# Goal-Directed Proof Search in Multiple-Conclusioned Intuitionistic Logic

## Abstract

A key property in the definition of logic programming languages is the completeness of goal-directed proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multiple-conclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contraction-free fragments of the logic.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.J.-M. Andreoli. Logic Programming with Focusing Proofs in Linear Logic.
*Journal of Logic and Computation*, 2(3), 1992.Google Scholar - 2.J.-M. Andreoli and R. Pareschi. Linear Objects: Logical Processes with Built-in Inheritance. In David H. D. Warren and Peter Szeredi, editors,
*Proceedings of the Seventh International Conference on Logic Programming*, pages 496–510, Jerusalem, 1990. The MIT Press.Google Scholar - 3.K. Clark. Negation as Failure. In H. Gallaie and J. Minker, editors,
*Logic and Databases*, pages 293–323. Plenum Press, 1978.Google Scholar - 4.R. Dyckhoff. Contraction-free Sequent Calculi for Intuitionistic Logic.
*Journal of Symbolic Logic*, 57:795–807, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 5.R. Dyckhoff. A Deterministic Terminating Sequent Calculus for Godel-Dummett logic.
*Logic Journal of the IGPL*, 7:319–326, 1999.zbMATHCrossRefMathSciNetGoogle Scholar - 6.D. Galmiche and G. Perrier. On Proof Normalisation in Linear Logic.
*Theoretical Computer Science*, 135:67–110, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 7.G. Gentzen. Untersuchungen üiber das logische Schliessen.
*Math. Zeit.*, 39:176–210,405-431, 1934.zbMATHCrossRefMathSciNetGoogle Scholar - 8.J. Harland. On Normal Forms and Equivalence for Logic Programs. In Krzysztof Apt, editor,
*Proceedings of the Joint International Conference and Symposium on Logic Programming*, pages 146–160, Washington, DC, 1992. ALP, MIT Press.Google Scholar - 9.J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability.
*Journal of Logic and Computation*, 4(1):69–88, January1994.Google Scholar - 10.J. Harland. On Goal-Directed Provability in Classical Logic.
*Computer Languages*, 23:161–178, 1997.zbMATHCrossRefGoogle Scholar - 11.J. Harland, D. Pym, and M. Winikoff. Programming in Lygon: An Overview. In M. Wirsing, editor,
*Lecture Notes in Computer Science*, pages 391–405. Springer, July1996.Google Scholar - 12.J. Hodas and D. Miller. Logic Programming in a Fragment of Intuitionistic Linear Logic.
*Information and Computation*, 110(2):327–365, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 13.N. Kobayash and A. Yonezawa. ACL-A Concurrent Linear Logic Programming Paradigm. In Dale Miller, editor,
*Logic Programming-Proceedings of the 1993 International Symposium*, pages 279–294, Vancouver, Canada, 1993. The MIT Press.Google Scholar - 14.T. Lutovac and J. Harland. Towards the Automation of the Design of Logic Programming Languages. Technical Report 97-30, Department of Computer Science, RMIT, 1997.Google Scholar
- 15.D. Miller. Forum: A Multiple-Conclusion Specification Logic.
*Theoretical Computer Science*, 165(1):201–232, 1996.zbMATHCrossRefMathSciNetGoogle Scholar - 16.D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov. Uniform Proofs as a Foundation for Logic Programming.
*Annals of Pure and Applied Logic*, 51:125–157, 1991.CrossRefMathSciNetzbMATHGoogle Scholar - 17.J. Minker and A. Rajasekar. A Fixpoint Semantics for Disjunctive Logic Programs.
*Journal of Logic Programming*, 9(1):45–74, July1990.Google Scholar - 18.G. Nadathur. Uniform Provability in Classical Logic.
*Journal of Logic and Computation*, 8(2):209–230, 1998.zbMATHCrossRefMathSciNetGoogle Scholar - 19.D. Pym and J. Harland. A Uniform Proof-theoretic Investigation of Linear Logic Programming.
*Journal of Logic and Computation*, 4(2): 175–207, April1994.Google Scholar - 20.E. Ritter, D. Pym, and L. Wallen. On the Intuitionistic Force of Classical Search, to appear in
*Theoretical Computer Science*, 1999.Google Scholar - 21.P. Volpe. Concurrent Logic Programming as Uniform Linear Proofs. In G. Levi and M. Rodríguez-Artalejo, editors,
*Proceedings of the Conference on Algebraic and Logic Programming*, pages 133–149. Springer, 1994.Google Scholar - 22.Lincoln Wallen.
*Automated Deduction in Nonclassical Logics*. MIT Press, 1990.Google Scholar - 23.M. Winikoff.
*Logic Programming with Linear Logic*. PhD thesis, University of Melbourne, 1997.Google Scholar