Combining Superposition and Induction: A Practical Realization

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8152)


We consider a proof procedure aiming at refuting clause sets containing arithmetic constants (or parameters), interpreted as natural numbers. The superposition calculus is enriched with a loop detection rule encoding a form of mathematical induction on the natural numbers (by “descente infinie”). This calculus and its theoretical properties are described in [2,16]. In the present paper, we focus on more practical aspects. We provide algorithms to apply the loop detection rule in an automatic and efficient way. We describe a research prototype implementing our technique and provide some preliminary experimental results.


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  1. 1.
    Althaus, E., Kruglov, E., Weidenbach, C.: Superposition modulo linear arithmetic SUP(LA). In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS, vol. 5749, pp. 84–99. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Aravantinos, V., Echenim, M., Peltier, N.: A resolution calculus for first-order schemata. Fundamenta Informaticae (accepted for publication, to appear, 2013)Google Scholar
  3. 3.
    Baaz, M., Leitsch, A.: Towards a clausal analysis of cut-elimination. J. Symb. Comput. 41(3-4), 381–410 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierachic first-order theories. Appl. Algebra Eng. Commun. Comput. 5, 193–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barthe, G., Stratulat, S.: Validation of the javacard platform with implicit induction techniques. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 337–351. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Baumgartner, P., Tinelli, C.: Model Evolution with Equality Modulo Built-in Theories. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 85–100. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Bouhoula, A., Kounalis, E., Rusinowitch, M.: SPIKE, an automatic theorem prover. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 460–462. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  8. 8.
    Comon, H.: Inductionless induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 14, pp. 913–962. North-Holland (2001)Google Scholar
  9. 9.
    Dunchev, T.: Automation of cut-elimination in proof schemata. PhD thesis, T.U. Vienna (2012)Google Scholar
  10. 10.
    Dunchev, T., Leitsch, A., Rukhaia, M., Weller, D.: Ceres for first-order schemata, Research Report (2013),
  11. 11.
    Falke, S., Kapur, D.: Rewriting induction + linear arithmetic = decision procedure. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 241–255. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiabiliby modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Giesl, J., Kapur, D.: Decidable classes of inductive theorems. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 469–484. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Giesl, J., Kapur, D.: Deciding inductive validity of equations. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 17–31. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Horbach, M., Weidenbach, C.: Superposition for fixed domains. ACM Trans. Comput. Logic 11(4), 1–35 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kersani, A., Peltier, N.: Completeness and Decidability Results for First-Order Clauses with Indices. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 58–75. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    McCune, W.: Prover9 and mace4 (2005–2010),
  18. 18.
    Robinson, A., Voronkov, A. (eds.): Handbook of Automated Reasoning. North-Holland (2001)Google Scholar
  19. 19.
    Rukhaia, M.: CERES in Proof Schemata. PhD thesis, T.U. Vienna (2012)Google Scholar
  20. 20.
    Stratulat, S.: Automatic ‘Descente infinie’ induction reasoning. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 262–276. Springer, Heidelberg (2005)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of Grenoble (LIG, CNRS)France

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