Proving Structural Properties of Sequent Systems in Rewriting Logic

  • Carlos Olarte
  • Elaine Pimentel
  • Camilo RochaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11152)


General and effective methods are required for providing good automation strategies to prove properties of sequent systems. Structural properties such as admissibility, invertibility, and permutability of rules are crucial in proof theory, and they can be used for proving other key properties such as cut-elimination. However, finding proofs for these properties requires inductive reasoning over the provability relation, which is often quite elaborated, exponentially exhaustive, and error prone. This paper aims at developing automatic techniques for proving structural properties of sequent systems. The proposed techniques are presented in the rewriting logic metalogical framework, and use rewrite- and narrowing-based reasoning. They have been fully mechanized in Maude and achieve a great degree of automation when used on several sequent systems, including intuitionistic and classical logics, linear logic, and normal modal logics.



The authors would like to thank the anonymous reviewers for their valuable comments on an earlier draft of this paper. The work of the three authors was supported by CAPES, Colciencias, and INRIA via the STIC AmSud project “EPIC: EPistemic Interactive Concurrency” (Proc. No 88881.117603/2016-01). The work of Pimentel and Olarte was also supported by CNPq and the project FWF START Y544-N23.


  1. 1.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Logic Comput. 2(3), 297–347 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bruni, R., Meseguer, J.: Semantic foundations for generalized rewrite theories. Theoret. Comput. Sci. 360(1–3), 386–414 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Logic 48, 551–577 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cervesato, I., Pfenning, F.: A linear logical framework. Inf. Comput. 179(1), 19–75 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. In: LICS, pp. 229–240. IEEE Computer Society Press (2008)Google Scholar
  6. 6.
    Clavel, M., et al.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007). Scholar
  7. 7.
    Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, North-Holland, pp. 68–131 (1969)Google Scholar
  8. 8.
    Girard, J.-Y.: Linear logic. Theoret. Comput. Sci. 50, 1–102 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lahav, O., Marcos, J., Zohar, Y.: Sequent systems for negative modalities. Logica Universalis 11(3), 345–382 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lellmann, B.: Linear nested sequents, 2-sequents and hypersequents. In: De Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 135–150. Springer, Cham (2015). Scholar
  11. 11.
    Lellmann, B., Pimentel, E.: Proof search in nested sequent calculi. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 558–574. Springer, Heidelberg (2015). Scholar
  12. 12.
    Lincoln, P., Mitchell, J., Scedrov, A., Shankar, N.: Decision problems for propositional linear logic. Ann. Pure Appl. Logic 56, 239–311 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maehara, S.: Eine darstellung der intuitionistischen logik in der klassischen. Nagoya Math. J. 7, 45–64 (1954)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Martí-Oliet, N., Meseguer, J.: Rewriting logic as a logical and semantic framework. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, pp. 1–87. Springer, Dordrecht (2002)zbMATHGoogle Scholar
  15. 15.
    Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theoret. Comput. Sci. 96(1), 73–155 (1992)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Miller, D., Pimentel, E.: A formal framework for specifying sequent calculus proof systems. Theoret. Comput. Sci. 474, 98–116 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Miller, D., Saurin, A.: From proofs to focused proofs: a modular proof of focalization in linear logic. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 405–419. Springer, Heidelberg (2007). Scholar
  18. 18.
    Nigam, V., Pimentel, E., Reis, G.: An extended framework for specifying and reasoning about proof systems. J. Logic Comput. 26(2), 539–576 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nigam, V., Reis, G., Lima, L.: Quati: an automated tool for proving permutation lemmas. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 255–261. Springer, Cham (2014). Scholar
  20. 20.
    Pfenning, F.: Structural cut elimination I. Intuitionistic and classical logic. Inf. Comput. 157(1/2), 84–141 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge University Press, New York (1996)zbMATHGoogle Scholar
  22. 22.
    Viry, P.: Equational rules for rewriting logic. Theoret. Comput. Sci. 285(2), 487–517 (2002)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Universidade Federal do Rio Grande do NorteNatalBrazil
  2. 2.Pontificia Universidad JaverianaCaliColombia

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