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On Combinatorial Proofs for Modal Logic

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Book cover Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

In this paper we extend Hughes’ combinatorial proofs to modal logics. The crucial ingredient for modeling the modalities is the use of a self-dual non-commutative operator that has first been observed by Retoré through pomset logic. Consequently, we had to generalize the notion of skew fibration from cographs to Guglielmi’s relation webs.

Our main result is a sound and complete system of combinatorial proofs for all normal and non-normal modal logics in the \(\mathsf {S4}\)-tesseract. The proof of soundness and completeness is based on the sequent calculus with some added features from deep inference.

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Notes

  1. 1.

    Note that \(\bot \) is only allowed directly inside a or \(\Diamond \). The main purpose of avoiding \(\bot \) as proper formula is to avoid the empty relation web (to be introduced in the next section). However, we do need formulas and \(\Diamond \bot \) in order to allow weakenings inside a or \(\Diamond \), which is needed to prove the decomposition theorem (stated in Theorem 2.2 below) which in turn is the basis for combinatorial proofs.

  2. 2.

    The logics defined by these systems can be seen as the “linear logic variants” of the standard modal logics \(\mathsf K\) and \(\mathsf K\mathsf {D}\).

  3. 3.

    Observe that all the logics in the \(\mathsf {S4}\)-tesseract are monotone. In fact, our methods can not be applied in presence of the rule . We therefore have to leave the investigation of combinatorial proofs for non-monotonic non-normal modal logics as an open problem for future research.

References

  1. Acclavio, M., Straßburger, L.: From syntactic proofs to combinatorial proofs. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 481–497. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_32

    Chapter  Google Scholar 

  2. Acclavio, M., Straßburger, L.: On combinatorial proofs for logics of relevance and entailment. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 1–16. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-59533-6_1

    Chapter  Google Scholar 

  3. Avron, A.: The method of hypersequents in the proof theory of propositional non-classical logics. In: Logic: From Foundations to Applications, European Logic Colloquium, pp. 1–32. Oxford University Press (1994)

    Google Scholar 

  4. Benjamin, R., Straßburger, L.: Towards a combinatorial proof theory. In: Cerrito, S., Popescu, A., (eds.) TABLEAUX 2019. LNAI, vol. 11714, pp. 259–276 (2019)

    Google Scholar 

  5. Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48(6), 551–577 (2009)

    Article  MathSciNet  Google Scholar 

  6. Brünnler, K., Tiu, A.F.: A local system for classical logic. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 347–361. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45653-8_24

    Chapter  MATH  Google Scholar 

  7. Chaudhuri, K., Marin, S., Straßburger, L.: Focused and synthetic nested sequents. In: Jacobs, B., Löding, C. (eds.) FoSSaCS 2016. LNCS, vol. 9634, pp. 390–407. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49630-5_23

    Chapter  MATH  Google Scholar 

  8. Chaudhuri, K., Marin, S., Straßburger, L.: Modular focused proof systems for intuitionistic modal logics. In: Kesner, D., Pientka, B. (eds.) FSCD 2016. LIPIcs, vol. 52, pp. 16:1–16:18. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)

    Google Scholar 

  9. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Logic 44(1), 36–50 (1979)

    Article  MathSciNet  Google Scholar 

  10. Goré, R., Postniece, L., Tiu, A.: On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. Log. Methods Comput. Sci. 7(2), 1–38 (2011)

    Article  MathSciNet  Google Scholar 

  11. Goré, R., Ramanayake, R., et al.: Labelled tree sequents, tree hypersequents and nested (deep) sequents. Adv. Modal Log. 9, 279–299 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Guglielmi, A.: A system of interaction and structure. ACM Trans. Comput. Log. 8(1), 1–64 (2007)

    Article  MathSciNet  Google Scholar 

  13. 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). https://doi.org/10.1007/3-540-44802-0_5

    Chapter  Google Scholar 

  14. Hein, R., Stewart, C.: Purity through unravelling. In: Structures and Deduction, pp. 126–143 (2005)

    Google Scholar 

  15. Hughes, D.: Proofs without syntax. Ann. Math. 164(3), 1065–1076 (2006)

    Article  MathSciNet  Google Scholar 

  16. Hughes, D.: Towards Hilbert’s 24\({}^{\text{ th }}\) problem: combinatorial proof invariants: (preliminary version). Electr. Notes Theor. Comput. Sci. 165, 37–63 (2006)

    Article  Google Scholar 

  17. Hughes, D.J.D.: First-order proofs without syntax, June 2019

    Google Scholar 

  18. Lellmann, B.: Hypersequent rules with restricted contexts for propositional modal logics. Theor. Comput. Sci. 656, 76–105 (2016)

    Article  MathSciNet  Google Scholar 

  19. Lellmann, B., Pimentel, E.: Modularisation of sequent calculi for normal and non-normal modalities. ACM Trans. Comput. Log. (TOCL) 20(2), 7 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Marin, S., Straßburger, L.: Label-free modular systems for classical and intuitionistic modal logics. In: Advances in Modal Logic 10 (2014)

    Google Scholar 

  21. Möhring, R.H.: Computationally tractable classes of ordered sets. In: Rival, I. (ed.) Algorithms and Order, pp. 105–194. Kluwer, Dordrecht (1989)

    Chapter  Google Scholar 

  22. Negri, S.: Proof analysis in modal logic. J. Philos. Log. 34(5–6), 507 (2005)

    Article  MathSciNet  Google Scholar 

  23. Retoré, C.: Pomset logic: a non-commutative extension of classical linear logic. In: de Groote, P., Roger Hindley, J. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 300–318. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62688-3_43

    Chapter  Google Scholar 

  24. Retoré, C.: Handsome proof-nets: perfect matchings and cographs. Theor. Comput. Sci. 294(3), 473–488 (2003)

    Article  MathSciNet  Google Scholar 

  25. Simpson, A.K.: The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh. College of Science and Engineering (1994)

    Google Scholar 

  26. Stewart, C., Stouppa, P.: A systematic proof theory for several modal logics. Adv. Modal Log. 5, 309–333 (2004)

    MATH  Google Scholar 

  27. Straßburger, L.: A characterization of medial as rewriting rule. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 344–358. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73449-9_26

    Chapter  MATH  Google Scholar 

  28. 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). https://doi.org/10.1007/978-3-642-37075-5_14

    Chapter  Google Scholar 

  29. Straßburger, L.: Combinatorial flows and their normalisation. In: Miller, D. (ed.) FSCD 2017. LIPIcs, vol. 84, pp. 31:1–31:17. Schloss Dagstuhl (2017)

    Google Scholar 

  30. Straßburger, L.: The problem of proof identity, and why computer scientists should care about Hilbert’s 24th problem. Philos. Trans. R. Soc. A 377(2140), 20180038 (2019)

    Article  MathSciNet  Google Scholar 

  31. Thiele, R.: Hilbert’s twenty-fourth problem. Am. Math. Mon. 110, 1–24 (2003)

    Article  MathSciNet  Google Scholar 

  32. Wansing, H.: Sequent calculi for normal modal propositional logics. J. Log. Comput. 4(2), 125–142 (1994)

    Article  MathSciNet  Google Scholar 

  33. Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol. 8, pp. 61–145. Springer, Dordrecht (2002)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Università Roma Tre, Rome, Italy

    Matteo Acclavio

  2. Inria Saclay, Palaiseau, France

    Lutz Straßburger

Authors
  1. Matteo Acclavio
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  2. Lutz Straßburger
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Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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Acclavio, M., Straßburger, L. (2019). On Combinatorial Proofs for Modal Logic. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_13

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  • DOI: https://doi.org/10.1007/978-3-030-29026-9_13

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