Abstract
This paper addresses 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. Since the equivalent output CNF may be exponential in the size of an input formula, we improve the translation using interpolation rules so that output CNF formulae are in linear size with respect to input ones; however, not equivalent - only equisatisfiable. 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 \(a\eqcirc b\) or a ≺ b, where \(\eqcirc \) and ≺ are interpreted by the equality and strict linear order on [0,1], respectively. A variant of the DPLL procedure, operating on order clausal theories, is proposed. The DPLL procedure is proved to be refutation sound and complete for countable order clausal theories. Finally, the validity problem of a formula (tautology checking) is reduced to the unsatisfiability of a finite order clausal theory.
Partially supported by the grants VEGA 1/0688/10, VEGA 1/0726/09, and Slovak Literary Fund.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aguzzoli, S., Ciabattoni, A.: Finiteness of infinite-valued Łukasiewicz logic. Journal of Logic, Language and Information 9, 5–29 (2000)
Anderson, R., 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)
Baaz, M., Ciabattoni, A., Fermüller, C.: Herbrand’s theorem for prenex gödel logic and its consequences for theorem proving. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 201–215. Springer, Heidelberg (2001)
Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation 4(3), 217–247 (1994)
Bachmair, L., Ganzinger, H.: Ordered chaining calculi for first-order theories of transitive relations. Journal of the ACM 45(6), 1007–1049 (1998)
Boy de la Tour, T.: An optimality result for clause form translation. Journal of Symbolic Computation 14(4), 283–301 (1992)
Davis, M., Putnam, H.: A computing procedure for quantification theory. Communications of the ACM 7, 201–215 (1960)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5, 394–397 (1962)
Guller, D.: Binary resolution over complete residuated Stone lattices. Fuzzy Sets and Systems 159(9), 1031–1041 (2008)
Guller, D.: On the refutational completeness of signed binary resolution and hyperresolution. Fuzzy Sets and Systems 160(8), 1162–1176 (2009)
Guller, D.: A DPLL procedure for the propositional Gödel logic. In: Proceedings of the ICFC Conference, INSTICC, pp. 31–42 (2010)
Hähnle, R.: Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence 12(3,4), 231–264 (1994)
Hähnle, R.: Short conjunctive normal forms in finitely-valued logics. Journal of Logic and Computation 4(6), 905–927 (1994)
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)
Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science 52, 145–153 (1987)
Nonnengart, A., Rock, G., Weidenbach, C.: On Generating Small Clause Normal Forms. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS (LNAI), vol. 1421, pp. 397–411. Springer, Heidelberg (1998)
Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. Journal of Symbolic Computation 2(3), 293–304 (1986)
Sheridan, D.: The optimality of a fast CNF conversion and its use with SAT. In: Online Proceedings of International Conference on the Theory and Applications of Satisfiability Testing (2004), http://www.satisfiability.org/SAT04/programme/114.pdf
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
Guller, D. (2012). On the Satisfiability and Validity Problems in the Propositional Gödel Logic. In: Madani, K., Dourado Correia, A., Rosa, A., Filipe, J. (eds) Computational Intelligence. IJCCI 2010. Studies in Computational Intelligence, vol 399. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27534-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-27534-0_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27533-3
Online ISBN: 978-3-642-27534-0
eBook Packages: EngineeringEngineering (R0)