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A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

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Abstract

Results of Schlipf (J Comput Syst Sci 51:64–86, 1995) and Fitting (Theor Comput Sci 278:25–51, 2001) show that the well-founded semantics of a finite predicate logic program can be quite complex. In this paper, we show that there is a close connection between the construction of the perfect kernel of a \(\Pi^0_1\) class via the iteration of the Cantor–Bendixson derivative through the ordinals and the construction of the well-founded semantics for finite predicate logic programs via Van Gelder’s alternating fixpoint construction. This connection allows us to transfer known complexity results for the perfect kernel of \(\Pi^0_1\) classes to give new complexity results for various questions about the well-founded semantics \({\mathit{wfs}}(P)\) of a finite predicate logic program P.

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References

  1. Andreka, H., Nemeti, I.: The generalized completeness of horn predicate logic as a programming language. Acta Cybern. 4, 3–10 (1978)

    MathSciNet  MATH  Google Scholar 

  2. Bendixson, L.: Quelques theéorèms de la th’eorie des ensumble de points. Acta Math. 2, 415–429 (1883)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cenzer, D., Mauldin, R.: On the Borel class of the derived set operator. Bull. Soc. Math. Fr. 110, 357–380 (1982)

    MathSciNet  MATH  Google Scholar 

  4. Cenzer, D., Mauldin, R.: On the Borel class of the derived set operator II. Bull. Soc. Math. Fr. 111, 367–372 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Cenzer, D., Clote, P., Smith, R., Soare, R., Wainer, S.: Members of countable \(\Pi^0_1\) classes. Ann. Pure Appl. Logic 31, 145–163 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cenzer, D., Remmel, J.B.: Index sets for \(\Pi^0_1\) classes. Ann. Pure Appl. Logic 93, 3–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cenzer, D., Remmel, J.B.: \(\Pi_1^0\) classes in mathematics. In: Handbook of Recursive Mathematics, pp. 623–821. Elsevier (1999)

  8. Cenzer, D., Remmel, J.B.: A connection between Cantor–Bendixson derivatives and the well-founded semantics of logic programs. In: ISAIM 2008, The Tenth International Symposium on Artificial Intelligence and Mathematics, Fort Lauderdale (2008)

  9. Cenzer, D., Remmel, J.B.: Effectively closed sets. In: ASL Lecture Notes in Logic (to appear)

  10. Denecker, M.: The well-founded semantics is the principle of inductive definition. In: Proceedings of the 6th European Workshop on Logics in Artificial Intelligence. LNAI, vol. 1489, pp. 1–16. Springer (1998)

  11. Denecker, M., Bruynooghe, M., Marek, V.W.: Logic programming revisited: logic programs as inductive definitions. ACM Trans. Comput. Logic 2, 623–654 (2001)

    Article  MathSciNet  Google Scholar 

  12. Fitting, M.C.: Fixpoint semantics for logic programming–a survey. Theor. Comput. Sci. 278, 25–51 (2001)

    Article  MathSciNet  Google Scholar 

  13. Gelfond, M., Lifschitz, V.: The stable semantics for logic programs. In: Kowalski, R., Bowen, K. (eds.) ICLP88, pp. 1070–1080 (1988)

  14. Hinman, P.G.: Recursion-Theoretic Hierarchies. Springer-Verlag (1978)

  15. Kuratowski, K.: Some problems concerning semi-continuous set-valued mappings. In: Set-Valued Mappings, Selections and Topological Properties of 2X. Lecture Notes in Math, vol. 171, pp. 45–48. Springer-Verlag (1970)

  16. Lempp, S.: Hyperarithmetical index sets in recursion theory. Trans. Amer. Math. Soc. 303, 559–583 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Marek, W., Nerode, A., Remmel, J.B.: The stable models of predicate logic programs. J. Logic Prog. 21(3), 129–154 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Schlipf, J.S.: The expressive power of the logic programming semantics. J. Comput. Syst. Sci. 51, 64–86 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Soare, R.: Recursively Enumerable Sets and Degrees. Springer-Verlag (1987)

  20. Smullyan, R.M.: First-order Logic. Springer-Verlag (1968)

  21. Tarski, A.: A lattice-theoretical theorem and its applications. Pac. J. Math. 5, 285–309 (1955)

    MathSciNet  MATH  Google Scholar 

  22. Van Gelder, A.: The alternating fixpoints of logic programs with negation. In: ACM Symposium on Principles of Database Systems, pp. 1–10 (1989)

  23. Van Gelder, A.: The alternating fixpoints of logic programs with negation. J. Comput. Syst. Sci. 47, 185–221 (1993)

    Article  MATH  Google Scholar 

  24. Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. J. ACM 38(3), 620–650 (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Florida, Gainesville, FL, 32611-8105, USA

    Douglas Cenzer

  2. University of California, San Diego, La Jolla, CA, 92093, USA

    Jeffrey B. Remmel

Authors
  1. Douglas Cenzer
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  2. Jeffrey B. Remmel
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Correspondence to Douglas Cenzer.

Additional information

Cenzer was partially supported by the NSF grant DMS-652372 and Remmel by the NSF grant DMS-0654060.

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Cenzer, D., Remmel, J.B. A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs. Ann Math Artif Intell 65, 1–24 (2012). https://doi.org/10.1007/s10472-012-9294-x

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