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Unification in Modal Logic

  • Philippe BalbianiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11600)

Abstract

Let \(\varphi _{1},\ldots ,\varphi _{n}\) and \(\psi \) be some formulas.

Keywords

Modal logics Unification problem Elementary unification Unification with constants Computability of unification Unification type 

References

  1. 1.
    Baader, F.: On the complexity of Boolean unification. Inf. Process. Lett. 67, 215–220 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baader, F., Borgwardt, S., Morawska, B.: Extending unification in \(\cal{EL}\) towards general TBoxes. In: Brewka, G. et al. (eds.) Principles of Knowledge Representation and Reasoning, pp. 568–572. AAAI Press (2012)Google Scholar
  3. 3.
    Baader, F., Ghilardi, S.: Unification in modal and description logics. Log. J. IGPL 19, 705–730 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baader, F., Morawska, B.: Unification in the description logic \(\cal{EL}\). In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 350–364. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02348-4_25CrossRefGoogle Scholar
  5. 5.
    Baader, F., Narendran, P.: Unification of concept terms in description logics. J. Symb. Comput. 31, 277–305 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Babenyshev, S., Rybakov, V.: Unification in linear temporal logic \(LTL\). Ann. Pure Appl. Log. 162, 991–1000 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Babenyshev, S., Rybakov, V., Schmidt, R., Tishkovsky, D.: A tableau method for checking rule admissibility in \(S4\). Electron. Notes Theor. Comput. Sci. 262, 17–32 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Balbiani, P.: Remarks about the unification type of some non-symmetric non-transitive modal logics. Log. J. IGPL (2018, to appear)Google Scholar
  9. 9.
    Balbiani, P., Gencer, Ç.: \(KD\) is nullary. J. Appl. Non Class. Log. 27, 196–205 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Balbiani, P., Gencer, Ç.: Unification in epistemic logics. J. Appl. Non Class. Log. 27, 91–105 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Balbiani, P., Tinchev, T.: Unification in modal logic \(Alt_{1}\). In: Advances in Modal Logic, pp. 117–134. College Publications (2016)Google Scholar
  12. 12.
    Balbiani, P., Tinchev, T.: Elementary unification in modal logic \(KD45\). J. Appl. Log. IFCoLog J. Log. Appl. 5, 301–317 (2018)zbMATHGoogle Scholar
  13. 13.
    Chagrov, A.: Decidable modal logic with undecidable admissibility problem. Algebra i Logika 31, 83–93 (1992)CrossRefGoogle Scholar
  14. 14.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  15. 15.
    Cintula, P., Metcalfe, G.: Admissible rules in the implication-negation fragment of intuitionistic logic. Ann. Pure Appl. Log. 162, 162–171 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dzik, W.: Unitary unification of \(S5\) modal logics and its extensions. Bull. Sect. Log. 32, 19–26 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dzik, W.: Unification Types in Logic. Wydawnicto Uniwersytetu Slaskiego, Katowice (2007)zbMATHGoogle Scholar
  18. 18.
    Dzik, W.: Remarks on projective unifiers. Bull. Sect. Log. 40, 37–46 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dzik, W., Wojtylak, P.: Projective unification in modal logic. Log. J. IGPL 20, 121–153 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fernández Gil, O.: Hybrid Unification in the Description Logic \(\cal{EL}\). Master thesis of Technische Universität Dresden (2012)Google Scholar
  21. 21.
    Gencer, Ç.: Description of modal logics inheriting admissible rules for \(K4\). Log. J. IGPL 10, 401–411 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gencer, Ç., de Jongh, D.: Unifiability in extensions of \(K4\). Log. J. IGPL 17, 159–172 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ghilardi, S.: Unification in intuitionistic logic. J. Symb. Log. 64, 859–880 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ghilardi, S.: Best solving modal equations. Ann. Pure Appl. Log. 102, 183–198 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ghilardi, S.: A resolution/tableaux algorithm for projective approximations in \(IPC\). Log. J. IGPL 10, 229–243 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ghilardi, S., Sacchetti, L.: Filtering unification and most general unifiers in modal logic. J. Symb. Log. 69, 879–906 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Iemhoff, R.: On the admissible rules of intuitionistic propositional logic. J. Symb. Comput. 66, 281–294 (2001)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Iemhoff, R., Metcalfe, G.: Proof theory for admissible rules. Ann. Pure Appl. Log. 159, 171–186 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jer̆ábek, E.: Complexity of admissible rules. Arch. Math. Log. 46, 73–92 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Jer̆ábek, E.: Blending margins: the modal logic \(K\) has nullary unification type. J. Log. Comput. 25, 1231–1240 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Martin, U., Nipkow, T.: Boolean unification – the story so far. J. Symb. Comput. 7, 275–293 (1989)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rozière, P.: Règles admissibles en calcul propositionnel intuitionniste. Thesis of the University Paris VII (1993)Google Scholar
  33. 33.
    Rybakov, V.: A criterion for admissibility of rules in the model system \(S4\) and the intuitionistic logic. Algebra Log. 23, 369–384 (1984)CrossRefGoogle Scholar
  34. 34.
    Rybakov, V.: Bases of admissible rules of the logics \(S4\) and \(Int\). Algebra Log. 24, 55–68 (1985)CrossRefGoogle Scholar
  35. 35.
    Rybakov, V.: Admissibility of Logical Inference Rules. Elsevier, Amsterdam (1997)zbMATHGoogle Scholar
  36. 36.
    Rybakov, V.: Construction of an explicit basis for rules admissible in modal system \(S4\). Math. Log. Q. 47, 441–446 (2001)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rybakov, V., Gencer, Ç., Oner, T.: Description of modal logics inheriting admissible rules for \(S4\). Log. J. IGPL 7, 655–664 (1999)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rybakov, V., Terziler, M., Gencer, Ç.: An essay on unification and inference rules for modal logics. Bull. Sect. Log. 28, 145–157 (1999)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rybakov, V., Terziler, M., Gencer, Ç.: Unification and passive inference rules for modal logics. J. Appl. Non Class. Log. 10, 369–377 (2000)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Rybakov, V., Terziler, M., Gencer, Ç.: On self-admissible quasi-characterizing inference rules. Stud. Logica. 65, 417–428 (2000)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wolter, F., Zakharyaschev, M.: Undecidability of the unification and admissibility problems for modal and description logics. ACM Trans. Comput. Log. 9, 25:1–25:20 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de recherche en informatique de ToulouseCNRS — Toulouse UniversityToulouseFrance

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