUR Workshop Proceedings. Rende, Italy.
  5. Cantone, D., Nicolosi Asmundo, M., & Orłowska, E. (2011). Dual tableau-based decision procedures for relational logics with restricted composition operator. Journal of Applied Non-classical Logics, 21(2), 177–200.CrossRefGoogle Scholar
  6. Cantone, D., Golińska-Pilarek, J., & Asmundo, M. N. (2014a). A relational dual tableau decision procedure for multimodal and description logics. In M. M. Polycarpou et al. (Eds.), 9th International Conference Proceedings of Hybrid Artificial Intelligence Systems HAIS 2014 (Vol. 8480, pp. 466–477). Lecture Notes in Computer Science. Salamanca: Springer.Google Scholar
  7. Cantone, D., Nicolosi Asmundo, M., & Orłowska, E. (2014b). In L. Giordano, V. Gliozzi, & G. Pozzato (Eds.), Proceedings of the 29th Italian Conference on Computational Logic (Vol. 1195, pp. 194–209). CEUR Workshop Proceedings. Torino, Italy.
  8. Cantone, D., Nicolosi Asmundo, M., & Orłowska, E. (2018). A dual tableau-based decision procedure for a relational logic with the universal relation (extended version). CoRR. arXiv:1802.07508.
  9. Dershowitz, N. & Jouannaud, J.-P. (1990). Rewrite systems. In Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B) (pp. 243–320).Google Scholar
  10. Fitting, M. (1996). First-order Logic and Automated Theorem Proving (2nd ed.). Graduate Texts in Computer Science. Berlin: Springer.CrossRefGoogle Scholar
  11. Formisano, A. & Nicolosi Asmundo, M. (2006). An efficient relational deductive system for propositional non-classical logics. Journal of Applied Non-classical Logics, 16(3–4), 367–408.CrossRefGoogle Scholar
  12. Glivenko, V. (1929). Sur quelques points de la logique de m. brouwer. Bulletins de la Classe des Sciences,5(15), 183–188.Google Scholar
  13. Golińska-Pilarek, J. & Orłowska, E. (2007). Tableaux and dual tableaux: Transformation of proofs. Studia Logica, 85(3), 291–310.CrossRefGoogle Scholar
  14. Golińska-Pilarek, J., Muñoz-Velasco, E., & Mora, A. (2011). A new deduction system for deciding validity in modal logic K. Logic Journal of the IGPL, 19(2), 425–434.CrossRefGoogle Scholar
  15. Golińska-Pilarek, J., Muñoz-Velasco, E., & Mora Bonilla, A. (2012). Relational dual tableau decision procedure for modal logic K. Logic Journal of the IGPL, 20(4), 747–756.CrossRefGoogle Scholar
  16. Golińska-Pilarek, J., Huuskonen, T., & Muñoz-Velasco, E. (2014). Relational dual tableau decision procedures and their applications to modal and intuitionistic logics. Annals of Pure and Applied Logic, 165(2), 409–427.CrossRefGoogle Scholar
  17. Konikowska, B. (2002). Rasiowa-Sikorski deduction systems in computer science applications. Theoretical Computer Science, 286(2), 323–366.CrossRefGoogle Scholar
  18. Mints, G. (1988). Gentzen-type systems and resolution rules. Part I. Propositional logic. In P. Martin-Löf & G. Mints (Eds.), Proceedings of COLOG-88, International Conference on Computer Logic (Vol. 417, pp. 198–231). Lecture Notes in Computer Science. Tallinn: Springer.Google Scholar
  19. Mora, A., Muñoz-Velasco, E., & Golińska-Pilarek, J. (2011). Implementing a relational theorem prover for modal logic. International Journal of Computer Mathematics, 88(9), 1869–1884.CrossRefGoogle Scholar
  20. Orłowska, E. (1988). Relational interpretation of modal logics. In H. Andreka, D. Monk, & I. Németi (Eds.), Algebraic Logic. Colloquia Mathematica Societatis Janos Bolyai (Vol. 54, pp. 443–471). Amsterdam: North Holland.Google Scholar
  21. Orłowska, E. & Golińska-Pilarek, J. (2011). Dual Tableaux: Foundations, Methodology, Case Studies. Trends in Logic. Dordrecht-Heidelberg-London-New York: Springer.CrossRefGoogle Scholar
  22. Rasiowa, H. & Sikorski, R. (1960). On the Gentzen theorem. Fundamenta Mathematicae, 48, 57–69.CrossRefGoogle Scholar
  23. Rasiowa, H. & Sikorski, R. (1963). The Mathematics of Metamathematics. Warszawa: Polish Scientific Publishers.Google Scholar
  24. Sattler, U. (1996). A concept language extended with different kinds of transitive roles. In G. Görz & S. Hölldobler (Eds.), 20th Annual German Conference on Artificial Intelligence Proceedings of KI-96: Advances in Artificial Intelligence (Vol. 1137, pp. 333–345). Lecture Notes in Computer Science. Dresden: Springer.Google Scholar
  25. Tarski, A. & Givant, S., (1987). Formalization of Set Theory Without Variables. Colloquium Publications. Providence: American Mathematical Society.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly

Personalised recommendations