Studia Logica

, Volume 102, Issue 6, pp 1245–1294 | Cite as

Hypersequent and Display Calculi – a Unified Perspective

  • Agata Ciabattoni
  • Revantha Ramanayake
  • Heinrich Wansing
Article

Abstract

This paper presents an overview of the methods of hypersequents and display sequents in the proof theory of non-classical logics. In contrast with existing surveys dedicated to hypersequent calculi or to display calculi, our aim is to provide a unified perspective on these two formalisms highlighting their differences and similarities and discussing applications and recent results connecting and comparing them.

Keywords

Proof theory Hypersequent calculi Display calculi 

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References

  1. 1.
    Andreoli J.-M.: Logic programming with focusing proofs in linear logic. J. Logic Comput. 2(3), 297–347 (1992)CrossRefGoogle Scholar
  2. 2.
    Avron A.: A constructive analysis of RM. J. of Symbolic Logic 52(4), 939–951 (1987)CrossRefGoogle Scholar
  3. 3.
    Avron A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4(3-4), 225–248 (1991)CrossRefGoogle Scholar
  4. 4.
    Avron, A., The method of hypersequents in the proof theory of propositional nonclassical logics. In Logic: from foundations to applications (Staffordshire, 1993), Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 1–32.Google Scholar
  5. 5.
    Avron A., Lev I.: Non-deterministic multi-valued structures. J. Logic Comput. 15, 241–261 (2005)CrossRefGoogle Scholar
  6. 6.
    Baaz M., Ciabattoni A., Fermüller C.G.: Hypersequent calculi for Gödel logics: A survey. J. Logic Comput. 13, 1–27 (2003)CrossRefGoogle Scholar
  7. 7.
    Baaz M., Ciabattoni A., Montagna F.: Analytic calculi for monoidal t-norm based logic. Fundamenta Informaticae 59(4), 315–332 (2004)Google Scholar
  8. 8.
    Baaz, M., and C. G. Fermüller, Analytic calculi for projective logics. In Tableaux ’99. LNAI, vol. 1617. Springer, 1999, pp. 36–50.Google Scholar
  9. 9.
    Baaz M., Fermüller C.G., Salzer G., Zach R.: Labeled calculi and finite-valued logics. Studia Logica 61(1), 7–33 (1998)CrossRefGoogle Scholar
  10. 10.
    Baaz M., Fermüller C.G., Zach R.: Elimination of cuts in first-order many-valued logics. J. of Inf. Proc. and Cybernetics 29, 333–355 (1994)Google Scholar
  11. 11.
    Baaz, M., O. Lahav, and A. Zamansky, Finite-valued semantics for canonical labelled calculi. J. of Automated Reasoning, 2013.Google Scholar
  12. 12.
    Baaz, M., and R. Zach, Hypersequents and the proof theory of intuitionistic fuzzy logic. In CSL’00. LNCS. Springer, 2000, pp. 187–201.Google Scholar
  13. 13.
    Baldi, P., A. Ciabattoni, and L. Spendier, Standard completeness for extensions of MTL: an automated approach. In WOLLIC 2012, vol. 7456 of LNCS. Springer, 2012, pp. 154–167.Google Scholar
  14. 14.
    Belnap N.D. Jr.: Tonk, plonk and plik. Analysis 22, 130–134 (1962)CrossRefGoogle Scholar
  15. 15.
    Belnap N.D. Jr.: Display logic. J. Philos. Logic 11(4), 375–417 (1982)CrossRefGoogle Scholar
  16. 16.
    Belnap, N. D., Jr., The display problem. In H. Wansing, (ed.), Proof Theory of Modal Logic. Kluwer, Dordrecht, 1996, pp. 79–92.Google Scholar
  17. 17.
    Blackburn, P., M. de Rijke, and I. Venema, Modal logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 2001.Google Scholar
  18. 18.
    Brotherston J.: Bunched logics displayed. Studia Logica 100(6), 1223–1254 (2012)CrossRefGoogle Scholar
  19. 19.
    Brotherston, J., and R. Goré, Craig interpolation in displayable logics. In Tableaux 2011, vol. 6793 of LNCS. Springer, 2011, pp. 88–103.Google Scholar
  20. 20.
    Brünnler, K., Deep sequent systems for modal logic. In Advances in modal logic. Vol. 6. Coll. Publ., London, 2006, pp. 107–119.Google Scholar
  21. 21.
    Ciabattoni, A., Automated generation of analytic calculi for logics with linearity. In CSL04. LNCS, vol. 3210. Springer, 2004, pp. 503–517.Google Scholar
  22. 22.
    Ciabattoni, A., A proof-theoretical investigation of global intuitionistic (fuzzy) logic. Archive of mathematical Logic 44:435–457, 2005.Google Scholar
  23. 23.
    Ciabattoni, A., C. G. Fermüller, and G. Metcalfe, Uniform rules and dialogue games for fuzzy logics. In LPAR 2004. LNCS, vol. 3452, 2004, pp. 496–510.Google Scholar
  24. 24.
    Ciabattoni, A., and M. Ferrari, Hypertableau and path-hypertableau calculi for some families of intermediate logics. In Tableaux 2000. LNAI, vol. 1847, 2000, pp. 160–175.Google Scholar
  25. 25.
    Ciabattoni A., Gabbay D., Olivetti N.: Cut-free proof systems for logics of weak excluded middle. Soft Computing 2(4), 147–156 (1999)CrossRefGoogle Scholar
  26. 26.
    Ciabattoni, A., N. Galatos, and K. Terui, From axioms to analytic rules in nonclassical logics. In LICS 2008, 2008, pp. 229–240.Google Scholar
  27. 27.
    Ciabattoni, A., N. Galatos, and K. Terui. Algebraic proof theory: hypersequents and hypercompletions. submitted. 2014.Google Scholar
  28. 28.
    Ciabattoni, A., P. Maffezioli, and L. Spendier, Hypersequent and labelled calculi for intermediate logics. In Tableaux 2013. LNCS, vol. 8123. Springer, 2013, pp. 81–96.Google Scholar
  29. 29.
    Ciabattoni, A., and G. Metcalfe, Bounded Łukasiewicz logics. In Tableaux 2003. LNCS, vol. 2796. Springer, 2003, pp. 32–47.Google Scholar
  30. 30.
    Ciabattoni A., Metcalfe G.: Density elimination. Theor. Comput. Sci. 403(2-3), 328–346 (2008)CrossRefGoogle Scholar
  31. 31.
    Ciabattoni A., Metcalfe G., Montagna F.: Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. Fuzzy Sets and Systems 161(3), 369–389 (2010)CrossRefGoogle Scholar
  32. 32.
    Ciabattoni A., Montagna F.: Proof theory for locally finite many-valued logics: Semi-projective logics. Theor. Comput. Sci. 480, 26–42 (2013)CrossRefGoogle Scholar
  33. 33.
    Ciabattoni, A., and R. Ramanayake, Structural rule extensions of display calculi: a general recipe. In WOLLIC 2013, vol. 8071 of LNCS. Springer, 2013, pp. 81–95.Google Scholar
  34. 34.
    Ciabattoni, A., L. Strassburger, and K. Terui, Expanding the realm of systematic proof theory. In CSL 2009. LNCS. Springer, 2009, pp. 163–178.Google Scholar
  35. 35.
    Clouston, R., R. Goré, and A. Tiu, Annotation-free sequent calculi for full intuitionistic linear logic. In CSL 2013, LNCS. Springer, 2013, pp. 197–214.Google Scholar
  36. 36.
    Conradie W., Palmigiano A.: Algorithmic correspondence and canonicity for distributive modal logic. Annals of Pure and Applied Logic 163(3), 338–376 (2012)CrossRefGoogle Scholar
  37. 37.
    Corsi G.: A cut-free calculus for Dummett’s LC quantified. Mathematical Logic Quarterly 35(4), 289–301 (1989)CrossRefGoogle Scholar
  38. 38.
    Dawson, J. E., and R. Goré, Formalised cut admissibility for display logic. In Theorem proving in higher order logics, vol. 2410 of LNCS. Springer, Berlin, 2002, pp. 131–147.Google Scholar
  39. 39.
    Dawson, J. E., and R. Goré, A new machine-checked proof of strong normalisation for display logic. In Electronic Notes in Theoretical Computer Science. Elsevier, 2002.Google Scholar
  40. 40.
    Demri S., Goré R.: Display calculi for nominal tense logics. J. Logic Comput. 12(6), 993–1016 (2002)CrossRefGoogle Scholar
  41. 41.
    Dummett, M., Frege: Philosophy of Language. Harper & Row, New York, 1973.Google Scholar
  42. 42.
    Dunn, J. M., A ‘Gentzen’ system for positive relevant implication. J. of Symbolic Logic 38:356-357, 1974. (Abstract).Google Scholar
  43. 43.
    Enderton H.: A mathematical introduction to logic. Academic Press, New York (1972)Google Scholar
  44. 44.
    Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for leftcontinuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)CrossRefGoogle Scholar
  45. 45.
    Fitting, M., Proof methods for modal and intuitionistic logics, vol. 169 of Synthese Library. D. Reidel Publishing Co., Dordrecht, 1983.Google Scholar
  46. 46.
    Gabbay, D., Labelled Deductive Systems, vol. 1—Foundations. Oxford University Press, 1996.Google Scholar
  47. 47.
    Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated lattices: an algebraic glimpse at substructural logics, vol. 151 of Studies in Logic and the Foundations of Mathematics. Elsevier B. V., Amsterdam, 2007.Google Scholar
  48. 48.
    Gentzen, G., Untersuchungen über das logische schließen. Mathematische Zeitschrift 39:176–210, 405–431, 1934/35. English translation in: American Philosophical Quarterly 1 (1964), 288–306 and American Philosophical Quarterly 2 (1965), 204–218, as well as in: The Collected Papers of Gerhard Gentzen, (ed. M. E. Szabo), Amsterdam, North Holland (1969), pp. 68–131.Google Scholar
  49. 49.
    Goldblatt, R., Topoi. The Categorical Analysis of Logic. North-Holland, Amsterdam, 1979.Google Scholar
  50. 50.
    Goré, R., Gaggles, Gentzen and Galois: how to display your favourite substructural logic. Log. J. IGPL 6(5):669–694, 1998.Google Scholar
  51. 51.
    Goré R.: Substructural logics on display. Log. J. IGPL 6(3), 451–504 (1998)CrossRefGoogle Scholar
  52. 52.
    Goré, R., L. Postniece, and A. Tiu, On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. Log. Methods Comput. Sci. 7(2):2:8, 38, 2011.Google Scholar
  53. 53.
    Goré, R., and R. Ramanayake, Labelled tree sequents, tree hypersequents and nested (deep) sequents. In Advances in modal logic. Volume 9. College Publications, London, 2012.Google Scholar
  54. 54.
    Goré R., Tiu A.: Classical modal display logic in the calculus of structures and minimal cut-free deep inference calculi for S5. J. Logic Comput. 17(4), 767–794 (2007)CrossRefGoogle Scholar
  55. 55.
    Grazl, N., Sequent calculi for multi-modal logic with interaction. In D. Grossi, O. Roy, and H. Huang, (eds.), Logic, Rationality, and Interaction, vol. 8196 of Lecture Notes in Computer Science Springer Berlin Heidelberg, 2013, pp. 124–134.Google Scholar
  56. 56.
    Guglielmi, A., Deep inference. http://alessio.guglielmi.name/res/cos/index.html. Accessed: 2013-10-14.
  57. 57.
    Guglielmi, A., A system of interaction and structure. TOCL, 8(1), 2007.Google Scholar
  58. 58.
    Hacking I.: What is logic?. J. Phil. 76, 285–319 (1979)CrossRefGoogle Scholar
  59. 59.
    Hähnle, R., Automated Deduction in Multiple-valued Logics. Oxford University Press, 1993.Google Scholar
  60. 60.
    Hájek, P., Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.Google Scholar
  61. 61.
    Horčík, R., Algebraic semantics: Semilinear FL-algebras. In Handbook of Mathematical Fuzzy Logic, vol 1, vol. 37 of Studies in Logic, Mathematical Logic and Foundations, pp. 175–214. College Publications, London, 2011.Google Scholar
  62. 62.
    Iemhoff, R., and G. Metcalfe, Hypersequent systems for the admissible rules of modal and intermediate logics. In LNCS, vol. 5407. Springer, 2009, pp. 230–245.Google Scholar
  63. 63.
    Iemhoff R., Metcalfe G.: Proof theory for admissible rules. Annals of Pure and Applied Logic 159(1-2), 171–186 (2009)CrossRefGoogle Scholar
  64. 64.
    Indrzejczak A.: Cut-free hypersequent calculus for S4.3. Bull. Sect. Logic Univ. Łódź 41(1-2), 89–104 (2012)Google Scholar
  65. 65.
    Kashima R.: Cut-free sequent calculi for some tense logics. Studia Logica 53(1), 119–135 (1994)CrossRefGoogle Scholar
  66. 66.
    Konikowska B.: Rasiowa-Sikorski Deduction Systems in CS applications. Theoretical Computer Science 286, 323–366 (2002)CrossRefGoogle Scholar
  67. 67.
    Kracht, M., Power and weakness of the modal display calculus. In Proof theory of modal logic (Hamburg, 1993), vol. 2 of Appl. Log. Ser.. Kluwer Acad. Publ., Dordrecht, 1996, pp. 93–121.Google Scholar
  68. 68.
    Kurz, A., G. Greco, and A. Palmigiani, Dynamic epistemic logic displayed. In Logic, Rationality, and Interaction, vol. 8196 of LNCS. Springer, 2013, pp. 135–148.Google Scholar
  69. 69.
    Lahav, O., From frame properties to hypersequent rules in modal logics. In LICS 2013, IEEE, 2013, pp. 408–417.Google Scholar
  70. 70.
    Leivant D.: Quantifiers as modal operators. Studia Logica 39, 145–158 (1980)CrossRefGoogle Scholar
  71. 71.
    Lellmann, B., and D. Pattinson, Correspondence between modal Hilbert axioms and sequent rules with an application to S5. In Tableaux 2013. LNCS, vol. 8123. Springer, pp. 219–233, 2013.Google Scholar
  72. 72.
    Metcalfe G., Montagna F.: Substructural fuzzy logics. J. of Symbolic Logic 72(3), 834–864 (2007)CrossRefGoogle Scholar
  73. 73.
    Metcalfe G., Olivetti N.: Towards a proof theory of Gödel modal logics. Logical Methods in Computer Science 7(2), 1–27 (2011)CrossRefGoogle Scholar
  74. 74.
    Metcalfe G., Olivetti N., Gabbay D.: Analytic proof calculi for product logics. Archive for Mathematical Logic 43(7), 859–889 (2004)CrossRefGoogle Scholar
  75. 75.
    Metcalfe G., Olivetti N., Gabbay D.: Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Transactions on Computational Logic 6(3), 578–613 (2005)CrossRefGoogle Scholar
  76. 76.
    Metcalfe, G., N. Olivetti, and D. Gabbay, Proof Theory for Fuzzy Logics, vol. 39 of Springer Series in Applied Logic. Springer, 2009.Google Scholar
  77. 77.
    Mints G.: Cut-elimination theorem in relevant logics. The Journal of Soviet Mathematics 6, 422–428 (1976)CrossRefGoogle Scholar
  78. 78.
    Negri S.: Proof analysis in modal logic. J. Philos. Logic 34(5-6), 507–544 (2005)CrossRefGoogle Scholar
  79. 79.
    Paoli, F., Substructural Logics: A Primer. Kluwer Academic Publ. Dordrecht, 2002.Google Scholar
  80. 80.
    Paoli F.: Implicational paradoxes and the meaning of logical constants. Australasian Journal of Philosophy 85, 553–579 (2007)CrossRefGoogle Scholar
  81. 81.
    Poggiolesi F.: A cut-free simple sequent calculus for modal logic S5. The Review of Symbolic Logic 1(1), 3–15 (2008)CrossRefGoogle Scholar
  82. 82.
    Poggiolesi, F., Gentzen Calculi for Modal Propositional Logic. Trends in Logic, Springer, 2010.Google Scholar
  83. 83.
    Poggiolesi, F., and G. Restall, Interpreting and applying proof theories for modal logic. In G. Restall and G. Russell, (eds.), New Waves in Philosophical Logic. Palgrave Macmillan, Basingstoke, 2012, pp. 39–62.Google Scholar
  84. 84.
    Pottinger, G., Uniform, cut-free formulations of T, S4 and S5 (abstract). J. of Symbolic Logic 48(3):900, 1983.Google Scholar
  85. 85.
    Prawitz, D., Natural Deduction. A Proof-theoretical Study. Almqvist and Wiksell, Stockholm, 1965.Google Scholar
  86. 86.
    Ramanayake, R., Embedding the hypersequent calculus in the display calculus. submitted. 2013.Google Scholar
  87. 87.
    Rauszer C.: A formalization of the propositional calculus of H – B logic. Studia Logica 33, 23–34 (1974)CrossRefGoogle Scholar
  88. 88.
    Restall, G., Display logic and gaggle theory. Rep. Math. Logic (29):133–146, 1995.Google Scholar
  89. 89.
    Restall G.: Displaying and deciding substructural logics. I. Logics with contraposition. J. Philos. Logic 27(2), 179–216 (1998)CrossRefGoogle Scholar
  90. 90.
    Restall G.: An Introduction to Substructural Logics. Routledge, London (1999)Google Scholar
  91. 91.
    Restall G.: A cut-free sequent system for two-dimensional modal logic, and why it matters. Ann. Pure Appl. Logic 163(11), 1611–1623 (2012)CrossRefGoogle Scholar
  92. 92.
    Rousseau, G., Sequents in many valued logic I, and II. Fundamenta Mathematicæ LX, LXVII:23–33, 125–131, 1967, 1970.Google Scholar
  93. 93.
    Schroeder-Heister, P., Sequent calculi and bidirectional natural deduction: On the proper basis of proof-theoretic semantics. In M. Peliš, (ed.), Logica Yearbook 2008. College Publications, London, 2008, pp. 237–251.Google Scholar
  94. 94.
    Schroeder-Heister, P., Proof-theoretic semantics. The Stanford Encyclopedia of Philosophy, E. N. Zalta (ed.), Spring 2013 Edition.Google Scholar
  95. 95.
    Standefer, S., Philosophical aspects of display logic. In M. Peliš, (ed.), Logica Yearbook 2009. College Publications, London, 2009, pp. 283–295.Google Scholar
  96. 96.
    Takeuti, G., Proof theory, vol. 81 of Studies in Logic and the Foundations of Mathematics. North Holland, Amsterdam, 1987.Google Scholar
  97. 97.
    Takeuti G., Titani T.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. of Symbolic Logic 49(3), 851–866 (1984)CrossRefGoogle Scholar
  98. 98.
    Tiu, A., A system of interaction and structure. II. The need for deep inference. Log. Methods Comput. Sci. 2(2):2:4, 24, 2006.Google Scholar
  99. 99.
    Tiu, A., A hypersequent system for Gödel-Dummett logic with non-constant domains. In Tableaux 2011. LNAI, pp. 248–262. Springer, 2011.Google Scholar
  100. 100.
    Troelstra, A. S., and H. Schwichtenberg, Basic proof theory, vol. 43 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, second edition, 2000.Google Scholar
  101. 101.
    van Benthem J.: Modal foundations of predicate logic. Log. J. IGPL 5, 259–286 (1997)CrossRefGoogle Scholar
  102. 102.
    Viganò, L., Labelled non-classical logics. Kluwer Academic Publishers, Dordrecht, 2000. With a foreword by Dov M. Gabbay.Google Scholar
  103. 103.
    Wansing H.: Sequent calculi for normal modal propositional logics. J. Logic Comput. 4(2), 125–142 (1994)CrossRefGoogle Scholar
  104. 104.
    Wansing H.: Modal tableaux based on residuation. J. Logic Comput. (7, 719–731 (1997)CrossRefGoogle Scholar
  105. 105.
    Wansing, H., Displaying Modal Logic. Springer, Trends in Logic, 1998.Google Scholar
  106. 106.
    Wansing H.: Translation of hypersequents into display sequents. Log. J. IGPL 6(5), 719–733 (1998)CrossRefGoogle Scholar
  107. 107.
    Wansing H.: Predicate logics on display. Studia Logica 62(1), 49–75 (1999)CrossRefGoogle Scholar
  108. 108.
    Wansing, H., Sequent systems for modal logics. In D. Gabbay and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol. 8. Kluwer, 2002, pp. 61–145.Google Scholar
  109. 109.
    Wansing H.: Constructive negation, implication, and co-implication. J. Appl. Non-Classical Logics 18(2-3), 341–364 (2008)CrossRefGoogle Scholar
  110. 110.
    Wansing, H., Prawitz, proofs, and meaning. In H. Wansing (ed.), Dag Prawitz on proofs and meaning. Springer, to appear 2014.Google Scholar
  111. 111.
    Wolter F.: On logics with coimplication. J. Philos. Logic 27(4), 353–387 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Agata Ciabattoni
    • 1
  • Revantha Ramanayake
    • 1
  • Heinrich Wansing
    • 2
  1. 1.Department of Computer LanguagesTechnische Universität WienWienAustria
  2. 2.Department of Philosophy IIRuhr University BochumBochumGermany

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