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Journal of Automated Reasoning

, Volume 62, Issue 1, pp 127–167 | Cite as

Goal-Oriented Proof-Search in Natural Deduction for Intuitionistic Propositional Logic

  • Mauro Ferrari
  • Camillo FiorentiniEmail author
Article
  • 104 Downloads

Abstract

We address the problem of proof-search in the natural deduction calculus for Intuitionistic propositional logic. Our aim is to improve the usual proof-search procedure where introduction rules are applied upwards and elimination rules downwards. In particular, we introduce \(\mathbf {Nbu} \), a variant of the usual natural deduction calculus for Intuitionistic Propositional Logic, and we show that it can be used as a base for a goal-oriented proof-search procedure. We also show that the implementation of our proof-search procedure is competitive with those based on sequent or tableaux calculi.

Keywords

Natural deduction Intuitionistic propositional logic Proof-search procedures 

Mathematics Subject Classification

03B20 03F07 03B35 

Notes

Acknowledgements

We thank the anonymous reviewers for their valuable remarks which helped us to improve the paper.

Supplementary material

References

  1. 1.
    Brock-Nannestad, T., Schürmann, C.: Focused natural deduction. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17, LNCS, vol. 6397, pp. 157–171. Springer, Berlin (2010)Google Scholar
  2. 2.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
  3. 3.
    Claessen, K., Rosén, D.: SAT modulo intuitionistic implications. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR-20, vol. 9450, pp. 622–637. Springer, Berlin (2015)Google Scholar
  4. 4.
    Dyckhoff, R., Lengrand, S.: A strongly focused calculus for intuitionistic logic. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE, LNCS, vol. 3988, pp. 173–185. Springer, Berlin (2006)Google Scholar
  5. 5.
    Dyckhoff, R., Pinto, L.: Permutability of inferences in intuitionistic sequent calculi. Technical Report CS/97/7, University of St Andrews (1997)Google Scholar
  6. 6.
    Dyckhoff, R., Pinto, L.: Cut-elimination and a permutation-free sequent calculus for intuitionistic logic. Stud. Log. 60(1), 107–118 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dyckhoff, R., Pinto, L.: Permutability of proofs in intuitionistic sequent calculi. Theor. Comput. Sci. 212(1–2), 141–155 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Englander, C., Dowek, G., Haeusler, E.H.: Yet another bijection between sequent calculus and natural deduction. Electron. Notes Theor. Comput. Sci. 312, 107–124 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferrari, M., Fiorentini, C.: Proof-search in natural deduction calculus for classical propositional logic. In: Nivelle, H.D. (ed.) TABLEAUX 2015, LNCS, vol. 9323, pp. 237–252. Springer, Berlin (2015)Google Scholar
  10. 10.
    Ferrari, M., Fiorentini, C., Fiorino, G.: FCube: an efficient prover for intuitionistic propositional logic. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17, LNCS, vol. 6397, pp. 294–301. Springer, Berlin (2010)Google Scholar
  11. 11.
    Ferrari, M., Fiorentini, C.: Simplification rules for intuitionistic propositional tableaux. ACM Trans. Comput. Log. TOCL 13(2), 14:1–14:23 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ferrari, M., Fiorentini, C., Fiorino, G.: A terminating evaluation-driven variant of G3i. In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013, LNCS, vol. 8123, pp. 104–118. Springer, Berlin (2013)Google Scholar
  13. 13.
    Ferrari, M., Fiorentini, C., Fiorino, G.: An evaluation-driven decision procedure for G3i. ACM Trans. Comput. Log. TOCL 16(1), 8:1–8:37 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ferrari, M., Fiorentini, C., Fiorino, G.: JTabWb: a Java framework for implementing terminating sequent and tableau calculi. Fundam. Inf. 150, 119–142 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gentzen, G., Gentzen, G.: Investigations into logical deduction. In: Szabo, M. (ed.) The Collected Works of Gerhard Gentzen, pp. 68–131. North-Holland, Amsterdam (1969)Google Scholar
  16. 16.
    Goré, R., Thomson, J., Wu, J.: A history-based theorem prover for intuitionistic propositional logic using global caching: IntHistGC system description. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) Automated Reasoning—7th International Joint Conference, IJCAR 2014, LNCS, vol. 8562, pp. 262–268. Springer, Berlin (2014)Google Scholar
  17. 17.
    Herbelin, H.: A lambda-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) CSL, LNCS, vol. 933, pp. 61–75. Springer, Berline (1994)Google Scholar
  18. 18.
    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, LNCS, vol. 1071, pp. 210–225. Springer, Berlin (1996)Google Scholar
  19. 19.
    Howard, W.: The formulae-as-types notion of construction. In: Seldin, J., Hindley, J. (eds.) To H.B. Curry: Essay on Combinatory Logic, Lambda-calculus and Formalism. Kluwer Academic Publishers, Hingham (1980)Google Scholar
  20. 20.
    Howe, J.: Proof search issues in some non-classical logics. Ph.D. thesis, School of Mathematical and Computational Sciences, University of St Andrews (1998)Google Scholar
  21. 21.
    Howe, J.W.: Two loop detection mechanisms: a comparison. In: Galmiche, D. (ed.) TABLEAUX 1997, LNCS, vol. 1227, pp. 188–200. Springer, Berlin (1997)Google Scholar
  22. 22.
    Liang, C., Miller, D.: On focusing and polarities in linear logic and intuitionistic logic (2008). http://www.lix.polytechnique.fr/~dale/papers/focusli.pdf
  23. 23.
    Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theor. Comput. Sci. 410(46), 4747–4768 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McLaughlin, S., Pfenning, F.: Imogen: focusing the polarized inverse method for intuitionistic propositional logic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR-15, LNCS, vol. 5330, pp. 174–181. Springer, Berlin (2008)Google Scholar
  25. 25.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Log. 51, 125–157 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mints, G., Steinert-Threlkeld, S.: ADC method of proof search for intuitionistic propositional natural deduction. J. Log. Comput. 26(1), 395–408 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pfenning, F.: Automated theorem proving. Lecture notes, Carnegie Mellon University (2004)Google Scholar
  28. 28.
    Prawitz, D.: Natural Deduction. Almqvist and Wiksell, Stockholm (1965)zbMATHGoogle Scholar
  29. 29.
    Raths, T., Otten, J., Kreitz, C.: The ILTP problem library for intuitionistic logic. J. Autom. Reason. 31, 261–271 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sahlin, D., Franzén, T., Haridi, S.: An intuitionistic predicate logic theorem prover. J. Log. Comput. 2(5), 619–656 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Santo, J.E.: The lambda-calculus and the unity of structural proof theory. Theory Comput. Syst. 45(4), 963–994 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Savnik, I.: Index data structure for fast subset and superset queries. In: Cuzzocrea, A., Kittl, C., Simos, D.E., Weippl, E., Xu, L. (eds.) Availability, Reliability, and Security in Information Systems and HCI, LNCS, vol. 8127, pp. 134–148. Springer, Berlin (2013)CrossRefGoogle Scholar
  33. 33.
    Sieg, W., Byrnes, J.: Normal natural deduction proofs (in classical logic). Stud. Log. 60(1), 67–106 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sieg, W., Cittadini, S.: Normal natural deduction proofs (in non-classical logics). In: Hutter, D., Stephan, W. (eds.) Mechanizing Mathematical Reasoning, LNCS, vol. 2605, pp. 169–191. Springer, Berlin (2005)CrossRefGoogle Scholar
  35. 35.
    Troelstra, A., Schwichtenberg, H.: Basic proof theory. In: Cambridge Tracts in Theoretical Computer Science (2nd ed.), vol. 43. Cambridge University Press (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dipartimento di Scienze Teoriche e ApplicateUniversità degli Studi dell’InsubriaVareseItaly
  2. 2.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanItaly

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