International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

Automated Reasoning with Analytic Tableaux and Related Methods pp 153-168

Disproving Using the Inverse Method by Iterative Refinement of Finite Approximations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9323)


In first-order logic, forward search using a complete strategy such as the inverse method can get stuck deriving larger and larger consequence sets when the goal query is unprovable. This is the case even in trivial theories where backward search strategies such as tableaux methods will fail finitely. We propose a general mechanism for bounding the consequence sets by means of finite approximations of infinite types. If the inverse method also implements forward subsumption and globalization, then the search space under this approximation is finite. We therefore obtain a type-directed iterative refinement algorithm for disproving queries.

The method has been implemented for intuitionistic first-order logic, and we discuss its performance on a variety of problems.


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  1. 1.
    Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. of Logic and Computation 3(4) (1994)Google Scholar
  2. 2.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, chapter 2, pp. 19–99. Elsevier Science, New York (2001)Google Scholar
  3. 3.
    Bonacina, M.P., Lynch, C., de Moura, L.M.: On deciding satisfiability by theorem proving with speculative inferences. J. of Automated Reasoning 47(2), 161–189 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chaudhuri, K.: The Focused Inverse Method for Linear Logic. PhD thesis, Carnegie Mellon University, Technical report CMU-CS-06-162, December 2006Google Scholar
  5. 5.
    Chaudhuri, K.: Magically constraining the inverse method using dynamic polarity assignment. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 202–216. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. of Automated Reasoning 40(2–3), 133–177 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chihani, Z., Miller, D., Renaud, F.: Foundational proof certificates in first-order logic. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 162–177. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Claessen, K., Sorensson, N.: New techniques that improve MACE-style finite model finding. In: Baumgartner, P., Fermueller, C. (eds.) Proceedings of the CADE-19 Workshop: Model Computation - Principles, Algorithms, Applications, Miami, USA (2003)Google Scholar
  9. 9.
    Degtyarev, A., Voronkov, A.: The inverse method. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), pp. 179–272. Elsevier and MIT Press (2001)Google Scholar
  10. 10.
    Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410(46), 4747–4768 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lynch, C.: Unsound theorem proving. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 473–487. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    McCune, W.: Mace4 reference manual and guide. Technical Report cs.SC/0310055 (2003)Google Scholar
  13. 13.
    McLaughlin, S., Pfenning, F.: Imogen: Focusing the polarized focused inverse method for intuitionistic propositional logic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 174–181. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    McLaughlin, S., Pfenning, F.: Efficient intuitionistic theorem proving with the polarized inverse method. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 230–244. Springer, Heidelberg (2009)Google Scholar
  15. 15.
    Raths, T., Otten, J., Kreitz, C.: The ILTP problem library for intuitionistic logic. Journal of Automated Reasoning 38(1), 261–271 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Inria and LIX/École PolytechniquePalaiseauFrance

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