Prime forms and minimal change in propositional belief bases

  • Jerusa Marchi
  • Guilherme Bittencourt
  • Laurent Perrussel
Article

Abstract

This paper proposes to use prime implicants and prime implicates normal forms to represent belief sets. This representation is used, on the one hand, to define syntactical versions of belief change operators that also satisfy the rationality postulates but present better complexity properties than those proposed in the literature and, on the other hand, to propose a new minimal distance that adopts as a minimal belief unit a “fact”, defined as a prime implicate of the belief set, instead of the usually adopted Hamming distance, i.e., the number of propositional symbols on which the models differ. Some experiments are also presented that show that this new minimal distance allows to define belief change operators that usually preserve more information of the original belief set.

Keywords

Belief change Minimal change Prime implicates Prime implicants Knowledge compilation 

Mathematics Subject Classifications (2010)

Primary 03B42; Secondary 68T30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alchourrón, C., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet functions for contraction and revision. J. Symb. Log. 50, 510–530 (1985)MATHCrossRefGoogle Scholar
  2. 2.
    Bienvenu, M.: Prime implicates and prime implicants: from propositional to modal logic. J. Artif. Intell. Res. 36, 71–128 (2009)MATHGoogle Scholar
  3. 3.
    Bienvenu, M., Herzig, A., Qi, G.: Prime implicate-based belief revision operators. In: Proc. of ECAI’08 (2008)Google Scholar
  4. 4.
    Bittencourt, G.: Advances in modeling adaptive and cognitive systems. In: Chap. A Memory Model for Cognitive Agents, pp. 60–76. UEFS (2010). ISBN 978-85-7395-194-3. http://www2.uefs.br/graco/amacs/
  5. 5.
    Bittencourt, G., Marchi, J.: Propositional reasoning for an embodied cognitive model. In: Bazzan, A.L.C., Labidi, S. (eds.) Proc. of the 17th Brazilian Symposium on Artificial Intelligence (SBIA’04), pp. 164–173. Springer, São Luís, Maranhão, Brasil (2004)Google Scholar
  6. 6.
    Bittencourt, G., Marchi, J.: Artificial cognition systems. In: Chap. An Embodied Logical Model for Cognition, pp. 27–63. IDEA Group Inc (2006)Google Scholar
  7. 7.
    Bittencourt, G., Marchi, J., Padilha, R.S.: A syntactic approach to satisfaction. In: Konev, B., Schimidt, R. (eds.) 4th Inter. Workshop on the Implementation of Logic (LPAR03), pp. 18–32. Univ. of Liverpool and Univ. of Manchester (2003)Google Scholar
  8. 8.
    Bittencourt, G., Perrussel, L., Marchi, J.: A syntactical approach to revision. In: Mántaras, R.L., Saitta, L. (eds.) Proc. of the 16th Europ. Conf. on Artificial Intelligence (ECAI’04), pp. 788–792. IOS Press, Valencia, Spain (2004)Google Scholar
  9. 9.
    Boutilier, C.: A unified model of qualitative belief change: a dynamical systems perspective. Artif. Intell. 98(1–2), 281–316 (1998). citeseer.ist.psu.edu/boutilier98unified.html MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cadoli, M., Donini, F.M.: A survey on knowledge compilation. AI Commun. 10(3–4), 137–150 (1997). citeseer.ist.psu.edu/cadoli98survey.html Google Scholar
  11. 11.
    Dalal, M.: Investigations into a theory of knowledge base revision: preliminary report. In: Rosenbloom, P., Szolovits, P. (eds.) Proceedings of the 7th National Conf. on Artificial Intelligence, vol. 2, pp. 475–479. AAAI Press, Menlo Park, California (1988). citeseer.nj.nec.com/dalal88investigations.html Google Scholar
  12. 12.
    Darwich, A., Marquis, P.: A knowledge compilation map. J. Artif. Intell. Res. 17, 229–264 (2002)Google Scholar
  13. 13.
    Darwiche, A., Marquis, P.: A perspective on knowledge compilation. In: IJCAI, pp. 175–182 (2001). citeseer.nj.nec.com/darwiche01perspective.html
  14. 14.
    del Val, A.: Syntactic characterizations of belief change operators. In: Bajcsy, R. (ed.) Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI’93), Chambéry, France, pp. 540–547. Morgan Kaufmann (1993)Google Scholar
  15. 15.
    Delgrande, J.P., Nayak, A.C., Pagnucco, M.: Conservative belief revision. In: McGuinness, D.L., Ferguson, G. (eds.) Proceedings of the Nineteenth National Conference on Artificial Intelligence, Sixteenth Conference on Innovative Applications of Artificial Intelligence, July 25–29, 2004, San Jose, California, USA, pp. 251–256. AAAI Press/The MIT Press (2004)Google Scholar
  16. 16.
    Doyle, J.: Rational belief revision. In: Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning (KR’91), Cambridge, MA, USA, April 22–25, 1991, pp. 163–174. Morgan Kaufmann Publishers (1991)Google Scholar
  17. 17.
    Fitting, M.: First-Order Logic and Automated Theorem Proving. Springer, New York (1990)MATHGoogle Scholar
  18. 18.
    Forbus, K.: Introducing actions into qualitative simulation. In: Proceedings IJCAI-89, pp. 1273–1278. Detroit, MI (1989)Google Scholar
  19. 19.
    Friedman, N., Halpern, J.Y.: A knowledge-based framework for belief change, part I: foundations. In: Fagin, R. (ed.) Theoretical Aspects of Reasoning about Knowledge: Proc. 5th Conference, pp. 44–64 (1994)Google Scholar
  20. 20.
    Friedman, N., Halpern, J.Y.: A knowledge-based framework for belief change, part II: revision and update. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) KR’94: Principles of Knowledge Representation and Reasoning, pp. 190–201. Morgan Kaufmann, San Francisco, California (1994). citeseer.nj.nec.com/friedman94knowledgebased.html Google Scholar
  21. 21.
    Gärdenfors, P.: Knowledge in Flux: Modelling the Dynamics of Epistemic States. Bradford Books, MIT Press (1988)Google Scholar
  22. 22.
    Gorogiannis, N., Ryan, M.: Implementation of belief change operators using bdds. Stud. Log. 70(1), 131–156 (2004)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Grove, A.: Two modellings for theory change. J. Philos. Logic 17, 157–170 (1988)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hansson, S.: A Textbook of Belief Dynamics. Theory Change and Database Updating. Kluwer (1999)Google Scholar
  25. 25.
    Herzig, A., Rifi, O.: Update operations: a review. In: Prade, H. (ed.) Proc. of the 13th European Conf. on Artificial Intelligence (ECAI’98), pp. 13–17. Wiley, Chichester (1998). citeseer.ist.psu.edu/herzig98update.html Google Scholar
  26. 26.
    Herzig, A., Rifi, O.: Propositional belief base update and minimal change. Artif. Intell. 115(1), 107–138 (1999). citeseer.nj.nec.com/herzig99propositional.html MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Jackson, P.: Computing prime implicants. In: Proceedings of the 10th International Conference on Automatic Deduction, Kaiserslautern, Germany. LNAI no. 449, pp. 543–557. Springer (1990)Google Scholar
  28. 28.
    Katsuno, H., Mendelzon, A.: On the difference between updating a knowledge base and revising it. In: Allen, J.F., Fikes, R., Sandewall, E. (eds.) KR’91: Principles of Knowledge Representation and Reasoning, pp. 387–394. Morgan Kaufmann, San Mateo, California (1991). citeseer.nj.nec.com/417296.html Google Scholar
  29. 29.
    Katsuno, H., Mendelzon, A.: Propositional knowledge base revision and minimal change. Artif. Intell. 52(3), 263–294 (1991)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Katsuno, H., Mendelzon, A.O.: A unified view of propositional knowledge base updates. In: Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI’89), Detroit, MI, USA, August 1989, pp. 1413–1419. Morgan Kaufmann (1989)Google Scholar
  31. 31.
    Kean, A., Tsiknis, G.: An incremental method for generating prime implicants/implicates. J. Symb. Comput. 9, 185–206 (1990)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Konieczny, S., Pérez, R.P.: Propositional belief base merging or how to merge beliefs/goals coming from several sources and some links with social choice theory. Eur. J. Oper. Res. 160(3), 785–802 (2005)MATHCrossRefGoogle Scholar
  33. 33.
    Liberatore, P., Schaerf, M.: Arbitration (or how to merge knowledge bases). IEEE Trans. Knowl. Data Eng. 10(1), 76–90 (1998)CrossRefGoogle Scholar
  34. 34.
    Makinson, D.: Propositional relevance through letter-sharing. Journal of Applied Logic 7, 377–387 (2009)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Manquinho, V.M., Flores, P.F., Marques-Silva, J.P., Oliveira, A.L.: Prime implicant computation using satisfiability algorithms. In: Proceedings of the IEEE International Conference on Tools with Artificial Intelligence (ICTAI’97), pp. 232–239. IEEE (1997)Google Scholar
  36. 36.
    Marchi, J., Bittencourt, G., Perrussel, L.: A syntactical approach to update. In: Proc. of Mexican International Conf. on Artificial Intelligence (MICAI’05). Springer, Monterrey, Mexico (2005)Google Scholar
  37. 37.
    Marchi, J., Bittencourt, G., Perrussel, L.: Perspectives on universal logic. In: Chap. Prime Forms and Belief Revision, pp. 365–377. Polimetrica (2007)Google Scholar
  38. 38.
    Morgan, C.G.: Probability, rational belief and belief change. In: Delgrande, J.P., Schaub, T. (eds.) 10th International Workshop on Non-Monotonic Reasoning (NMR 2004), Whistler, Canada, June 6–8, 2004, Proceedings, pp. 297–305 (2004)Google Scholar
  39. 39.
    Nebel, B.: A knowledge level analysis of belief revision. In: Principles of Knowledge Representation and Reasoning: Proceedings of the 1st International Conference (KR’89), pp. 301–311 (1989)Google Scholar
  40. 40.
    Nebel, B.: Belief revision and default reasoning: syntax-based approaches. In: Allen, J.A., Fikes, R., Sandewall, E. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the 2nd International Conference, pp. 417–428. Morgan Kaufmann, San Mateo (1991). citeseer.ist.psu.edu/nebel91belief.html Google Scholar
  41. 41.
    Nebel, B.: Base revision operations and schemes: semantics, representation, and complexity. In: Proceedings of the 11th European Conference on Artificial Intelligence (ECAI’94), pp. 341–345 (1994)Google Scholar
  42. 42.
    Pagnucco, M.: The Role of Abductive Reasoning Within the Process of Belief Revision. Ph.D. Thesis, Department of Computer Science, University of Sydney (1996)Google Scholar
  43. 43.
    Pagnucco, M.: Knowledge compilation for belief change. In: Proceedings of the 19th Australian Joint Conference on Artificial Intelligence (AI06). Lecture Notes in Artificial Intelligence, vol. 4304, pp. 90–99. Springer (2006)Google Scholar
  44. 44.
    Parikh, R.: Beliefs, Belief Revision, and Splitting Languages, vol. 2, pp. 266–278. Center for the Study of Language and Information, Stanford, CA, USA (1999)Google Scholar
  45. 45.
    Perrussel, L., Marchi, J., Bittencourt, G.: Quantum-based belief merging. In: Proceedings of the 11th Ibero-American Conference on AI (IBERAMIA’08). Lecture Notes in Computer Science, vol. 5290, pp. 21–30. Springer (2008)Google Scholar
  46. 46.
    Ramesh, A., Becker, G., Murray, N.V.: CNF and DNF considered harmful for computing prime implicants/implicates. J. Autom. Reason. 18(3), 337–356 (1997). citeseer.nj.nec.com/516217.html MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Revesz, P.Z.: On the semantics of arbitration. Int. J. Algebra Comput. 7(2), 133–160 (1995)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Satoh, K.: Nonmonotonic reasoning by minimal belief revision. In: FGCS, pp. 455–462 (1988)Google Scholar
  49. 49.
    Schrag, R., Crawford, J.M.: Implicates and prime implicates in random 3-SAT. Artif. Intell. 81(1–2), 199–222 (1995). citeseer.ist.psu.edu/article/schrag95implicate.html MathSciNetGoogle Scholar
  50. 50.
    Sloan, R.H., Szörényi, B., Turán, G.: On k-term dnf with the largest number of prime implicants. SIAM J. Discrete Math. 21(4), 987–998 (2008). doi:10.1137/050632026 CrossRefGoogle Scholar
  51. 51.
    Socher, R.: Optimizing the clausal normal form transformation. J. Autom. Reason. 7(3), 325–336 (1991)MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Winslett, M.: Reasoning about action using a possible models approach. In: Proceedings of the 7th National Conf. on Artificial Intelligence, pp. 89–93 (1988)Google Scholar
  53. 53.
    Zhuang, Z., Pagnucco, M., Meyer, T.: Implementing iterated belief change via prime implicates. In: Proceeding of the 20th Australian Joint Conference on Artificial Intelligence, pp. 507–518 (2007)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Jerusa Marchi
    • 1
  • Guilherme Bittencourt
    • 2
  • Laurent Perrussel
    • 3
  1. 1.Departamento de Ciência da ComputaçãoUniversidade Federal de LavrasLavrasBrazil
  2. 2.Departamento de Automação e SistemasUniversidade Federal de Santa CatarinaFlorianópolisBrazil
  3. 3.IRIT—Institut de Recherche en Informatique de ToulouseUniversité de ToulouseToulouse Cedex 9France

Personalised recommendations