Sequent Calculus in the Topos of Trees

  • Ranald CloustonEmail author
  • Rajeev Goré
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


Nakano’s “later” modality, inspired by Gödel-Löb provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic KM lin . We give a sound and cut-free complete sequent calculus for KM lin via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.


Modal Logic Intuitionistic Logic Sequent Calculus Internal Logic Classical Propositional Logic 
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  1. 1.
    Appel, A.W., Melliès, P.A., Richards, C.D., Vouillon, J.: A very modal model of a modern, major, general type system. In: POPL, pp. 109–122 (2007)Google Scholar
  2. 2.
    Bengtson, J., Jensen, J.B., Sieczkowski, F., Birkedal, L.: Verifying object-oriented programs with higher-order separation logic in Coq. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 22–38. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Birkedal, L., Møgelberg, R.E.: Intensional type theory with guarded recursive types qua fixed points on universes. In: LICS, pp. 213–222 (2013)Google Scholar
  4. 4.
    Birkedal, L., Møgelberg, R.E., Schwinghammer, J., Støvring, K.: First steps in synthetic guarded domain theory: Step-indexing in the topos of trees. LMCS 8(4) (2012)Google Scholar
  5. 5.
    Birkedal, L., Schwinghammer, J., Støvring, K.: A metric model of lambda calculus with guarded recursion. In: FICS, pp. 19–25 (2010)Google Scholar
  6. 6.
    Bizjak, A., Birkedal, L., Miculan, M.: A model of countable nondeterminism in guarded type theory. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 108–123. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  7. 7.
    Boolos, G.: The logic of provability. CUP (1995)Google Scholar
  8. 8.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. OUP (1997)Google Scholar
  9. 9.
    Clouston, R., Bizjak, A., Grathwohl, H.B., Birkedal, L.: Programming and reasoning with guarded recursion for coinductive types. In: Pitts, A. (ed.) FoSSaCS 2015. LNCS, vol. 9034, pp. 407–421. Springer, Heidelberg (2015)Google Scholar
  10. 10.
    Clouston, R., Goré, R.: Sequent calculus in the topos of trees. arXiv:1501.03293, extended version (2015)Google Scholar
  11. 11.
    Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  12. 12.
    Corsi, G.: Semantic trees for Dummett’s logic LC. Stud. Log. 45(2), 199–206 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dreyer, D., Ahmed, A., Birkedal, L.: Logical step-indexed logical relations. In: LICS, pp. 71–80 (2009)Google Scholar
  14. 14.
    Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Arch. Math. Log. 51(1-2), 71–92 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ferrari, M., Fiorentini, C., Fiorino, G.: Contraction-free linear depth sequent calculi for intuitionistic propositional logic with the subformula property and minimal depth counter-models. J. Autom. Reason. 51(2), 129–149 (2013)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Fiorino, G.: Terminating calculi for propositional Dummett logic with subformula property. J. Autom. Reason. 52(1), 67–97 (2014)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Garg, D., Genovese, V., Negri, S.: Countermodels from sequent calculi in multi-modal logics. In: LICS, pp. 315–324 (2012)Google Scholar
  18. 18.
    Hirai, Y.: A lambda calculus for Gödel–Dummett logic capturing waitfreedom. In: Schrijvers, T., Thiemann, P. (eds.) FLOPS 2012. LNCS, vol. 7294, pp. 151–165. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Hobor, A., Appel, A.W., Nardelli, F.Z.: Oracle semantics for concurrent separation logic. In: Drossopoulou, S. (ed.) ESOP 2008. LNCS, vol. 4960, pp. 353–367. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Ishigaki, R., Kikuchi, K.: Tree-sequent methods for subintuitionistic predicate logics. In: Olivetti, N. (ed.) TABLEAUX 2007. LNCS (LNAI), vol. 4548, pp. 149–164. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Krishnaswami, N.R., Benton, N.: A semantic model for graphical user interfaces. In: ICFP, pp. 45–57 (2011)Google Scholar
  22. 22.
    Krishnaswami, N.R., Benton, N.: Ultrametric semantics of reactive programs. In: LICS, pp. 257–266 (2011)Google Scholar
  23. 23.
    Litak, T.: A typing system for the modalized Heyting calculus. In: COS (2013)Google Scholar
  24. 24.
    Litak, T.: Constructive modalities with provability smack, author’s cut v. 2.03 (2014) (retrieved from author’s website)Google Scholar
  25. 25.
    Milius, S., Litak, T.: Guard your daggers and traces: On the equational properties of guarded (co-) recursion. arXiv:1309.0895 (2013)Google Scholar
  26. 26.
    Muravitsky, A.: Logic KM: A biography. Outstanding Contributions to Logic 4, 155–185 (2014)CrossRefGoogle Scholar
  27. 27.
    Nakano, H.: A modality for recursion. In: LICS, pp. 255–266 (2000)Google Scholar
  28. 28.
    Pottier, F.: A typed store-passing translation for general references. In: POPL, pp. 147–158 (2011)Google Scholar
  29. 29.
    Restall, G.: Subintuitionistic logics. NDJFL 34(1), 116–129 (1994)MathSciNetGoogle Scholar
  30. 30.
    Rowe, R.N.: Semantic Types for Class-based Objects. Ph.D. thesis, Imperial College London (2012)Google Scholar
  31. 31.
    Sonobe, O.: A Gentzen-type formulation of some intermediate propositional logics. J. Tsuda College 7, 7–14 (1975)MathSciNetGoogle Scholar
  32. 32.
    Troelstra, A., Schwichtenberg, H.: Basic Proof Theory. CUP (1996)Google Scholar
  33. 33.
    Wolter, F., Zakharyaschev, M.: Intuitionistic modal logics. In: Logic and Foundations of Mathematics, pp. 227–238 (1999)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceAarhus UniversityAarhusDenmark
  2. 2.Research School of Computer ScienceAustralian National UniversityCanberraAustralia

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