Hyperresolution for Propositional Product Logic

Dušan Guller

Abstract

We provide the foundations of automated deduction in the propositional product logic. Particularly, we generalise the hyperresolution principle for the propositional product logic. We propose translation of a formula to an equivalent satisfiable finite order clausal theory, which consists of order clauses - finite sets of order literals of the augmented form: e1 @ e2 where e1 is either a truth constant, 0, 1, or a conjunction of powers of propositional atoms, and @ is a connective from =, <. = and < are interpreted by the equality and strict linear order on [0,1], respectively. We devise a hyperresolution calculus over order clausal theories, which is refutation sound and complete for the finite case. By means of the translation and calculus, we solve the deduction problem T |= phi for a finite theory T and a formula phi.

References

  1. Baaz, M., Ciabattoni, A., and Ferm üller, C. G. (2012). Theorem proving for prenex Gödel logic with Delta: checking validity and unsatisfiability. Logical Methods in Computer Science, 8(1).
  2. Baaz, M. and Ferm üller, C. G. (2010). A resolution mechanism for prenex Gödel logic. In CSL 2010, volume 6247 of Lecture Notes in Computer Science, pages 67-79. Springer.
  3. Bachmair, L. and Ganzinger, H. (1994). Rewrite-based equational theorem proving with selection and simplification. J. Log. Comput., 4(3):217-247.
  4. Bachmair, L. and Ganzinger, H. (1998). Ordered chaining calculi for first-order theories of transitive relations. J. ACM, 45(6):1007-1049.
  5. 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.
  6. Bongini, M., Ciabattoni, A., and Montagna, F. (2016). Proof search and co-NP completeness for manyvalued logics. Fuzzy Sets and Systems, 292:130-149.
  7. Davis, M., Logemann, G., and Loveland, D. (1962). A machine program for theorem-proving. Commun. ACM, 5(7):394-397.
  8. Davis, M. and Putnam, H. (1960). A computing procedure for quantification theory. J. ACM, 7(3):201-215.
  9. Esteva, F. and Godo, L. (2001). Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124(3):271-288.
  10. Guller, D. (2009). On the refutational completeness of signed binary resolution and hyperresolution. Fuzzy Sets and Systems, 160(8):1162 - 1176.
  11. Guller, D. (2010). A DPLL procedure for the propositional Gödel logic. In ICFC 2010, pages 31-42. SciTePress.
  12. Guller, D. (2012a). On the satisfiability and validity problems in the propositional Gödel logic. In Computational Intelligence - IJCCI 2010, volume 399 of Studies in Computational Intelligence, pages 211-227. Springer.
  13. Guller, D. (2012b). An order hyperresolution calculus for Gödel logic - General first-order case. InFCTA 2012, pages 329-342. SciTePress.
  14. Guller, D. (2013). A DPLL procedure for the propositional product logic. In FCTA 2013, pages 213-224. SciTePress.
  15. Guller, D. (2014). An order hyperresolution calculus for Gödel logic with truth constants. In FCTA 2014, pages 37-52. SciTePress.
  16. Guller, D. (2015a). An order hyperresolution calculus for Gödel logic with truth constants and equality, strict order, Delta. In FCTA 2015, pages 31-46. SciTePress.
  17. Guller, D. (2015b). Unsatisfiable formulae of Gödel logic with truth constants and P, ?, ? are recursively enumerable. In BRICS Congress, CCI 2015, Part III, volume 9142 of Lecture Notes in Computer Science, pages 242-250. Springer.
  18. Guller, D. (2016a). On the deduction problem in Gödel and Product logics. In Computational Intelligence - IJCCI 2013, volume 613 of Studies in Computational Intelligence, pages 299-321. Springer.
  19. Guller, D. (2016b). Unsatisfiable formulae of Gödel logic with truth constants and Delta are recursively enumerable. In Computational Intelligence - IJCCI 2014, volume 620 of Studies in Computational Intelligence, pages 213-234. Springer.
  20. Hájek, P. (2001). Metamathematics of Fuzzy Logic. Trends in Logic. Springer.
  21. Klement, E. and Mesiar, R. (2005). Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier.
  22. Klement, E., Mesiar, R., and Pap, E. (2013). Triangular Norms. Trends in Logic. Springer.
  23. Marchioni, E. and Metcalfe, G. (2010). Interpolation properties for uninorm based logics. In ISMVL 2010, pages 205-210. IEEE Computer Society.
  24. Marques-Silva, J. P. and Sakallah, K. A. (1999). Grasp: A search algorithm for propositional satisfiability. Computers, IEEE Transactions on, 48(5):506-521.
  25. Mostert, P. S. and Shields, A. L. (1957). On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics, pages 117-143.
  26. 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.
  27. Robinson, J. A. (1965a). Automatic deduction with hyperresolution. Internat. J. Comput. Math., 1(3):227-234.
  28. Robinson, J. A. (1965b). A machine-oriented logic based on the resolution principle. J. ACM, 12(1):23-41.
  29. Schöning, U. and Torán, J. (2013). The Satisfiability Problem: Algorithms and Analyses. Mathematik für Anwendungen. Lehmanns Media.
  30. Silva, J. P. M. and Sakallah, K. A. (1996). GRASP - a new search algorithm for satisfiability. In ICCAD 1996, pages 220-227.
Download


Paper Citation


in Harvard Style

Guller D. (2016). Hyperresolution for Propositional Product Logic . In Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016) ISBN 978-989-758-201-1, pages 30-41. DOI: 10.5220/0006044300300041


in Bibtex Style

@conference{fcta16,
author={Dušan Guller},
title={Hyperresolution for Propositional Product Logic},
booktitle={Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016)},
year={2016},
pages={30-41},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006044300300041},
isbn={978-989-758-201-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 8th International Joint Conference on Computational Intelligence - Volume 2: FCTA, (IJCCI 2016)
TI - Hyperresolution for Propositional Product Logic
SN - 978-989-758-201-1
AU - Guller D.
PY - 2016
SP - 30
EP - 41
DO - 10.5220/0006044300300041