Abstract
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.
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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)
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)
Birkedal, L., Møgelberg, R.E.: Intensional type theory with guarded recursive types qua fixed points on universes. In: LICS, pp. 213–222 (2013)
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)
Birkedal, L., Schwinghammer, J., Støvring, K.: A metric model of lambda calculus with guarded recursion. In: FICS, pp. 19–25 (2010)
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)
Boolos, G.: The logic of provability. CUP (1995)
Chagrov, A., Zakharyaschev, M.: Modal Logic. OUP (1997)
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)
Clouston, R., Goré, R.: Sequent calculus in the topos of trees. arXiv:1501.03293, extended version (2015)
Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)
Corsi, G.: Semantic trees for Dummett’s logic LC. Stud. Log. 45(2), 199–206 (1986)
Dreyer, D., Ahmed, A., Birkedal, L.: Logical step-indexed logical relations. In: LICS, pp. 71–80 (2009)
Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Arch. Math. Log. 51(1-2), 71–92 (2012)
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)
Fiorino, G.: Terminating calculi for propositional Dummett logic with subformula property. J. Autom. Reason. 52(1), 67–97 (2014)
Garg, D., Genovese, V., Negri, S.: Countermodels from sequent calculi in multi-modal logics. In: LICS, pp. 315–324 (2012)
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)
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)
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)
Krishnaswami, N.R., Benton, N.: A semantic model for graphical user interfaces. In: ICFP, pp. 45–57 (2011)
Krishnaswami, N.R., Benton, N.: Ultrametric semantics of reactive programs. In: LICS, pp. 257–266 (2011)
Litak, T.: A typing system for the modalized Heyting calculus. In: COS (2013)
Litak, T.: Constructive modalities with provability smack, author’s cut v. 2.03 (2014) (retrieved from author’s website)
Milius, S., Litak, T.: Guard your daggers and traces: On the equational properties of guarded (co-) recursion. arXiv:1309.0895 (2013)
Muravitsky, A.: Logic KM: A biography. Outstanding Contributions to Logic 4, 155–185 (2014)
Nakano, H.: A modality for recursion. In: LICS, pp. 255–266 (2000)
Pottier, F.: A typed store-passing translation for general references. In: POPL, pp. 147–158 (2011)
Restall, G.: Subintuitionistic logics. NDJFL 34(1), 116–129 (1994)
Rowe, R.N.: Semantic Types for Class-based Objects. Ph.D. thesis, Imperial College London (2012)
Sonobe, O.: A Gentzen-type formulation of some intermediate propositional logics. J. Tsuda College 7, 7–14 (1975)
Troelstra, A., Schwichtenberg, H.: Basic Proof Theory. CUP (1996)
Wolter, F., Zakharyaschev, M.: Intuitionistic modal logics. In: Logic and Foundations of Mathematics, pp. 227–238 (1999)
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Clouston, R., Goré, R. (2015). Sequent Calculus in the Topos of Trees. In: Pitts, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2015. Lecture Notes in Computer Science(), vol 9034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46678-0_9
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