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International Joint Conference on Automated Reasoning

IJCAR 2018: Automated Reasoning pp 279–294Cite as

A Generic Framework for Implicate Generation Modulo Theories

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

Abstract

The clausal logical consequences of a formula are called its implicates. The generation of these implicates has several applications, such as the identification of missing hypotheses in a logical specification. We present a procedure that generates the implicates of a quantifier-free formula modulo a theory. No assumption is made on the considered theory, other than the existence of a decision procedure. The algorithm has been implemented (using the solvers MiniSAT, CVC4 and Z3) and experimental results show evidence of the practical relevance of the proposed approach.

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Notes

  1. 1.

    Note that this ordering is not necessarily related to the ordering \(\prec \) on clauses.

  2. 2.

    A refined comparison of the set of generated \((\mathcal{T},\mathcal{A})\)-implicates modulo theory entailment is left for future work.

References

  1. Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 171–177. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_14

    Chapter  Google Scholar 

  2. Barrett, C., Stump, A., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2010). www.SMT-LIB.org

  3. Bienvenu, M.: Prime implicates and prime implicants in modal logic. In: Proceedings of the National Conference on Artificial Intelligence, vol. 22, p. 379. AAAI Press/MIT Press, Menlo Park, Cambridge, London (1999, 2007)

    Google Scholar 

  4. Blackburn, P., Van Benthem, J., Wolter, F.: Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3. Elsevier, Amsterdam (2007). ISSN 1570–2464

    MATH  Google Scholar 

  5. De Kleer, J.: An improved incremental algorithm for generating prime implicates. In: Proceedings of the National Conference on Artificial Intelligence, p. 780. Wiley (1992)

    Google Scholar 

  6. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    Chapter  Google Scholar 

  7. Dillig, I., Dillig, T., McMillan, K.L., Aiken, A.: Minimum satisfying assignments for SMT. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 394–409. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31424-7_30

    Chapter  Google Scholar 

  8. Echenim, M., Peltier, N.: A superposition calculus for abductive reasoning. J. Autom. Reason. 57(2), 97–134 (2016)

    Article  MathSciNet  Google Scholar 

  9. Echenim, M., Peltier, N., Tourret, S.: An approach to abductive reasoning in equational logic. In: Proceedings of International Conference on Artificial Intelligence, IJCAI 2013, pp. 3–9. AAAI (2013)

    Google Scholar 

  10. Echenim, M., Peltier, N., Tourret, S.: A rewriting strategy to generate prime implicates in equational logic. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 137–151. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_10

    Chapter  Google Scholar 

  11. Echenim, M., Peltier, N., Tourret, S.: Quantifier-free equational logic and prime implicate generation. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 311–325. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21401-6_21

    Chapter  Google Scholar 

  12. Echenim, M., Peltier, N., Tourret, S.: Prime implicate generation in equational logic. J. Artif. Intell. Res. 60, 827–880 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_37

    Chapter  Google Scholar 

  14. Fredkin, E.: Trie memory. Commun. ACM 3(9), 490–499 (1960)

    Article  Google Scholar 

  15. Jackson, P.: Computing prime implicates incrementally. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 253–267. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55602-8_170

    Chapter  Google Scholar 

  16. Kean, A., Tsiknis, G.: An incremental method for generating prime implicants/implicates. J. Symb. Comput. 9(2), 185–206 (1990)

    Article  MathSciNet  Google Scholar 

  17. Knill, E., Cox, P.T., Pietrzykowski, T.: Equality and abductive residua for Horn clauses. Theoret. Comput. Sci. 120(1), 1–44 (1993)

    Article  MathSciNet  Google Scholar 

  18. 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). https://doi.org/10.1007/3-540-54507-7_12

    Chapter  Google Scholar 

  19. Marquis, P.: Consequence finding algorithms. In: Kohlas, J., Moral, S. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems. HAND, vol. 5, pp. 41–145. Springer, Dordrecht (2000). https://doi.org/10.1007/978-94-017-1737-3_3

    Chapter  Google Scholar 

  20. Matusiewicz, A., Murray, N.V., Rosenthal, E.: Prime implicate tries. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS, vol. 5607, pp. 250–264. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02716-1_19

    Chapter  Google Scholar 

  21. Matusiewicz, A., Murray, N.V., Rosenthal, E.: Tri-based set operations and selective computation of prime implicates. In: Kryszkiewicz, M., Rybinski, H., Skowron, A., Raś, Z.W. (eds.) ISMIS 2011. LNCS, vol. 6804, pp. 203–213. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21916-0_23

    Chapter  Google Scholar 

  22. Mayer, M.C., Pirri, F.: First order abduction via tableau and sequent calculi. Log. J. IGPL 1(1), 99–117 (1993)

    Article  MathSciNet  Google Scholar 

  23. Mishchenko, A.: An introduction to zero-suppressed binary decision diagrams. Technical report, Proceedings of the 12th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning (2001)

    Google Scholar 

  24. Nabeshima, H., Iwanuma, K., Inoue, K., Ray, O.: SOLAR: an automated deduction system for consequence finding. AI Commun. 23(2), 183–203 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Previti, A., Ignatiev, A., Morgado, A., Marques-Silva, J.: Prime compilation of non-clausal formulae. In: Proceedings of the 24th International Conference on Artificial Intelligence, pp. 1980–1987. AAAI Press (2015)

    Google Scholar 

  26. Quine, W.: A way to simplify truth functions. Am. Math. Mon. 62(9), 627–631 (1955)

    Article  MathSciNet  Google Scholar 

  27. Riazanov, A., Voronkov, A.: Vampire 1.1 (system description). In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 376–380. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45744-5_29

    Chapter  MATH  Google Scholar 

  28. Schulz, S.: System description: E 1.8. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 735–743. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45221-5_49

    Chapter  Google Scholar 

  29. Simon, L., Del Val, A.: Efficient consequence finding. In: Proceedings of the 17th International Joint Conference on Artificial Intelligence, pp. 359–370 (2001)

    Google Scholar 

  30. Tison, P.: Generalization of consensus theory and application to the minimization of boolean functions. IEEE Trans. Electron. Comput. 4, 446–456 (1967)

    Article  Google Scholar 

  31. Weidenbach, C., Afshordel, B., Brahm, U., Cohrs, C., Engel, T., Keen, E., Theobalt, C., Topić, D.: System description: Spass version 1.0.0. In: CADE 1999. LNCS, vol. 1632, pp. 378–382. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48660-7_34

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Univ. Grenoble Alpes, CNRS, LIG, 38000, Grenoble, France

    Mnacho Echenim, Nicolas Peltier & Yanis Sellami

Authors
  1. Mnacho Echenim
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  2. Nicolas Peltier
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  3. Yanis Sellami
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Correspondence to Nicolas Peltier .

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Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

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Echenim, M., Peltier, N., Sellami, Y. (2018). A Generic Framework for Implicate Generation Modulo Theories. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_19

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  • DOI: https://doi.org/10.1007/978-3-319-94205-6_19

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