Skip to main content
SpringerLink
Log in

An Algorithm for Dual Transformation in First-Order Logic

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

The transformation between conjunctive and disjunctive canonical forms is useful in domains such as theorem proving, function minimization, and knowledge representation. In this paper, we present a concurrent algorithm for this transformation, suitable for first-order logic theories. The proposed algorithm use the holographic relation between these normal forms in order to avoid the generation of noncondensed and subsumed (dual) clauses. We also stress the facts that, in first-order logic, this transformation is asymmetric and that disjunctive normal form, in some special cases, may be not unique, depending on choices about which subsumptions are allowed or not. The algorithm, which is part of a theorem-proving knowledge representation project, has been implemented and tested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, P. B.: Theorem proving via general matings, J. ACM 28(2) (1981), 193–214.

    Google Scholar 

  2. Baader, F. and Siekmann, J. H.: Unification theory, in D. M. Gabbay, C. J. Hogger, and J. A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming-Logical Foundations, Vol. 2, Oxford University Press, 1994, pp. 41–125.

    Google Scholar 

  3. Bibel, W. and Eder, E.: Methods and calculi for deduction, in D. M. Gabbay, C. J. Hogger, and J. A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming-Logical Foundations, Vol. 1, Oxford University Press, 1993, pp. 67–182.

    Google Scholar 

  4. Bittencourt, G.: In the quest of the missing link, in Proceedings of IJCAI 15, Nagoya, Japan, August 23–29, Morgan Kaufmann, 1997, pp. 310–315.

  5. Bittencourt, G.: Concurrent inference through dual transformation, Logic Journal of the Interest Group in Pure and Applied Logics (IGPL) 6(6) (1998), 795–834.

    Google Scholar 

  6. Bittencourt, G. and Tonin, I.: A multi-agent approach to first-order logic, in Proceedings of 8 th Portuguese Conference on Artificial Intelligence (EPIA'97), Coimbra, Portugal, October 6–9. Springer-Verlag, 1997.

  7. Bittencourt, G. and Tonin, I.: A proof strategy based on a dual representation, in E. Roanes-Lozano and J. Calmet (eds.), Proceedings of the Fifth International Conference Artificial Intelligence and Symbolic Computation Theory, Implementations and Applications (AISC'2000), Lecture Notes in Artificial Intelligence, Springer, 2000.

  8. da Costa, A. C. P. L. and Bittencourt, G.: Dynamic social knowledge: A cooperation strategy for cognitive multi-agent systems, in Third International Conference on Multi-Agent Systems (ICMAS'98), IEEE Computer Society, 1998, pp. 415–416.

  9. da Costa, A. C. P. L. and Bittencourt, G.: From a concurrent agent architecture to an autonomous concurrent agent architecture, IJCAI'99 Third RoboCup Workshop, Stockolm, Sweden, July 31-August 1, 1999.

  10. Chang, C.-L. and Lee, R. C.-T.: Symbolic Logic and Mechanical Theorem Proving, Academic Press, Computer Science Classics, 1973.

  11. Dahl, V.: Logic programming as representation of knowledge, IEEE Computer (October 1983), 106–111.

  12. Eder, E.: Consolution and its relation to resolution, in Proceedings of IJCAI 12, 1991.

  13. Eisinger, N. and Ohlbach, H. J.: Deduction systems based on resolution, in D. M. Gabbay, C. J. Hogger, and J. A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming-Logical Foundations, Vol. 1, Oxford University Press, 1993, pp. 183–271.

    Google Scholar 

  14. Fitting, M.: First-Order Logic and Automated Theorem Proving, Springer-Verlag, 1990.

  15. Gottlob, G. and Fermüller, C. G.: Removing redundancy from a clause, Artificial Intelligence 61 (1993), 263–289.

    Google Scholar 

  16. Gottlob, G. and Leitsch, A.: On the efficiency of subsumption algorithms, J. ACM 32(2) (1985), 280–295.

    Google Scholar 

  17. Gallaire, H. and Minker, J.: Logic and Datbases, Plenum Press, 1978.

  18. Genesereth, M. R. and Nilsson, N. J.: Logical Foundations of Artificial Intelligence, Morgan Kaufmann Publishers Inc., 1987.

  19. Jackson, P.: Computing prime implicants, in Proceedings of the 10th International Conference on Automatic Deduction, Kaiserslautern, Germany, LNAI 449, Springer-Verlag, 1990, pp. 543–557.

  20. Kowalsky, R. A.: A proof procedure using connection graphs, J. ACM 22(4) (1975), 572–595.

    Google Scholar 

  21. Kean, A. and Tsiknis, G.: An incremental method for generating prime implicants/implicates, J. Symbolic Comput. 9 (1990), 185–206.

    Google Scholar 

  22. Loveland, D.: Automated Theorem Proving: A Logical Basis, North-Holland, 1978.

  23. MaClachlan, R. A.: CMU Common Lisp User's Manual, Carnegie Mellon University, Pittsburgh, 1992.

    Google Scholar 

  24. Nilsson, N. J.: Problem Solving Methods in Artificial Intelligence, McGraw-Hill, 1971.

  25. Nonnengart, A., Rock, G. and Weidenbach, C.: On generating small clause normal forms, in C. Kirchner and H. Kirchner (eds.), 15th International Conference on Automated Deduction, CADE-15, LNAF 1421, Springer-Verlag, 1998, pp. 397–411.

  26. Quine, W. V. O.: The problem of simplifying truth functions, Amer. Math. Monthly 59 (1952), 521–531.

    Google Scholar 

  27. Quine, W. V. O.: On cores and prime implicants of truth functions, Amer. Math. Monthly 66 (1959), 755–760.

    Google Scholar 

  28. Quinn, M. J.: Desingning Efficient Algorithms for Parallel Computers, McGraw-Hill International, 1987.

  29. Reiter, R.: A logic for default reasoning, Artificial Intelligence 13(1–2) (April 1980), 81–132.

    Google Scholar 

  30. Robinson, J. A.: A machine-oriented logic based on the resolution principle, J. ACM 12(1) (January 1965), 23–41.

    Google Scholar 

  31. Slagle, J. R., Chang, C. L. and Lee, R. C. T.: A new algorithm for generating prime implicants, IEEE Trans. Comput. 19(4) (1970), 304–310.

    Google Scholar 

  32. Smullyan, R. M.: First Order Logic, Springer-Verlag, 1968.

  33. Socher, R.: Optimizing the clausal normal form transformation, J. Automated Reasoning 7(3) (1991), 325–336.

    Google Scholar 

  34. Steele, Jr., G. L.: Common LISP, the Language, Digital Press, 1984.

  35. Tammet, T.: Towards efficient subsumption, in C. Kirchner and H. Kirchner (eds.), 15 th International Conference on Automated Deduction, CADE-15, Lecture Notes in Artificial Intelligence 1421, Springer-Verlag, 1998, pp. 427–441.

  36. Tison, P.: Generalized consensus theory and application to the minimization of Boolean functions, IEEE Trans. Comput. 16(4) (1967), 446–456.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Automação e Sistemas, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brazil

    Guilherme Bittencourt & Isabel Tonin

Authors
  1. Guilherme Bittencourt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Isabel Tonin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bittencourt, G., Tonin, I. An Algorithm for Dual Transformation in First-Order Logic. Journal of Automated Reasoning 27, 353–389 (2001). https://doi.org/10.1023/A:1011917032470

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011917032470

Search

Navigation