A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC

Dušan Guller

Abstract

In the paper, we investigate the satisfiability and validity problems of a formula in the propositional Gödel logic. Our approach is based on the translation of a formula to an equivalent CNF one which contains literals of the augmented form: either a or a→b or (a→b)→b, where a, b are propositional atoms or the propositional constants 0, 1. A CNF formula is further translated to an equisatisfiable finite order clausal theory which consists of order clauses, finite sets of order literals of the forms a ≖ b or a ≺ b. ≖ and ≺ are interpreted by the equality and strict linear order on [0,1], respectively. A variant of the DPLL procedure for deciding the satisfiability of a finite order clausal theory is proposed. The DPLL procedure is proved to be refutation sound and complete. Finally, we reduce the validity problem of a formula (tautology checking) to the unsatisfiability of a finite order clausal theory.

References

  1. Aguzzoli, S. and Ciabattoni, A. Finiteness of infinite-valued Lukasiewicz logic. Journal of Logic, Language and Information, 9:5-29, 2000.
  2. Anderson, R. and Bledsoe, W. W. A linear format for resolution with merging and a new technique for establishing completeness. Journal of the ACM, 17(3):525- 534, 1970.
  3. Baaz, M., Fermüller, C. G. and Ciabattoni, A. Herbrand's theorem for prenex Gödel logic and its consequences for theorem proving. Proceedings of the LPAR conference, LNCS vol. 2250, Springer-Verlag, 201-215, 2001.
  4. Bachmair, L. and Ganzinger, H. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217- 247, 1994.
  5. Bachmair, L. and Ganzinger, H. Ordered chaining calculi for first-order theories of transitive relations. Journal of the ACM, 45(6):1007-1049, 1998.
  6. Beckert, B., Hähnle, R. and Manyà, F. The SAT problem of signed CNF formulas. In Labelled Deduction, Basin, D., D'Agostino, M., Gabbay, D., Matthews, S. and Viganó, L., eds., Applied Logic Series, vol. 17, Kluwer Academic Publishers, 61-82, 2000.
  7. Biere, A., Heule, M., van Maaren, H. and Walsh, T., eds. Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications Series, vol. 185, IOS Press, 2009.
  8. Boy de la Tour, T. An optimality result for clause form translation. Journal of Symbolic Computation, 14(4):283-301, 1992.
  9. Davis, M. and Putnam, H. A computing procedure for quantification theory. Communications of the ACM, 7:201- 215, 1960.
  10. Davis, M., Logemann, G. and Loveland, D. A machine program for theorem-proving. Communications of the ACM, 5:394-397, 1962.
  11. Dixon, H. E., Ginsberg, M. L., Luks, E. M. and Parkes, A. J. Generalizing Boolean satisfiability II: Theory. Journal of Artificial Intelligence Research, 22:481-534, 2004.
  12. Dixon, H. E., Ginsberg, M. L. and Parkes, A. J. Generalizing Boolean satisfiability I: Background and survey of existing work. Journal of Artificial Intelligence Research, 21:193-243, 2004.
  13. Gomes, C. P., Kautz, H., Sabharwal, A. and Selman, B. Satisfiability solvers. In Handbook of Knowledge Representation, Harmelen, F. v., Lifschitz, V. and Porter, B., eds., Elsevier Science Publishers, Part I, Chap. 3, 2007.
  14. Guller, D. On the refutational completeness of signed binary resolution and hyperresolution. Fuzzy Sets and Systems, 160(8):1162-1176, 2009.
  15. Hähnle, R. Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(3,4):231-264, 1994.
  16. Hähnle, R. Short conjunctive normal forms in finitelyvalued logics. Journal of Logic and Computation, 4(6):905-927, 1994.
  17. Hähnle, R. Exploiting data dependencies in many-valued logics. Journal of Applied Non-Classical Logics, 6(1):49-69, 1996.
  18. Hähnle, R. Proof theory of many-valued logic - linear optimization - logic design: Connections and interactions. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 1(3):107-119, 1997.
  19. Kautz, H. and Selman, B. The state of SAT. Discrete Applied Mathematics, 155(12):1514-1524, 2007.
  20. Manyà, F., Béjar, R. and Escalada-Imaz, G. The satisfiability problem in regular CNF-formulas. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 2(3):116-123, 1998.
  21. Mundici, D. Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science, 52:145- 153, 1987.
  22. Nonnengart, A., Rock, G. and Weidenbach, Ch. On generating small clause normal forms. Proceedings of the CADE conference, LNAI vol. 1421, Springer-Verlag, 397-411, 1998.
  23. Plaisted, D. A. and Greenbaum, S. A structure-preserving clause form translation. Journal of Symbolic Computation, 2(3):293-304, 1986.
  24. Sheridan, D. The optimality of a fast CNF conversion and its use with SAT. Online Proceedings of International Conference on the Theory and Applications of Satisfiability Testing, www.satisfiability.org/SAT04/programme/114.pdf, 2004.
Download


Paper Citation


in Harvard Style

Guller D. (2010). A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC . In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation - Volume 1: ICFC, (IJCCI 2010) ISBN 978-989-8425-32-4, pages 31-42. DOI: 10.5220/0003061700310042


in Bibtex Style

@conference{icfc10,
author={Dušan Guller},
title={A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC},
booktitle={Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation - Volume 1: ICFC, (IJCCI 2010)},
year={2010},
pages={31-42},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003061700310042},
isbn={978-989-8425-32-4},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation - Volume 1: ICFC, (IJCCI 2010)
TI - A DPLL PROCEDURE FOR THE PROPOSITIONAL GÖDEL LOGIC
SN - 978-989-8425-32-4
AU - Guller D.
PY - 2010
SP - 31
EP - 42
DO - 10.5220/0003061700310042