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CDCL-Based Abstract State Transition System for Coherent Logic

  • Mladen Nikolić
  • Predrag Janičić
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7362)

Abstract

We present a new, CDCL-based approach for automated theorem proving in coherent logic — an expressive semi-decidable fragment of first-order logic that provides potential for obtaining human readable and machine verifiable proofs. The approach is described by means of an abstract state transition system, inspired by existing transition systems for SAT and represents its faithful lifting to coherent logic. The presented transition system includes techniques from which CDCL SAT solvers benefited the most (backjumping and lemma learning), but also allows generation of human readable proofs. In contrast to other approaches to theorem proving in coherent logic, reasoning involved need not to be ground. We prove key properties of the system, primarily that the system yields a semidecision procedure for coherent logic. As a consequence, the semidecidability of another fragment of first order logic which is a proper superset of coherent logic is also proven.

Keywords

coherent logic CDCL SAT solving abstract state transition systems machine verifiable proofs readable proofs 

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References

  1. 1.
    Baumgartner, P., Tinelli, C.: The Model Evolution Calculus as a First-Order DPLL Method. Artificial Intelligence 172(4-5) (2008)Google Scholar
  2. 2.
    Bezem, M., Coquand, T.: Automating Coherent Logic. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 246–260. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    M. Bezem. On the Undecidability of Coherent Logic. Processes, Terms and Cycles (2005)Google Scholar
  4. 4.
    Bezem, M., Hendriks, D.: On the Mechanization of the Proof of Hessenberg’s Theorem in Coherent Logic. J. of Automated Reasoning 40(1) (2008)Google Scholar
  5. 5.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. IOS Press (2009)Google Scholar
  6. 6.
    Davis, M., Putnam, H.: A Computing Procedure for Quantification Theory. J. of ACM 7(3) (1960)Google Scholar
  7. 7.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5(7) (1962)Google Scholar
  8. 8.
    Fisher, J., Bezem, M.: Skolem Machines and Geometric Logic. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) ICTAC 2007. LNCS, vol. 4711, pp. 201–215. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Janičić, P., Kordić, S.: EUCLID — the geometry theorem prover. FILOMAT 9(3) (1995)Google Scholar
  10. 10.
    Krstić, S., Goel, A.: Architecting Solvers for SAT Modulo Theories: Nelson-Oppen with DPLL. In: Konev, B., Wolter, F. (eds.) FroCos 2007. LNCS (LNAI), vol. 4720, pp. 1–27. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Marić, F.: Formalization and Implementation of Modern SAT Solvers. J. of Automated Reasoning 43(1) (2009)Google Scholar
  12. 12.
    Marić, F., Janičić, P.: Formalization of Abstract State Transition Systems for SAT. Logical Methods in Computer Science 7(3) (2011)Google Scholar
  13. 13.
    Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: From an abstract Davis–Putnam–Logemann–Loveland procedure to DPLL(T). J. of ACM 53(6) (2006)Google Scholar
  14. 14.
    Nikolić, M., Marić, F., Janičić, P.: Instance-Based Selection of Policies for SAT Solvers. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 326–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    de Nivelle, H., Meng, J.: Geometric Resolution: A Proof Procedure Based on Finite Model Search. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 303–317. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Piskač, R., de Moura, L., Bjorner, N.: Deciding effectively propositional logic using DPLL and substitution sets. J. of Automated Reasoning 44 (2010)Google Scholar
  17. 17.
    Polonsky, A.: Proofs, Types, and Lambda Calculus. PhD thesis, University of Bergen (2010)Google Scholar
  18. 18.
    Stojanović, S., Pavlović, V., Janičić, P.: Automated Generation of Formal and Readable Proofs in Geometry Using Coherent Logic. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) ADG 2010. LNCS, vol. 6877, Springer, Heidelberg (2011)Google Scholar
  19. 19.
    Wenzel, M.: Isar - A Generic Interpretative Approach to Readable Formal Proof Documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 167–183. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mladen Nikolić
    • 1
  • Predrag Janičić
    • 1
  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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