Intuitionistic Decision Procedures Since Gentzen

Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)


Gentzen solved the decision problem for intuitionistic propositional logic in his doctoral thesis [31]; this paper reviews some of the subsequent progress. Solutions to the problem are of importance both for general philosophical reasons and because of their use in implementations of proof assistants (such as Coq [4], widely used in software verification) based on intuitionistic logic.


Intuitionistic Logic Initial Sequent Proof Theory Sequent Calculus Proof Assistant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Thanks are especially due to Gerhard Jäger and Helmut Schwichtenberg, whose scientific encouragement over the years has been substantial; and to Grisha Mints, now, alas, no longer with us, for helpful comments on historical matters—regrettably not all incorporated (thanks to a failure of technology).


  1. 1.
    P. Abate, R. Goré, The Tableaux Work Bench, in Proceedings of IJCAR 2003. LNCS, vol. 2796 (Springer, 2003), pp. 230–236Google Scholar
  2. 2.
    R. Antonsen, A. Waaler, A labelled system for IPL with variable splitting, in Proceedings of CADE 2007. LNAI, vol. 4603 (Springer, 2007), pp. 132–146Google Scholar
  3. 3.
    A. Avellone, G. Fiorino, U. Moscato, An implementation of a \(O(n\ log\ n)\)-SPACE decision procedure for propositional intuitionistic logic, in 3rd International Workshop on the Implementation of Logics (Tbilisi, Georgia, Oct 2002)Google Scholar
  4. 4.
    B. Barras, S. Boutin, et al., The Coq proof assistant reference manual, version 6.2.1. Technical Report, INRIA (2000).
  5. 5.
    E.W. Beth, The Foundations of Mathematics (North-Holland, 1959)Google Scholar
  6. 6.
    M. Bezem, T. Coquand, Automating coherent logic, in Proceedings of LPAR 2005. LNCS, vol. 3835 (Springer, 2005), pp. 246–260Google Scholar
  7. 7.
    K. Brünnler, M. Lange, Cut-free sequent systems for temporal logic. J. Logic Algebraic Program. 76, 216–225 (2008)Google Scholar
  8. 8.
    J. Caldwell, Decidability extracted: synthesizing “correct-by-construction” decision procedures from constructive proofs. Ph.D. dissertation (Cornell University, 1998)Google Scholar
  9. 9.
    L. Catach, TABLEAUX: a general theorem prover for modal logics. J. Autom. Reason. 7, 489–510 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    G. Corsi, G. Tassi, Intuitionistic logic freed of all metarules. J. Symb. Logic 72, 1204–1218 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Curry, Foundations of Mathematical Logic (Dover Publications, 1963)Google Scholar
  12. 12.
    K. Dos̆en, A note on Gentzen’s decision procedure for intuitionistic propositional logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 33, 453–456 (1987)Google Scholar
  13. 13.
    A.G. Dragalin, Mathematical Intuitionism, Translations of Mathematical Monographs 67 (Trans. E. Mendelson). (American Mathematical Society, Providence, R.I., 1988)Google Scholar
  14. 14.
    R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic. J. Symb. Logic 57, 795–807 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    R. Dyckhoff, Yet Another Proof Engine, MS (available from the author) (2014)Google Scholar
  16. 16.
    R. Dyckhoff, D. Kesner, S. Lengrand, Strong cut-elimination systems for Hudelmaier’s depth-bounded sequent calculus for implicational logic, in IJCAR 2006 Proceedings. LNCS, vol. 4130, pp. 347–361 (2006)Google Scholar
  17. 17.
    R. Dyckhoff, S. Lengrand, LJQ: a strongly focused calculus for intuitionistic logic, in Proceedings of Computability in Europe 2006. LNCS, vol. 3988 (Springer, 2006), pp. 173–185Google Scholar
  18. 18.
    R. Dyckhoff, S. Negri, Admissibility of structural rules for contraction-free systems of intuitionistic logic. J. Symb. Logic 65, 1499–1518 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R. Dyckhoff, S. Negri, Decision methods for linearly ordered Heyting algebras. Arch. Math. Log. 45, 411–422 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Dyckhoff, S. Negri, Proof analysis for intermediate logics. Arch. Math. Log. 51, 71–92 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    R. Dyckhoff, S. Negri, Geometrisation of first-order logic, B. Symb. Logic 21, 123–163 (2015) Geometrisation of First-order Formulae, Submitted June 2014Google Scholar
  22. 22.
    R. Dyckhoff, S. Negri, Coherentisation of accessibility conditions in labelled sequent calculi, in Extended Abstract (2 pp.), Gentzen Systems and Beyond 2014, Informal Proceedings, ed. by R. Kuznets & G. Metcalfe. Vienna Summer of Logic, July 2014Google Scholar
  23. 23.
    U. Egly, S. Schmitt, On intuitionistic proof transformations, their complexity, and application to constructive program synthesis. Fundamenta Informaticae 39, 59–83 (1999)Google Scholar
  24. 24.
    M. Ferrari, C. Fiorentini, G. Fiorino, Simplification rules for intuitionistic propositional tableaux. ACM Trans. Comput. Log. 13, 14:1–14:23 (2012)Google Scholar
  25. 25.
    M. Ferrari, C. Fiorentini, G. Fiorino, Contraction-free linear depth sequent calculi for intuitionistic propositional logic with the subformula property and minimal depth counter-models. J. Autom. Reason. 51, 129–149 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    G. Fiorino, Decision procedures for propositional intermediate logics. Ph.D. thesis (Milan University, 2000)Google Scholar
  27. 27.
    M. Fitting, Intuitionistic Logic, Model Theory and Forcing (North-Holland, 1969)Google Scholar
  28. 28.
    M. Fitting, Proof Methods for Modal and Intuitionistic Logic (Reidel, 1983)Google Scholar
  29. 29.
    T. Franzén, Algorithmic aspects of intuitionistic propositional logic, I and II, SICS Research Reports R870X and R8906, 1987 and 1989Google Scholar
  30. 30.
    D. Garg, V. Genovese, S. Negri, Countermodels from sequent calculi in multi-modal logics, in Proceedings of LICS 2012 (IEEE, 2012), pp. 315–324Google Scholar
  31. 31.
    G. Gentzen, Untersuchungen über das logische Schliessen. Math. Zeitschrift 39, 176–210, 405–431 (1935)Google Scholar
  32. 32.
    R. Goré, Tableau methods for modal and temporal logics, in Handbook of Tableau Methods (Kluwer, 1999), pp. 297–396Google Scholar
  33. 33.
    R. Goré, J. Thomson, BDD-based automated reasoning in propositional non-classical logics: progress report, in Proceedings of PAAR-2012, EPiC Series 21 (EasyChair, 2013), pp 43–57Google Scholar
  34. 34.
    R. Goré, J. Thomson, J. Wu, A history-based theorem prover for intuitionistic propositional logic using global caching: IntHistGC system description, in Proceedings of IJCAR 2014. LNAI, vol. 8562 (Springer, 2014), pp. 262–268Google Scholar
  35. 35.
    J. Goubault-Larrecq, Implementing tableaux by decision diagrams, Unpublished note, Institut für Logik, Komplexität und Deduktionssysteme, Universität Karlsruhe, 47 pp. (1996)Google Scholar
  36. 36.
    H. Herbelin, Séquents qu’on calcule. Thèse de Doctorat, Université Paris 7 (1995)Google Scholar
  37. 37.
    A. Heuerding, M. Seyfried, H. Zimmermann, Efficient loop-check for backward proof search in some non-classical logics, in Proceedings of Tableaux 1996. LNAI, vol. 1071 (Springer, 1996), pp. 210–225Google Scholar
  38. 38.
    J. Howe, Two loop-detection mechanisms: a comparison, in Proceedings of Tableaux 1997. LNCS, vol. 1227 (Springer, 1997), pp. 188–200Google Scholar
  39. 39.
    J. Hudelmaier, A Prolog program for intuitionistic propositional logic, SNS-Bericht 88–28, Tübingen (1988)Google Scholar
  40. 40.
    J. Hudelmaier, Bounds for cut elimination in intuitionistic propositional logic. Arch. Math. Logic 31, 331–353 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    J. Hudelmaier, An \(O(n\ log\ n)\)-SPACE decision procedure for intuitionistic propositional logic. J. Logic Comput. 3, 63–76 (1993)Google Scholar
  42. 42.
    O. Ketonen, Untersuchungen zum Prädikatenkalkül. Annales Acad. Sci. Fenn, Ser. A.I. 23 (1944)Google Scholar
  43. 43.
    S.C. Kleene, Introduction to Metamathematics (North-Holland, 1952)Google Scholar
  44. 44.
    D. Larchey-Wendling, D. Mery, D. Galmiche, STRIP: structural sharing for efficient proof-search, in Proceedings of IJCAR 2001. LNCS, vol. 2083 (Springer, 2001), pp. 696–700Google Scholar
  45. 45.
    P. Lincoln, A. Scedrov, N. Shankar, Linearizing intuitionistic implication. Ann. Pure Appl. Logic 60, 151–177 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    O. Gasquet, A. Herzig, D. Longin, M. Sahade, LoTREC: logical tableaux research engineering companion, in Proceedings of Tableaux 2005. LNCS, vol. 3702 (Springer, 2005), pp. 318–322Google Scholar
  47. 47.
    S. Maehara, Eine Darstellung der Intuitionistischen Logik in der Klassischen. Nagoya Math. J. 7, 45–64 (1954)MathSciNetzbMATHGoogle Scholar
  48. 48.
    S.Yu. Maslov, An inverse method of establishing deducibility in the classical predicate calculus. Dokl. Akad. Nauk. SSSR 159, 17–20 (translated as Soviet Math. Dokl. 5, 1420) (1964)Google Scholar
  49. 49.
    S. McLaughlin, F. Pfenning, Imogen: focusing the polarized inverse method for intuitionistic propositional logic, in Proceedings of LPAR’08. LNCS, vol. 5330 (Springer, 2008), pp. 174–181Google Scholar
  50. 50.
    G. Mints, Gentzen-type systems and resolution rule. Part I. LNCS 417, 198–231 (1990)MathSciNetzbMATHGoogle Scholar
  51. 51.
    G. Mints, Complexity of subclasses of the intuitionistic propositional calculus, Programming Logic (ed. by B. Nordström). BIT 31, 64–69 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    G. Mints, A Short Introduction to Intuitionistic Logic, CSLI Stanford Lecture Notes (Springer, 2000)Google Scholar
  53. 53.
    S. Negri, Proofs and countermodels in non-classical logics. Logica Universalis 8, 25–60 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    K. Ono, Logische Untersuchungen über die Grundlagen der Mathematik. J. Fac. Sci. Imperial Univ. Tokyo. Section I. Math. Astron. Phys. Chem. 3, 329–389 (1938)zbMATHGoogle Scholar
  55. 55.
    J. Otten, Clausal connection-based theorem proving in intuitionistic first-order logic, in Proceedings of Tableaux 2005. LNCS, vol. 3702 (Springer, 2005), pp 245–261Google Scholar
  56. 56.
    J. Otten, The ILTP Library.
  57. 57.
    L. Pinto, R. Dyckhoff, Loop-free construction of counter-models for intuitionistic propositional logic, Symposia Gaussiana, Conf. A, ed. by M. Behara, R. Fritsch, R.G. Lintz (Walter de Gruyter & Co, Berlin, 1995), pp. 225–232Google Scholar
  58. 58.
    J. von Plato, Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics Springer (to appear, 2016)Google Scholar
  59. 59.
    S.L. Read, Semantic Pollution and Syntactic Purity, R. Symb. Logic 8, 649–691 (2015)Google Scholar
  60. 60.
    D. Sahlin, T. Franzén, S. Haridi, An intuitionistic predicate logic theorem prover. J. Logic Comput. 2, 619–656 (1992)Google Scholar
  61. 61.
    G. Sambin, S. Valentini, The modal logic of provability. The sequential approach. J. Philos. Logic 11, 311–342 (1982)Google Scholar
  62. 62.
    R. Schmidt, D. Tishkovsky, Automated synthesis of tableau calculi. Logical Methods Comput. Sci. 7, 32 (2011)Google Scholar
  63. 63.
    K. Schütte, Vollstandige Systeme modaler und intuitionistischer Logik, Ergebnisse der Mathematik (Springer, 1968)Google Scholar
  64. 64.
    W. Sieg, S. Cittadini, Normal natural deduction proofs (in non-classical logics), in LNAI 2605 (Springer, 2005), pp. 169–191Google Scholar
  65. 65.
    R. Statman, Intuitionistic propositional logic is polynomial-space complete. Theoret. Comput. Sci. 9, 67–72 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    T. Tammet, A resolution theorem prover for intuitionistic logic, in CADE-13. LNCS, vol. 1104 (Springer, 1996), pp. 2–16Google Scholar
  67. 67.
    N. Tennant, Autologic (Edinburgh University Press, 1992)Google Scholar
  68. 68.
    A.S. Troelstra, H. Schwichtenberg, Basic Proof Theory (Cambridge, 2001)Google Scholar
  69. 69.
    J. Underwood, A constructive completeness proof for intuitionistic propositional calculus, TR 90–1179, Department of Computer Science, Cornell University, 1990; also in Proceedings of the Workshop on Analytic Tableaux (Marseille, 1993)Google Scholar
  70. 70.
    N.N. Vorob’ev, The derivability problem in the constructive propositional calculus with strong negation. Doklady Akademii Nauk SSSR 85, 689–692 (1952)MathSciNetGoogle Scholar
  71. 71.
    N.N. Vorob’ev, A new algorithm for derivability in the constructive propositional calculus. AMS Transl. Ser. 2(94), 37–71 (1970)zbMATHGoogle Scholar
  72. 72.
    A. Waaler, L. Wallen, Tableaux methods in intuitionistic logic, in Handbook of Tableaux Methods, ed. by M. D’Agostino, D.M. Gabbay, R. Hähnle, J. Posegga (Kluwer, Dordrecht, 1999), pp. 255–296Google Scholar
  73. 73.
    K. Weich, Improving proof search in intuitionistic propositional logic. Munich Ph.D. thesis, also from Logos Verlag Berlin (2001)Google Scholar
  74. 74.
    K. Weich, Decision procedures for intuitionistic propositional logic by program extraction, in Proceedings of Tableaux 1998. LNCS, vol. 1397 (Springer, 1998), pp. 292–306Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of St. AndrewsSt. AndrewsUK

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