Journal of Automated Reasoning

, Volume 51, Issue 2, pp 129–149

# Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

Article

## Abstract

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke counter-model for σ. Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.

### Keywords

Intuitionistic propositional logic Sequent calculi Subformula property Decision procedures Counter-models generation

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### References

1. 1.
Bozzato, L., Ferrari, M., Fiorentini, C., Fiorino, G.: A decidable constructive description logic. In: Janhunen, T., Niemelä, I. (eds.) Logics in Artificial Intelligence, JELIA 2010, vol. 6341, pp. 51–63. Springer, New York (2010)
2. 2.
Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)
3. 3.
Corsi, G., Tassi, G.: Intuitionistic logic freed of all metarules. J. Symb. Log. 72(4), 1204–1218 (2007)
4. 4.
Dyckhoff, R.: Contraction-free sequent calculi for intuitionistic logic. J. Symb. Log. 57(3), 795–807 (1992)
5. 5.
Ferrari, M., Fiorentini, C., Fiorino, G.: A tableau calculus for propositional intuitionistic logic with a refined treatment of nested implications. J. Appl. Non-Class. Log. 19(2), 149–166 (2009)
6. 6.
Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Works of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
7. 7.
Goré, R., Postniece, L.: Combining derivations and refutations for cut-free completeness in bi-intuitionistic logic. J. Log. Comput. 20(1), 233–260 (2010)
8. 8.
Heuerding, A., Seyfried, M., Zimmermann, H.: Efficient loop-check for backward proof search in some non-classical propositional logics. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds.) TABLEAUX, Lecture Notes in Computer Science, vol. 1071, pp. 210–225. Springer, New York (1996)Google Scholar
9. 9.
Hirokawa, S., Nagano, D.: Long normal form proof search and counter-model generation. Electron. Notes Theor. Comput. Sci. 37, 11 (2000)
10. 10.
Howe, J.M.: Two loop detection mechanisms: a comparision. In: Galmiche, D. (ed.) TABLEAUX, Lecture Notes in Computer Science, vol. 1227, pp. 188–200. Springer, New York (1997)Google Scholar
11. 11.
Hudelmaier, J.: An O(n logn)-space decision procedure for intuitionistic propositional logic. J. Log. Comput. 3(1), 63–75 (1993)
12. 12.
Larchey-Wendling, D., Méry, D., Galmiche, D.: Strip: Structural sharing for efficient proof-search. In: Goré, R., Leitsch, A., ipkow, T. (eds.) IJCAR, Lecture Notes in Computer Science, vol. 2083, pp. 696–700. Springer, New York (2001)Google Scholar
13. 13.
Miglioli, P., Moscato, U., Ornaghi, M.: An improved refutation system for intuitionistic predicate logic. J. Autom. Reason. 12, 361–373 (1994)
14. 14.
Miglioli, P., Moscato, U., Ornaghi, M.: Avoiding duplications in tableau systems for intuitionistic logic and Kuroda logic. Log. J. IGPL 5(1), 145–167 (1997)
15. 15.
Pinto, L., Dyckhoff, R.: Loop-free construction of counter-models for intuitionistic propositional logic. In: Behara, Fritsch, Lintz (eds.) Symposia Gaussiana, Conference A, pp. 225–232. Walter de Gruyter, Berlin (1995)Google Scholar
16. 16.
Statman, R.: Intuitionistic logic is polynomial-space complete. Theor. Comp. Sci. 9(1), 67–72 (1979)
17. 17.
Stoughton, A.: Porgi: a proof-or-refutation generator for intuitionistic propositional logic. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE, Lecture Notes in Computer Science, vol. 1104, pp. 109–116. Springer, New York (1996)Google Scholar
18. 18.
Svejdar, V.: On sequent calculi for intuitionistic propositional logic. Comment. Math. Univ. Carol. 47(1), 159–173 (2006)
19. 19.
Troelstra, A.S., Schwichtenberg, H.: Basic proof theory. In: Cambridge Tracts in Theoretical Computer Science, vol. 43. Cambridge University Press, Cambridge (1996)Google Scholar
20. 20.
Vorob’ev, N.N.: A new algorithm of derivability in a constructive calculus of statements. In: Sixteen Papers on Logic and Algebra, American Mathematical Society Translations, Series 2, vol. 94, pp. 37–71. American Mathematical Society, Providence (1970)Google Scholar
21. 21.
Waaler, A., Wallen, L.: Tableaux for intuitionistic logics. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 255–296. Kluwer Academic, Dordrecht (1999)

## Authors and Affiliations

• Mauro Ferrari
• 1
• Camillo Fiorentini
• 2
• Guido Fiorino
• 3
1. 1.Dipartimento di Informatica e ComunicazioneUniversità degli Studi dell’InsubriaVareseItaly
2. 2.Dipartimento di Scienze dell’InformazioneUniversità degli Studi di MilanoMilanoItaly
3. 3.Dipartimento di Metodi Quantitativi per le Scienze Economiche AziendaliUniversità degli Studi di Milano-BicoccaMilanoItaly