An Order Hyperresolution Calculus for Gödel Logic with Truth Constants

Dušan Guller

Abstract

We have generalised the well-known hyperresolution principle to the first-order G¨odel logic for the general case. This paper is a continuation of our work. We propose a modification of the hyperresolution calculus suitable for automated deduction with explicit partial truth. We expand the first-order G¨odel logic by a countable set of intermediate truth constants ¯ c, c 2 (0;1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form e1  e2 where  is a connective either P or . P and  are interpreted by the equality and standard strict linear order on [0;1], respectively. We shall investigate the so-called canonical standard completeness, where the semantics of the first-order G¨odel logic is given by the standard G-algebra and truth constants are interpreted by themselves. The modified hyperresolution calculus is refutation sound and complete for a countable order clausal theory under certain condition for suprema and infima of sets of the truth constants occurring in the theory.

References

  1. Biere, A., Heule, M. J., van Maaren, H., and Walsh, T. (2009). Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications. IOS Press, Amsterdam.
  2. Davis, M., Logemann, G., and Loveland, D. (1962). A machine program for theorem-proving. Commun. ACM, 5(7):394-397.
  3. Davis, M. and Putnam, H. (1960). A computing procedure for quantification theory. J. ACM, 7(3):201-215.
  4. de la Tour, T. B. (1992). An optimality result for clause form translation. J. Symb. Comput., 14(4):283-302.
  5. Esteva, F., Gispert, J., Godo, L., and Noguera, C. (2007a). Adding truth-constants to logics of continuous tnorms: axiomatization and completeness results. Fuzzy Sets and Systems, 158(6):597-618.
  6. Esteva, F., Godo, L., and Montagna, F. (2001). The L? and L? 12 logics: two complete fuzzy systems joining
  7. Lukasiewicz and Product logics. Arch. Math. Log., 40(1):39-67.
  8. Esteva, F., Godo, L., and Noguera, C. (2007b). On completeness results for the expansions with truthconstants of some predicate fuzzy logics. In Stepnicka, M., Novák, V., and Bodenhofer, U., editors, New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, 2007, Volume 2: Regular Sessions, pages 21-26. Universitas Ostraviensis.
  9. Esteva, F., Godo, L., and Noguera, C. (2009). First-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties. Ann. Pure Appl. Logic, 161(2):185-202.
  10. Esteva, F., Godo, L., and Noguera, C. (2010a). Expanding the propositional logic of a t-norm with truthconstants: completeness results for rational semantics. Soft Comput., 14(3):273-284.
  11. Esteva, F., Godo, L., and Noguera, C. (2010b). On expansions of WNM t-norm based logics with truthconstants. Fuzzy Sets and Systems, 161(3):347-368.
  12. Gallier, J. H. (1985). Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row Publishers, Inc., New York, NY, USA.
  13. Guller, D. (2010). A DPLL procedure for the propositional Gödel logic. In Filipe, J. and Kacprzyk, J., editors, ICFC-ICNC 2010 - Proceedings of the International Conference on Fuzzy Computation and International Conference on Neural Computation, [parts of the International Joint Conference on Computational Intelligence IJCCI 2010], Valencia, Spain, October 24-26, 2010, pages 31-42. SciTePress.
  14. Guller, D. (2012). An order hyperresolution calculus for Gödel logic - General first-order case. In Rosa, A. C., Correia, A. D., Madani, K., Filipe, J., and Kacprzyk, J., editors, IJCCI 2012 - Proceedings of the 4th International Joint Conference on Computational Intelligence, Barcelona, Spain, 5 - 7 October, 2012, pages 329-342. SciTePress.
  15. Hähnle, R. (1994). Short conjunctive normal forms in finitely valued logics. J. Log. Comput., 4(6):905-927.
  16. Hájek, P. (2001). Metamathematics of Fuzzy Logic. Trends in Logic. Springer.
  17. Nonnengart, A., Rock, G., and Weidenbach, C. (1998). On generating small clause normal forms. In Kirchner, C. and Kirchner, H., editors, Automated Deduction - CADE-15, 15th International Conference on Automated Deduction, Lindau, Germany, July 5-10, 1998, Proceedings, volume 1421 of Lecture Notes in Computer Science, pages 397-411. Springer.
  18. Novák, V., Perfilieva, I., and Moc?ko?r, J. (1999). Mathematical Principles of Fuzzy Logic. The Springer International Series in Engineering and Computer Science. Springer US.
  19. Pavelka, J. (1979). On fuzzy logic I, II, III. Semantical completeness of some many-valued propositional calculi. Mathematical Logic Quarterly, 25(2529):45-52, 119-134, 447-464.
  20. Plaisted, D. A. and Greenbaum, S. (1986). A structurepreserving clause form translation. J. Symb. Comput., 2(3):293-304.
  21. Robinson, J. A. (1965a). Automatic deduction with hyperresolution. Internat. J. Comput. Math., 1(3):227-234.
  22. Robinson, J. A. (1965b). A machine-oriented logic based on the resolution principle. J. ACM, 12(1):23-41.
  23. SavickÉ, P., Cignoli, R., Esteva, F., Godo, L., and Noguera, C. (2006). On Product logic with truth-constants. J. Log. Comput., 16(2):205-225.
Download


Paper Citation


in Harvard Style

Guller D. (2014). An Order Hyperresolution Calculus for Gödel Logic with Truth Constants . In Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014) ISBN 978-989-758-053-6, pages 37-52. DOI: 10.5220/0005073700370052


in Bibtex Style

@conference{fcta14,
author={Dušan Guller},
title={An Order Hyperresolution Calculus for Gödel Logic with Truth Constants},
booktitle={Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)},
year={2014},
pages={37-52},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005073700370052},
isbn={978-989-758-053-6},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Fuzzy Computation Theory and Applications - Volume 1: FCTA, (IJCCI 2014)
TI - An Order Hyperresolution Calculus for Gödel Logic with Truth Constants
SN - 978-989-758-053-6
AU - Guller D.
PY - 2014
SP - 37
EP - 52
DO - 10.5220/0005073700370052