Abstract
Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system \({ G3K} \) of Negri for the modal logic \(K\) and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in \({ G3K} \) corresponds to a bipole—a pair of a positive and a negative phases—in \({ LKF} \). Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Note that, for simplicity, as in [14], we restrict to the case where only a single variable is bound to each existential quantifier.
- 2.
In fact, it is possible to show that every modal formula can be translated into a formula in the fragment of first-order logic which uses only two variables [2]. By the decidability of such a fragment, an easy proof of the decidability of propositional modal logic follows.
- 3.
Note that in \({ LKF} \) we consider one-sided sequents and the one we propose is in fact a polarization of the negation of the axiom.
- 4.
We note that in this way, we provide no information on which substitution term to use in case of existential quantifiers, and let such terms be reconstructed by the checker. In order to obtain a completely faithful encoding of the original \({ G3K} \) proof, the label term used for instantiating \(\lozenge \)-formulas should also be contained in the proof certificate and the expert predicate for the \(\exists \) should take that into account.
References
Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Logic Comput. 2(3), 297–347 (1992)
Blackburn, P., Van Benthem, J.: Modal logic: a semantic perspective. In: Handbook of Modal Logic, pp. 1–82. Elsevier (2007)
Chihani, Z., Miller, D., Renaud, F.: Foundational proof certificates in first-order logic. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 162–177. Springer, Heidelberg (2013)
Chihani, Z., Libal, T., Reis, G.: The Proof Certifier Checkers. To appear in Tableaux, System Description (2015)
Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Arch. Math. Logic 51(1–2), 71–92 (2012)
Dyckhoff, R., Negri, S.: Geometrisation of first-order logic. Bull. Symbolic Logic 21, 123–163 (2015)
Fitting, M.: Modal proof theory. In: Wolter, F., Blackburn, P., van Benthem, J. (eds.) Handbook of Modal Logic, pp. 85–138. Elsevier, New York (2007)
Gabbay, D.M.: Labelled Deductive Systems. Clarendon Press, Oxford (1996)
Girard, J.-Y.: On the meaning of logical rules I: syntax vs. semantics. In: Berger, U., Schwichtenberg, H. (eds.) Computational Logic. NATO ASI, pp. 215–272. Springer, Heidelberg (1999)
Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theo. Comput. Sci. 410(46), 4747–4768 (2009)
Miller, D.: A proposal for broad spectrum proof certificates. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 54–69. Springer, Heidelberg (2011)
Miller, D., Nadathur, G.: Programming with Higher-Order Logic. Cambridge University Press, Cambridge (2012)
Miller, D., Pimentel, E.: A formal framework for specifying sequent calculus proof systems. Theo. Comput. Sci. 474, 98–116 (2013)
Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–544 (2005)
Negri, S., von Plato, J.: Cut elimination in the presence of axioms. Bull. Symbolic Logic 4(4), 418–435 (1998)
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
Sahlqvist, H.: Completeness and correspondence in first and second order semantics for modal logic. In: Kanger, S., (ed.) Proceedings of the Third Scandinavian Logic Symposium, pp. 110–143, North Holland (1975)
Simpson, A.K.: The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, School of Informatics, University of Edinburgh (1994)
Viganò, L.: Labelled Non-Classical Logics. Kluwer Academic Publishers, Dordrecht (2000)
Acknowledgments
This work was carried out during the tenure of an ERCIM Alain Bensoussan Fellowship Programme by the second author and was funded by the ERC Advanced Grant ProofCert.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Miller, D., Volpe, M. (2015). Focused Labeled Proof Systems for Modal Logic. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_19
Download citation
DOI: https://doi.org/10.1007/978-3-662-48899-7_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48898-0
Online ISBN: 978-3-662-48899-7
eBook Packages: Computer ScienceComputer Science (R0)