Skip to main content

How Hard is it to Revise a Belief Base?

  • Chapter
  • First Online:
Book cover Belief Change

Part of the book series: Handbook of Defeasible Reasoning and Uncertainty Management Systems ((HAND,volume 3))

Abstract

If a new piece of information contradicts our previously held beliefs, we have to revise our beliefs. This problem of belief revision 1 arises in a number of areas in Computer Science and Artificial Intelligence, e.g., in updating logical database [Fagin et al., 1983], in hypothetical reasoning [Ginsberg, 1986], and in machine learning [Wrobel, 1994]. Most of the research in this area is influenced by work in philosophical logic, in particular by Gärdenfors and his colleagues [Alchourrón et al., 1985; Gärdenfors, 1988; Gärdenfors, 1992a], who developed the theory of belief revision. Here we will focus on the computational aspects of this theory, surveying results that address the issue of the computational complexity of belief revision.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. E. Alchourrón and D. Makinson. On the logic of theory change: contraction functions and their associated revision functions. Theoria, 48: 14–37, 1982.

    MathSciNet  MATH  Google Scholar 

  2. C. E. Alchourrón and D. Makinson. On the logic of theory change: Safe contractions. Studia Logica, 44: 405–422, 1985.

    MathSciNet  MATH  Google Scholar 

  3. C. E. Alchourrón and D. Makinson. Maps between some different kinds of contraction function: The finite case. Studia Logica, 45(2): 187–198, 1986.

    MathSciNet  MATH  Google Scholar 

  4. C. E. Alchourrón, P. Gärdenfors and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50(2):510–530, June 1985.

    MathSciNet  MATH  Google Scholar 

  5. J. L. Balcázar, J. Diaz and J. Gabarró. Structural Complexity I. Springer-Verlag, Berlin, Heidelberg, New York, 1988.

    MATH  Google Scholar 

  6. J. L. Balcázar, J. Diaz and J. Gabarró. Structural Complexity II. Springer-Verlag, Berlin, Heidelberg, New York, 1990.

    MATH  Google Scholar 

  7. S. Benferhat, C. Cayrol, D. Dubois, J. Lang and H. Prade. Inconsistency management and prioritized syntax-based revision. In Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 640–647, Chambery, France, August 1993.

    Google Scholar 

  8. S. Benferhat, D. Dubois and H. Prade. How to infer from inconsistent beliefs without revising. In Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI-95), pages 1449–1455, Montreal, Canada, August 1995.

    Google Scholar 

  9. S. Benferhat, D. Dubois and H. Prade. Étude comparative d’inférences tolérantes à l’inconsistance — some syntactic approaches to the handling of inconsistent knowledge bases: a comparative study. Technical Report IRIT/94-55-R, IRIT, Toulouse, France, 1995.

    Google Scholar 

  10. A. Borgida. Language features for flexible handling of exceptions in information systems. ACM Transactions on Database Systems, 10(4):565–603, December 1985.

    Google Scholar 

  11. G. Brewka. Preferred subtheories: An extended logical framework for default reasoning. In Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI-89), pages 1043–1048, Detroit, MI, August 1989. Morgan Kaufmann.

    Google Scholar 

  12. G. Brewka. Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge University Press, Cambridge, UK, 1991.

    MATH  Google Scholar 

  13. M. Cadoli and M. Schaerf. A survey of complexity results for nonmonotonic logics. The Journal of Logic Programming, 17: 127–160, 1993.

    MathSciNet  MATH  Google Scholar 

  14. M. Cadoli and M. Schaerf. Tractable reasoning via approximation. Artificial Intelligence, 74(2):249–310, 1995.

    MathSciNet  MATH  Google Scholar 

  15. M. Cadoli, F. Donini, P. Liberatore and M. Schaerf. The size of a revised knowledge base. In Proceedings of the 14th ACM SIGACT-S1GMOD-SIGART Symposium on Principles of Database-Systems (PODS-95), pages 151–162, 1995.

    Google Scholar 

  16. J.-Y. Cai, T. Gundermann, J. Hartmanis, L. A. Hemachandra, V. Sewelson, K. W. Wagner and G. Wechsung. The Boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6):1232–1252, December 1988.

    MathSciNet  MATH  Google Scholar 

  17. C. Cayrol and M.-C. Lagasquie-Schiex. Comparaison de relations d’inférence nonmonotone: Etude de complexité. Research Report IRIT/93-23-R, IRIT, Toulouse, France, 1993.

    Google Scholar 

  18. M. Dalal. Investigations into a theory of knowledge base revision: Preliminary report. In Proceedings of the 7th National Conference of the American Association for Artificial Intelligence (AAAI-88), pages 475–479, Saint Paul, MI, August 1988.

    Google Scholar 

  19. J. de Kleer. Using crude probability estimates to guide diagnosis. Artificial Intelligence, 45: 381–391, 1990.

    Google Scholar 

  20. A. del Val. Computing knowledge base updates. In B. Nebel, W. Swartout, and C. Rich, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the 3rd International Conference (KR-92), pages 740–752, Cambridge, MA, October 1992. Morgan Kaufmann.

    Google Scholar 

  21. A. del Val. On the relation between the coherence and foundations theories of belief revision. In Proceedings of the 12th National Conference of the American Association for Artificial Intelligence (AAAI-94), pages 909–914, Seattle, WA, July 1994. MIT Press.

    Google Scholar 

  22. W. F. Dowling and J. H. Gallier. Linear time algorithms for testing the satisfiability of prepositional Horn formula. The Journal of Logic Programming, 3: 267–284, 1984.

    MATH  Google Scholar 

  23. D. Dubois and H. Prade. Epistemic entrenchment and possibilistic logic. Artificial Intelligence, 50: 223–239, 1991.

    MathSciNet  MATH  Google Scholar 

  24. D. Dubois and H. Prade. Belief change and possibility theory. In Gärdenfors [1992a], pages 142–182.

    Google Scholar 

  25. D. Dubois, J. Lang and H. Prade. Possibilistic logic. In D. M. Gabbay, C. J. Hogger, J. A. Robinson, and D. Nute, editors, Handbook of Logic in Artificial Intelligence and Logic Programming-Vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning, pages 439–513. Oxford University Press, Oxford, UK, 1994.

    Google Scholar 

  26. T. Eiter and G. Gottlob. On the complexity of propositional knowledge base revision, updates, and counterfactuals. Artificial Intelligence, 57: 227–270, 1992.

    MathSciNet  MATH  Google Scholar 

  27. T. Eiter and G. Gottlob. The complexity of nested counterfactuals and iterated knowledge base revisions. In Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 526–533, Chambery, France, August 1993.

    Google Scholar 

  28. T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. S1AM Journal on Computing, 24(6):421–457, 1995.

    MathSciNet  MATH  Google Scholar 

  29. R. Fagin, J. D. Ullman and M. Y. Vardi. On the semantics of updates in databases. In 2nd ACM SIGACT-SIGMOD Symposium on Principles of Database Systems, pages 352–365, Atlanta, Ga., 1983.

    Google Scholar 

  30. R. Fagin, G. M. Kuper, J. D. Ullman, and M. Y. Vardi. Updating logical databases. Advances in Computing Research, 3: 1–18, 1986.

    Google Scholar 

  31. L. Farinas del Cerro and A. Herzig. A conditional logic for updating in the possible models approach. In B. Nebel and L. Dreschler-Fischer, editors, KI-94: Advances in Artificial Intelligence, pages 237–247, Saarbrücken, Germany, 1993. Springer-Verlag.

    Google Scholar 

  32. K. D. Forbus. Introducing actions into qualitative simulation. In Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI-89), pages 1273–1278, Detroit, MI, August 1989. Morgan Kaufmann.

    Google Scholar 

  33. A. Fuhrmann. Theory contraction through base contraction. Journal of Philosophical Logic, 20: 175–203, 1991.

    MathSciNet  MATH  Google Scholar 

  34. D. M. Gabbay, L. Giordano, A. Martelli and N. Olivetti. Conditional logic programming. In P. Van Hentenryck, editor, Proceedings of the Eleventh International Conference on Logic Programming (ICLP-94), pages 272–289, Santa Marherita Ligure, Italy, 1994. MIT Press.

    Google Scholar 

  35. P. Gärdenfors and D. Makinson. Revision of knowledge systems using epistemic entrenchment. In Theoretical Aspects of Reasoning about Knowledge: Proceedings of the Second Conference (TARK-88). Morgan Kaufmann, Asilomar, CA, 1988.

    Google Scholar 

  36. P. Gärdenfors and H. Rott. Belief revision. In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming. Vol. 4, chapter 4.2. Oxford University Press, Oxford, UK, 1995.

    Google Scholar 

  37. P. Gärdenfors. An epistemic approach to conditionals. American Philosophical Quaterly, 18: 203–211, 1981.

    Google Scholar 

  38. P. Gärdenfors. Belief revision and the Ramsey test for conditionals. The Philosophical Review, XCV(1):81–93, January 1986.

    Google Scholar 

  39. P. Gärdenfors. Knowledge in Flux—Modeling the Dynamics of Epistemic States. MIT Press, Cambridge, MA, 1988.

    Google Scholar 

  40. P. Gärdenfors. Belief revision and nonmonotonic logic: Two sides of the same coin? In Proceedings of the 9th European Conference on Artificial Intelligence (ECAI-90), pages 768–773, Stockholm, Sweden, August 1990. Pitman.

    Google Scholar 

  41. P. Gärdenfors, editor. Belief Revision, volume 29 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, UK, 1992.

    Google Scholar 

  42. P. Gärdenfors. Belief revision: An introduction. In Gärdenfors [1992a], pages 1–28.

    Google Scholar 

  43. M. R. Garey and D. S. Johnson. Computers and Intractability—A Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA, 1979.

    Google Scholar 

  44. M. L. Ginsberg. Counterfactuals. Artificial Intelligence, 30(1):35–79, October 1986.

    MathSciNet  MATH  Google Scholar 

  45. [Gogic et al., 1994] G. Gogic, C. H. Papadimitriou and M. Sideri. Incremental recompilation of knowledge. In Proceedings of the 12th National Conference of the American Association for Artificial Intelligence (AAAI-94), pages 922–927, Seattle, WA, July 1994. MIT Press.

    Google Scholar 

  46. G. Gottlob. Complexity results for nonmonotonic logics. Journal for Logic and Computation, 2(3):397–425, 1992.

    MathSciNet  MATH  Google Scholar 

  47. G. Granne and A. O. Mendelzon. Updates and subjunctive queries. Information and Computation, 116(2):183–198, 1995.

    MathSciNet  MATH  Google Scholar 

  48. S. O. Hannson. Belief base dynamics. Doctoral dissertation, Uppsala University, Sweden, 1991.

    Google Scholar 

  49. S. O. Hansson. Reversing the Levi identity. Journal of Philosophical Logic, 22: 637–669, 1993.

    MathSciNet  MATH  Google Scholar 

  50. S. O. Hansson. Theory and base contraction unified. Journal of Symbolic Logic, 58: 602–625, 1993.

    MathSciNet  MATH  Google Scholar 

  51. S. O. Hansson. Kernel contraction. Journal of Symbolic Logic, 59: 845–859, 1994.

    MathSciNet  MATH  Google Scholar 

  52. S. O. Hansson. Taking belief bases seriously. In Prawitz, editor, Logic and Philosophy of Science in Uppsala, pages 13–28. Kluwer, Dordrecht, Holland, 1994.

    Google Scholar 

  53. S. O. Hansson. Revision of belief sets and belief bases. In this volume. 1996.

    Google Scholar 

  54. D. S. Johnson. A catalog of complexity classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, Vol. A, pages 67–161. MIT Press, 1990.

    Google Scholar 

  55. J. Kadin. p NP[o(log n)] and sparse Turing-complete sets for NP. Journal of Computer and System Sciences, 39(3):282–298, December 1989.

    MathSciNet  MATH  Google Scholar 

  56. R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Conference Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing (STOC-80), pages 302–309, Los Angeles, California, 28–30 April 1980.

    Google Scholar 

  57. H. Katsuno and A. O. Mendelzon. Propositional knowledge base revision and minimal change. Artificial Intelligence, 52: 263–294, 1991.

    MathSciNet  MATH  Google Scholar 

  58. H. Katsuno and A. O. Mendelzon. On the difference between updating a knowledge base and revising it. In Gärdenfors [1992a], pages 183–203.

    Google Scholar 

  59. A. Kratzer. Partition and revision: The semantics of counterfactuals. Journal of Philosophical Logic, 10: 201–216, 1981.

    MathSciNet  MATH  Google Scholar 

  60. S. Kraus, D. Lehmann and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44: 167–207, 1990.

    MathSciNet  MATH  Google Scholar 

  61. D. Lehman and M. Magidor. What does a conditional knowledge base entail? Artificial Intelligence, 55: 1–60, 1992.

    MathSciNet  MATH  Google Scholar 

  62. D. Lehmann. Another perspective on default reasoning. Technical report, Hebrew University, Jerusalem, Israel, 1993.

    Google Scholar 

  63. I. Levi. Subjunctives, dispositions and chances. Synthese, 34: 423–455, 1977.

    MATH  Google Scholar 

  64. D. Makinson and P. Gärdenfors. Relations between the logic of theory change and nonmonotonic logic. In A. Fuhrmann and M. Morreau, editors, The Logic of Theory Change, volume 465 of Lecture Notes in Artificial Intelligence. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    MATH  Google Scholar 

  65. A. C. Nayak. Foundational belief change. Journal of Philosophical Logic, 23: 495–533, 1994.

    MathSciNet  MATH  Google Scholar 

  66. B. Nebel. A knowledge level analysis of belief revision. In R. Brachman, H. J. Levesque, and R. Reiter, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the 1st International Conference (KR-89), pages 301–311, Toronto, ON, May 1989. Morgan Kaufmann.

    Google Scholar 

  67. B. Nebel. Reasoning and Revision in Hybrid Representation Systems, volume 422 of Lecture Notes in Artificial Intelligence. Springer-Verlag, Berlin, Heidelberg, New York, 1990.

    Google Scholar 

  68. B. Nebel. Belief revision and default reasoning: Syntax-based approaches. In J. A. Allen, R. Fikes, and E. Sandewall, editors, Principles of Knowledge Representation and Reasoning: Proceedings of the 2nd International Conference (KR-9I), pages 417–428, Cambridge, MA, April 1991. Morgan Kaufmann.

    Google Scholar 

  69. B. Nebel. Syntax-based approaches to belief revision. In Gärdenfors [1992a], pages 52–88.

    Google Scholar 

  70. B. Nebel. Base revision operations and schemes: Semantics, representation, and complexity. In Proceedings of the 11th European Conference on Artificial Intelligence (ECAI-94), pages 341–345, Amsterdam, The Netherlands, August 1994. Wiley.

    Google Scholar 

  71. B. Nebel. Artificial intelligence: A computational perspective. In G. Brewka, editor, Essentials in Knowledge Representation, Studies in Logic, Language and Information, pages 237–266. CSLI Publications, Stanford, CA, 1996.

    Google Scholar 

  72. C. H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, MA, 1994.

    MATH  Google Scholar 

  73. J. Pollock. Subjunctive Reasoning. Reidel, Dordrecht, Holland, 1976.

    Google Scholar 

  74. D. Poole. A logical framework for default reasoning. Artificial Intelligence, 36: 27–47, 1988.

    MathSciNet  MATH  Google Scholar 

  75. R. Reiter. A theory of diagnosis from first principles. Artificial Intelligence, 32(1):57–95, April 1987.

    MathSciNet  MATH  Google Scholar 

  76. N. Rescher. Hypothetical Reasoning. North-Holland, Amsterdam, Holland, 1964.

    MATH  Google Scholar 

  77. N. Rescher. The Coherence Theory of Truth. Oxford University Press, Oxford, UK, 1973.

    Google Scholar 

  78. H. Rott. A nonmonotonic conditional logic for belief revision I. In A. Fuhrmann and M. Morreau, editors, The Logic of Theory Change, volume 465 of Lecture Notes in Artificial Intelligence, pages 135–183. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    MATH  Google Scholar 

  79. H. Rott. Two methods of constructing contractions and revisions of knowledge systems. Journal of Philosophical Logic, 20: 149–173, 1991.

    MathSciNet  MATH  Google Scholar 

  80. H. Rott. On the logic of theory change: More maps between different kinds of contraction functions. In Gärdenfors [1992a], pages 122–141.

    Google Scholar 

  81. H. Rott. Belief contraction in the context of the general theory of rational choice. Journal of Symbolic Logic, 58, December 1993.

    Google Scholar 

  82. K. Satoh. Nonmonotonic reasoning by minimal belief revision. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 455–462, Tokyo, Japan, 1988. Springer-Verlag.

    Google Scholar 

  83. B. Selman and H. A. Kautz. Knowledge compilation using Horn approximations. In Proceedings of the 9th National Conference of the American Association for Artificial Intelligence (AAAI-91), pages 904–909, Anaheim, CA, July 1991. MIT Press.

    Google Scholar 

  84. B. Selman and H. A. Kautz. Domain-independent extensions to GSAT: Solving large structured satisfiability problems. In Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 290–295, Chambery, France, August 1993.

    Google Scholar 

  85. B. Selman, H. J. Levesque and D. Mitchell. A new method for solving hard satisfiability problems. In Proceedings of the 10th National Conference of the American Association for Artificial Intelligence (AAAI-92), pages 440–446, San Jose, CA, July 1992. MIT Press.

    Google Scholar 

  86. L. Stockmeyer. Classifying the computational complexity of problems. Journal of Symbolic Logic, 52(1): 1–43, 1987.

    MathSciNet  MATH  Google Scholar 

  87. F. Veltman. Prejudices, presuppositions, and the theory of counterfactuals. In J. Groenendijk and M. Stokhof, editors, Amsterdam Papers of Formal Grammar, volume I, pages 248–281. Centrale Interfaculteit, Universiteit Amsterdam, Amsterdam, The Netherlands, 1976.

    Google Scholar 

  88. K. W. Wagner and G. Wechsung. Computational Complexity. Reidel, Dordrecht, Holland, 1988.

    Google Scholar 

  89. K. W. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51: 53–80, 1987.

    MathSciNet  MATH  Google Scholar 

  90. K. W. Wagner. Bounded query classes Bounded query classes. SIAM Journal on Computing, 19(5):833–846, 1990.

    MathSciNet  MATH  Google Scholar 

  91. A. Weber. Updating propositional formulas. In L. Kerschberg, editor, Expert Database Systems—Proceedings From the 1st International Conference, pages 487–500. Benjamin/Cummings, 1986.

    Google Scholar 

  92. M.-A. Williams. On the logic of theory base change. In C. MacNish, D. Pearce, and L.M. Pereira, editors, Logics in Artificial Intelligence — European Workshop JELIA′94, pages 86–105, York, UK, 1994. Springer-Verlag.

    Google Scholar 

  93. M.-A. Williams. Transmutations of Knowledge Systems. PhD thesis, University of Sydney, Australia, 1994.

    Google Scholar 

  94. M. S. Winslett. Reasoning about action using a possible models approach. In Proceedings of the 7th National Conference of the American Association for Artificial Intelligence (AAAI-88), pages 89–93, Saint Paul, MI, August 1988.

    Google Scholar 

  95. M. S. Winslett. Updating Logical Databases. Cambridge University Press, Cambridge, UK, 1990.

    MATH  Google Scholar 

  96. S. Wrobel. Concept Formation and Knowledge Revision. Kluwer, Dordrecht, Holland, 1994.

    Google Scholar 

  97. C. K. Yap. Some consequences of non-uniform conditions on uniform classes. Theoretical Computer Science, 26: 287–300, 1993.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Nebel, B. (1998). How Hard is it to Revise a Belief Base?. In: Dubois, D., Prade, H. (eds) Belief Change. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5054-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5054-5_3

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6123-0

  • Online ISBN: 978-94-011-5054-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics