Logic for Programming, Artificial Intelligence, and Reasoning

Logic for Programming, Artificial Intelligence, and Reasoning pp 558-574 | Cite as

Proof Search in Nested Sequent Calculi

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9450)


We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic \(\mathsf {K}\), a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent calculi for the considered non-normal modal logics.


  1. 1.
    Andreoli, J.M.: Logic programming with focusing proofs in linear logic. J. Logic Comput. 2(3), 297–347 (1992)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009)MATHCrossRefGoogle Scholar
  3. 3.
    Chaudhuri, K., Guenot, N., Straßburger, L.: The focused calculus of structures. In: Bezem, M. (ed.) CSL 2011, pp. 159–173. Leibniz International Proceedings in Informatics (2011)Google Scholar
  4. 4.
    Chellas, B.F.: Modal Logic. Cambridge University Press, Cambridge (1980)MATHCrossRefGoogle Scholar
  5. 5.
    Demri, S.: Complexity of simple dependent bimodal logics. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS, vol. 1847, pp. 190–204. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Goré, R., Ramanayake, R.: Labelled tree sequents, tree hypersequents and nested (deep) sequents. In: AiML, vol. 9, pp. 279–299 (2012)Google Scholar
  7. 7.
    Guglielmi, A., Straßburger, L.: Non-commutativity and MELL in the calculus of structures. In: Fribourg, L. (ed.) CSL 2001. LNCS, vol. 2142, pp. 54–68. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  8. 8.
    Lavendhomme, R., Lucas, T.: Sequent calculi and decision procedures for weak modal systems. Studia Logica 65, 121–145 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lellmann, B.: Linear nested sequents, 2-sequents and hypersequents. In: De Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 135–150. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  10. 10.
    Lellmann, B., Pattinson, D.: Constructing cut free sequent systems with context restrictions based on classical or intuitionistic logic. In: Lodaya, K. (ed.) ICLA 2013. LNCS (LNAI), vol. 7750, pp. 148–160. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Masini, A.: 2-sequent calculus: a proof theory of modalities. Ann. Pure Appl. Logic 58, 229–246 (1992)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mendler, M., Scheele, S.: Cut-free Gentzen calculus for multimodal CK. Inf. Comput. (IANDC) 209, 1465–1490 (2011)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Miller, D., Pimentel, E.: A formal framework for specifying sequent calculus proof systems. Theor. Comput. Sci. 474, 98–116 (2013)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Negri, S., van Plato, J.: Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar
  15. 15.
    Nigam, V., Miller, D.: A framework for proof systems. J. Autom. Reasoning 45(2), 157–188 (2010)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Nigam, V., Pimentel, E., Reis, G.: An extended framework for specifying and reasoning about proof systems. J. Logic Comput. (2014). doi:10.1093/logcom/exu029, http://logcom.oxfordjournals.org/content/early/2014/06/06/logcom.exu029.abstract
  17. 17.
    Poggiolesi, F.: The method of tree-hypersequents for modal propositional logic. In: Makinson, D., Malinowski, J., Wansing, H. (eds.) Towards Mathematical Philosophy. Trends in Logic, vol. 28, pp. 31–51. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Straßburger, L.: Cut elimination in nested sequents for intuitionistic modal logics. In: Pfenning, F. (ed.) FOSSACS 2013. LNCS, vol. 7794, pp. 209–224. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Computer LanguagesTU WienViennaAustria
  2. 2.Departamento de MatemáticaUFRNNatalBrazil

Personalised recommendations