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

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## 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

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## References

- 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.Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)zbMATHGoogle Scholar
- 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.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.Dyckhoff, R., Pinto, L.: Permutability of inferences in intuitionistic sequent calculi. Technical Report CS/97/7, University of St Andrews (1997)Google Scholar
- 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.Dyckhoff, R., Pinto, L.: Permutability of proofs in intuitionistic sequent calculi. Theor. Comput. Sci.
**212**(1–2), 141–155 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theor. Comput. Sci.
**410**(46), 4747–4768 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.Pfenning, F.: Automated theorem proving. Lecture notes, Carnegie Mellon University (2004)Google Scholar
- 28.Prawitz, D.: Natural Deduction. Almqvist and Wiksell, Stockholm (1965)zbMATHGoogle Scholar
- 29.Raths, T., Otten, J., Kreitz, C.: The ILTP problem library for intuitionistic logic. J. Autom. Reason.
**31**, 261–271 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Sahlin, D., Franzén, T., Haridi, S.: An intuitionistic predicate logic theorem prover. J. Log. Comput.
**2**(5), 619–656 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Sieg, W., Byrnes, J.: Normal natural deduction proofs (in classical logic). Stud. Log.
**60**(1), 67–106 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Troelstra, A., Schwichtenberg, H.: Basic proof theory. In: Cambridge Tracts in Theoretical Computer Science (2nd ed.), vol. 43. Cambridge University Press (2000)Google Scholar

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