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LINEAR TIME IN HYPERSEQUENT FRAMEWORK

Published online by Cambridge University Press:  29 March 2016

ANDRZEJ INDRZEJCZAK
ANDRZEJ INDRZEJCZAK*
Affiliation:
DEPARTMENT OF LOGIC INSTITUTE OF PHILOSOPHY UNIVERSITY OF LODZ KOPCIŃSKIEGO 16/18 90–232 ŁÓDŹ, POLANDE-mail: [email protected]
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Abstract

Hypersequent calculus (HC), developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi (SC). In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and provide a cut-free HC formalization of Kt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite (multi)sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules (three for future-necessity and three dual rules for past-necessity). Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents.

Keywords

03F0303B44proof theorytemporal logichypersequent calculi
Type
Communication
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 1 , March 2016 , pp. 121 - 144
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Avron, Arnon, A constructive analysis of RM. The Journal of Symbolic Logic, vol. 52 (1987), pp. 939–951.Google Scholar
Avron, Arnon, Using hypersequents in proof systems for non-classical logics. Annals of Mathematics and Artificial Intelligence, vol. 4 (1991), pp. 225–248.Google Scholar
Avron, Arnon, The method of hypersequents in the proof theory of logic, Logic: From Foundations to Applications (Hodges, W. et al. ., editors), Oxford Science Publication, Oxford, 1996, pp. 1–32.Google Scholar
Avron, Arnon, A simple proof of completeness and cut-elimination for propositional Gödel logic. Journal of Logic and Computation, vol. 21 (2011), no. 5, pp. 813–821.CrossRefGoogle Scholar
Baaz, Mathias and Ciabattoni, Agata, A Schütte-Tait style cut-elimination proof for first-order Gödel logic, Automated Reasoning with Tableaux and Related Methods (Tableaux ‘02), LNAI, vol. 2381, Springer, Dordrecht, 2002, pp. 24–38.Google Scholar
Baaz, Mathias, Ciabattoni, Agata, and Fermüller, Christian. Hypersequent calculi for Gödel logics—a survey. Journal of Logic and Computation, vol. 13 (2003), pp. 1–27.Google Scholar
Baldoni, Matteo, Normal multimodal logics: Automatic deduction and logic programming extension, Ph.D. Thesis, University of Torino, 1998.Google Scholar
Bednarska, Kaja and Indrzejczak, Andrzej, Hypersequent calculi for S5—the methods of cut-elimination. Logic and Logical Philosophy, vol. 24 (2015), no. 3, pp. 277–311.Google Scholar
Belnap, Nuel D., Display logic. Journal of Philosophical Logic, vol. 11 (1982), pp. 375–417.Google Scholar
Brünnler, Kai, Deep sequent systems for modal logic. Archive for Mathematical Logic, vol. 48 (2009), no. 6, pp. 551–577.Google Scholar
Catach, Laurent, TABLEAUX: A general theorem prover for modal logics. Journal of Automated Reasoning, vol. 7 (1991), no. 4, pp. 489–510.Google Scholar
Ciabattoni, Agata and Ferrari, Marco, Hypertableau and path-hypertableau calculi for some families of intermediate logics, Automated Reasoning with Tableaux and Related Methods (Tableaux ‘00), LNAI, vol. 2381, Springer, Dordrecht, 2000, pp. 24–38.Google Scholar
Ciabattoni, Agata and Ramanayake, Revantha, Structural extensions of display calculi: A general recipe, Proceedings of WOLLIC, Springer, Dordrecht, 2013.Google Scholar
Ciabattoni, Agata, Ramanayake, Revantha, and Wansing, Heinrich, Hypersequent and display calculi—a unified perspective. Studia Logica, vol. 102 (2014), no. 6, pp. 1245–1294.Google Scholar
Dos̆en, Kosta, Sequent-systems for modal logic. The Journal of Symbolic Logic, vol. 50 (1985), pp. 149–159.CrossRefGoogle Scholar
Fitting, Melvin, Intuitionistic Logic Model Theory and Forcing, North-Holland, Amsterdam, 1969.Google Scholar
Gallier, Jean H., Logic for Computer Science, Harper and Row, New York, 1986.Google Scholar
Goldblatt, Robert, Logics of Time and Computation, CSLI Lecture Notes, Stanford University, Stanford, 1987.Google Scholar
Goré, Rajev, Tableau methods for modal and temporal logics, Handbook of Tableau Methods (D’Agostino, M. et al. ., editors), Kluwer Academic Publishers, Dordrecht, 1999, pp. 297–396.Google Scholar
Hodges, Wilfried, Elementary predicate logic, Handbook of Philosophical Logic, vol. I (Gabbay, D. and Guenther, F., editors), Kluwer Academic Publishers, Dordrecht, 1983, pp. 1–132.Google Scholar
Horrocks, Ian, Satler, Uve, and Tobies, Stephen, Practical reasoning for very expressive description logics. Logic Journal of the IGPL, vol. 8 (2000), no. 3, pp. 239–263.CrossRefGoogle Scholar
Indrzejczak, Andrzej, Natural deduction system for tense logics. Bulletin of the Section of Logic, vol. 23 (1994), no. 4, pp. 173–179.Google Scholar
Indrzejczak, Andrzej, A labelled natural deduction system for linear temporal logic. Studia Logica, vol. 75 (2003), no. 3, pp. 345–376.CrossRefGoogle Scholar
Indrzejczak, Andrzej, Natural Deduction, Hybrid Systems and Modal Logics, Springer, Heidelberg, Dordrecht, 2010.Google Scholar
Indrzejczak, Andrzej, Cut-free hypersequent calculus for S4.3. Bulletin of the Section of Logic, vol. 41 (2012), nos. 1–2, pp. 89–104.Google Scholar
Indrzejczak, Andrzej, Cut-free hypersequent calculus for the basic temporal logic of linear frames, Short Papers Presented at AiML, Groningen, 2014, pp. 29–33.Google Scholar
Indrzejczak, Andrzej, Eliminability of cut in hypersequent calculi for some modal logics of linear frames. Information Processing Letters, vol. 115 (2015) ,no. 2, pp. 75–81.Google Scholar
Kashima, Ryo, Cut-free sequent calculi for some tense logics. Studia Logica, vol. 53 (1994), pp. 119–135.Google Scholar
Kleene, Stephen C., Mathematical Logic, Willey, New York, 1967.Google Scholar
Kracht, Marcus, Power and weakness of the modal display calculus, Proof Theory of Modal Logic (Wansing, H., editor), Kluwer Academic Publishers, Dordrecht, 1996, pp. 95–122.Google Scholar
Kurokawa, Hidenori, Hypersequent calculi for modal logics extending S4, New Frontiers in Artificial Intelligence (2013), Springer, Dordrecht, 2014, pp. 51–68.Google Scholar
Lahav, Ori, From frame properties to hypersequent rules in modal logics, Proceedings of LICS, IEEE Publications, New Orleans, 2013, pp. 408–417.Google Scholar
Lellmann, Bjorn, Axioms vs hypersequent rules with context restrictions, Proceedings of IJCAR, Springer, Dordrecht, 2014, pp. 307–321.Google Scholar
Marx, Marteen, Mikulas, Sabolcs, and Reynolds, Mike, The mosaic method for temporal logics, Automated Reasoning with Analytic Tableaux and Related Methods, Proc. of International Conference TABLEAUX 2000, Saint Andrews, Scotland, LNAI 1847 (Dyckhoff, R., editor), Springer, Dordrecht, 2000, pp. 324–340.Google Scholar
Metcalfe, George, Olivetti, Niccola, and Gabbay, Dov, Proof Theory for Fuzzy Logics, Springer, Dordrecht, 2008.Google Scholar
Negri, Sara, Proof analysis in modal logic. Journal of Philosophical Logic, vol. 34 (2005), pp. 507–544.Google Scholar
Negri, Sara, Kripke completeness revisited, Acts of Knowledge—History, Philosophy and Logic (Primero, G. and Rahman, S., editors), College Publications, London, 2009, pp. 247–282.Google Scholar
Negri, Sara and von Plato, Jan, Structural Proof Theory, Cambridge University Press, Cambridge, 2001.Google Scholar
Poggiolesi, Francesca, A cut-free simple sequent calculus for modal logic S5. Review of Symbolic Logic, vol. 1 (2008), pp. 3–15.CrossRefGoogle Scholar
Poggiolesi, Francesca, Gentzen Calculi for Modal Propositional Logics, Springer, Dordrecht, 2011.Google Scholar
Pottinger, Garrel, Uniform cut-free formulations of T, S4 and S5 (abstract). The Journal of Symbolic Logic, vol. 48 (1983), p. 900.Google Scholar
Ramanayake, Revantha, Embedding the hypersequent calculus in the display calculus. Journal of Logic and Computation, vol. 25 (2015), no. 3, pp. 921–942.Google Scholar
Rescher, Nicolas and Urquhart, Alasdair, Temporal Logic, Springer, Berlin, 1971.Google Scholar
Restall, Greg, Proofnets for S5: Sequents and circuits for modal logic, Lecture Notes in Logic, vol. 28, Cambridge University Press, Cambridge, 2007, pp. 151–172.Google Scholar
Schütte, Kurt, Proof Theory, Springer, Berlin, 1977.Google Scholar
Stouppa, Phinniki, A deep inference system for the modal logic S5. Studia Logica, vol. 85 (2007), pp. 199–214.Google Scholar
Tait, William W., Normal derivability in classical logic, The Sintax and Semantics of Infinitary Languages, LNM, vol. 72, 1968, pp. 204–236.Google Scholar
Takeuti, Gaisi, Proof Theory, North-Holland, Amsterdam, 1987.Google Scholar
Benthem, Johan van, The Logic of Time, Kluwer Academic Publishers, Dordrecht, 1983.Google Scholar
Wansing, Heinrich, Displaying Modal Logics, Kluwer Academic Publishers, Dordrecht, 1999.Google Scholar
Zeman, Jay, Modal Logic, Oxford University Press, Oxford, 1973.Google Scholar