Abstract
Proof systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labeling approach lies at the heart of most resolution, natural deduction, and tableau systems for hybrid logic. But there is another, less well-known approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was first introduced and explored by Jerry Seligman in the setting of natural deduction and sequent calculus in the late 1990s. The purpose of this paper is to introduce a Seligman-style tableau system.
The most obvious feature of Seligman-style systems is that they work with arbitrary formulas, not just formulas prefixed by @-operators. To achieve this in a tableau system, we introduce a rule called GoTo which allows us to “jump to a named world” on a tableau branch, thereby creating a local proof context (which we call a block) on that branch. To the surprise of some of the authors (who have worked extensively on developing the labeling approach) Seligman-style inference is often clearer: not only is the approach more modular, individual proofs can be more direct. We briefly discuss termination and extensions to richer logics, and relate our system to Seligman’s original sequent calculus.
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References
Areces, C., Heguiabehere, J.: Direct Resolution for Modal-like Logics. In: Proceedings of the 3rd International Workshop on the Implementation of Logics, Tbilisi, Georgia, pp. 3–16 (2002)
Areces, C., Gorín, D.: Ordered Resolution with Selection for \(\mathcal{H}(@)\). In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 125–141. Springer, Heidelberg (2005)
Areces, C., Gorín, D.: Resolution with Order and Selection for Hybrid Logics. J. Autom. Reasoning 46(1), 1–42 (2011)
Bierman, G., de Paiva, V.: On an Intuitionistic Modal Logic. Studia Logica 65, 383–416 (2000)
Blackburn, P.: Internalizing labelled deduction. Journal of Logic and Computation 10(1), 137–168 (2000)
Blackburn, P., Jørgensen, K.F.: Indexical Hybrid Tense Logic. In: Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (eds.) Advances in Modal Logic, vol. 9, pp. 144–160 (2012)
Blackburn, P., Marx, M.: Tableaux for quantified hybrid logic. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 38–52. Springer, Heidelberg (2002)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Blackburn, P., Tzakova, M.: Hybrid Languages and Temporal Logic. Logic Journal of the IGPL 7(1), 27–54 (1999)
Bolander, T., Blackburn, P.: Termination for Hybrid Tableaus. Journal of Logic and Computation 17(3), 517–554 (2007)
Bolander, T., Braüner, T.: Tableau-Based Decision Procedures for Hybrid Logic. Journal of Logic and Computation 16, 737–763 (2006)
Braüner, T.: Two natural deduction systems for hybrid logic: A comparison. Journal of Logic, Language and Information 13, 1–23 (2004)
Braüner, T.: Hybrid Logic and its Proof-Theory. Applied Logic Series, vol. 37. Springer (2011)
Braüner, T.: Hybrid-logical Reasoning in False-Belief Tasks. In: Schipper, B. (ed.) Proceedings of Fourteenth Conference on Theoretical Aspects of Rationality and Knowledge (TARK), pp. 186–195 (2013), http://tark.org
Braüner, T., Ghilardi, S.: First-order modal logic. In: Handbook of Modal Logic, pp. 549–620. Elsevier (2007)
Chadha, R., Macedonio, D., Sassone, V.: A hybrid intuitionistic logic: Semantics and decidability. Journal of Logic and Computation 16, 27–59 (2006)
Fitting, M., Mendelsohn, R.: First-Order Modal Logic. Springer (1998)
Galmiche, D., Salhi, Y.: Sequent calculi and decidability for intuitionistic hybrid logic. Information and Computation 209, 1447–1463 (2011)
Götzmann, D., Kaminski, M., Smolka, G.: Spartacus: A Tableau Prover for Hybrid Logic. Electr. Notes Theor. Comput. Sci. 262, 127–139 (2010)
Hoffmann, G., Areces, C.: HTab: A Terminating Tableaux System for Hybrid Logic. In: Proceedings of Methods for Modalities, vol. 5 (November 2007)
Hoffmann, G.: Tâches de raisonnement en logiques hybrides. Ph.D. thesis, Université Henri Poincaré - Nancy I (December 2010), http://tel.archives-ouvertes.fr/tel-00541664
Jia, L.-m., Walker, D.W.: Modal proofs as distributed programs (extended abstract). In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 219–233. Springer, Heidelberg (2004)
Kushida, H., Okada, M.: A Proof-Theoretic Study of the Correspondence of Hybrid Logic and Classical Logic. Journal of Logic, Language and Information 16, 35–61 (2007)
Seligman, J.: The Logic of Correct Description. In: de Rijke, M. (ed.) Advances in Intensional Logic. Applied Logic Series, vol. 7, pp. 107–135. Kluwer (1997)
Seligman, J.: Internalisation: The Case of Hybrid Logics. Journal of Logic and Computation 11, 671–689 (2001); special Issue on Hybrid Logics. Areces, C., Blackburn, P. (eds.)
Tzakova, M.: Tableau Calculi for Hybrid Logics. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 278–292. Springer, Heidelberg (1999)
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Blackburn, P., Bolander, T., Braüner, T., Jørgensen, K.F. (2013). A Seligman-Style Tableau System. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2013. Lecture Notes in Computer Science, vol 8312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45221-5_11
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