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A Calculus for Generating Ground Explanations

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 7364)

Abstract

We present a modification of the superposition calculus that is meant to generate explanations why a set of clauses is satisfiable. This process is related to abductive reasoning, and the explanations generated are clauses constructed over so-called abductive constants. We prove the correctness and completeness of the calculus in the presence of redundancy elimination rules, and develop a sufficient condition guaranteeing its termination; this sufficient condition is then used to prove that all possible explanations can be generated in finite time for several classes of clause sets, including many of interest to the SMT community. We propose a procedure that generates a set of explanations that should be useful to a human user and conclude by suggesting several extensions to this novel approach.

Keywords

  • Inference Rule
  • Unit Clause
  • Abductive Reasoning
  • Empty Clause
  • Ground Substitution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work has been partly funded by the project ASAP of the French Agence Nationale de la Recherche (ANR-09-BLAN-0407-01).

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References

  1. Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. ACM Transactions on Computational Logic 10(1), 129–179 (2009)

    CrossRef  MathSciNet  Google Scholar 

  2. Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. Information and Computation 183(2), 140–164 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)

    Google Scholar 

  4. Bachmair, L., Ganzinger, H., Waldmann, U.: Superposition with Simplification as a Decision Procedure for the Monadic Class with Equality. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 83–96. Springer, Heidelberg (1993)

    CrossRef  Google Scholar 

  5. Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierachic first-order theories. Appl. Algebra Eng. Commun. Comput. 5, 193–212 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Bonacina, M.P., Echenim, M.: On variable-inactivity and polynomial T-satisfiability procedures. Journal of Logic and Computation 18(1), 77–96 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Bonacina, M.P., Echenim, M.: Theory decision by decomposition. Journal of Symbolic Computation 45(2), 229–260 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Echenim, M., Peltier, N.: A calculus for generating ground explanations (technical report). CoRR, abs/1201.5954 (2012), http://arxiv.org/1201.5954

  9. Eiter, T., Gottlob, G.: The complexity of logic-based abduction. J. ACM 42(1), 3–42 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Jackson, P.: Computing prime implicates. In: ACM Conference on Computer Science, pp. 65–72 (1992)

    Google Scholar 

  11. Leitsch, A.: The resolution calculus. Texts in Theoretical Computer Science. Springer (1997)

    Google Scholar 

  12. Marquis, P.: Extending Abduction from Propositional to First-order Logic. In: Jorrand, P., Kelemen, J. (eds.) FAIR 1991. LNCS, vol. 535, pp. 141–155. Springer, Heidelberg (1991)

    CrossRef  Google Scholar 

  13. McCarthy, J.: Computer programs for checking mathematical proofs. In: Recursive Function Theory, pp. 219–228. Providence, Rhode Island (1962); Proc. of Symposia in Pure Mathematics, vol. 5. American Mathematical Society

    Google Scholar 

  14. Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 371–443. Elsevier and MIT Press (2001)

    Google Scholar 

  15. Tran, D.-K., Ringeissen, C., Ranise, S., Kirchner, H.: Combination of convex theories: Modularity, deduction completeness, and explanation. J. Symb. Comput. 45(2), 261–286 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Y\({\textsc{ices}}\), http://yices.csl.sri.com

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Echenim, M., Peltier, N. (2012). A Calculus for Generating Ground Explanations. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_17

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  • DOI: https://doi.org/10.1007/978-3-642-31365-3_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31364-6

  • Online ISBN: 978-3-642-31365-3

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