On the Expressibility of Stable Logic Programming

  • V. W. Marek
  • J. B. Remmel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2173)


Schlipf [Sch95] proved that Stable Logic Programming (SLP) solves all NP decision problems. We extend Schlipf’s result to prove that SLP solves all search problems in the class NP. Moreover, we do this in a uniform way as defined in [MT99]. Specifically, we show that there is a single DATALOG⌝ program P Trg such that given any Turing machine M, any polynomial p with non-negative integer coefficients and any input σ of size n over a fixed alphabet ∑, there is an extensional database edb M,p,σ such that there is a one-to-one correspondence between the stable models of edb M,p,σ ∪ PTrg and the accepting computations of the machine M that reach the final state in at most p(n) steps. Moreover, edb M,p,σ can be computed in polynomial time from p, σ and the description of M and the decoding of such accepting computations from its corresponding stable model of edb M,p,σP Trg can be computed in linear time. A similar statement holds for Default Logic with respect to∑stackP stack2-search problems.

We also show that there is single program Meta which is a metainterpreter for SLP programs. That is, for any program Q, there there is an encoding of Q as an extensional data base edb Q such that the stable models of Metaedb Q are in one-to-one correspondence with the stable models of Q.


Logic Program Turing Machine Logic Programming Stable Model Function Symbol 
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  1. Ap90.
    K. R. Apt. Logic Programming. in Handbook of Theoretical Computer Science, pp. 475–574. Elsevier, 1990. 107Google Scholar
  2. AB90.
    K. R. Apt and H. A. Blair. Arithmetical classification of perfect models of stratified programs. Fundamenta Informaticae, 12:1–17, 1990. 109MathSciNetGoogle Scholar
  3. ABW88.
    K. Apt, H. A. Blair, and A. Walker. Towards a theory of declarative knowledge. In J. Minker, editor, Foundations of deductive databases and logic programming, pages 89–142, Los Altos, CA, 1988. Morgan Kaufmann. 109Google Scholar
  4. AP93.
    K. R. Apt and D. Pedreschi. Reasoning about termination of pure Prolog programs. Information and Computation 106:109–157, 1994. 108CrossRefMathSciNetGoogle Scholar
  5. AP94.
    K. R. Apt and A. Pellegrini. On the occur-check free pure Prolog programs. ACM Toplas 16:687–726, 1994. 108CrossRefGoogle Scholar
  6. AHV95.
    S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases. Addison-Wesley Publishing Company, 1995. 110Google Scholar
  7. BMS95.
    H. A. Blair, W. Marek, and J. Schlipf. The expressiveness of locally stratified programs. Annals of Mathematics and Artificial Intelligence, 15:209–229, 1995. 109zbMATHCrossRefMathSciNetGoogle Scholar
  8. BK82.
    K. A. Bowen and R. A. Kowalski. Amalgamating language and metalanguage in Logic Programming. In: Logic Programming, pp. 153–172, Academic Press, 1982.Google Scholar
  9. CP98.
    M. Cadoli and L. Palipoli. Circumscribing datalog: expressive power and complexity. Theoretical Computer Science, 193:215–244, 1998. 107zbMATHCrossRefMathSciNetGoogle Scholar
  10. CEG97.
    M. Cadoli, T. Eiter and G. Gottlob. Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering, 9:448–463, 1997. 116CrossRefGoogle Scholar
  11. CR99.
    D. Cenzer and J. B. Remmel. ?0 1 Classes in Mathematics. Handbook of Recursive Mathematics pp. 623–821, Elsevier 1999. 109Google Scholar
  12. CMT96.
    P. Cholewiński, W. Marek, and M. Truszczyński. Default reasoning system DeReS. In Proceedings of KR-96, pages 518–528. Morgan Kaufmann, 1996. 107, 108Google Scholar
  13. Co71.
    S. Cook. The complexity of theorem-proving procedures. Proceedings of Third Annual ACM Symposium on Theory of Computing pp. 151–158. 1971. 111Google Scholar
  14. DEGV99.
    E. Dantsin, T. Eiter, G. Gottloband A. Voronkov. Complexity and Expressive Power of Logic Programming, Technical Report of Institut für Informationssysteme, Technische Universität Wienff, INFSYS RR-1843-99-05, 1999, To appear in: ACM Computing Surveys. 111Google Scholar
  15. DK89.
    P. M. Dung and K. Kanchanasut. On the generalized predicate completion of non-Horn programs, In: Logic programming, Proceedings of the North American Conference, pp. 587–603, MIT Press, 1989.Google Scholar
  16. EFLP01.
    T. Eiter, W. Faber, N. Leone and G. Pfeifer, Computing Preferred and Weakly Preferred Answer Sets by Meta-Interpretation in Answer Set Programming, In: Proceedings AAAI 2001 Spring Symposium on Answer Set Programming: Towards Efficient and Scalable Knowledge Representation and Reasoning, Stanford, CA (Workshop Technical Report SS-01-01), AAAI Press, 2001. 118Google Scholar
  17. ELM+97.
    T. Eiter, N. Leone, C. Mateis, G. Pfeifer, and F. Scarcello. A deductive system for non-monotonic reasoning. In Proceedings of the 4th International Conference on Logic Programming and Nonmonotonic Reasoning, pages 363–374, 1997. Springer LN in Computer Science 1265. 107, 108Google Scholar
  18. GJ79.
    M. R. Garey and D. S. Johnson. Computers and intractability; a guide to the theory of NP-completeness. W. H. Freeman, 1979. 110Google Scholar
  19. GL88.
    M. Gelfond and V. Lifschitz. The stable semantics for logic programs. In Proceedings of the 5th International Symposium on Logic Programming, pages 1070–1080, Cambridge, MA, 1988. MIT Press. 107, 109Google Scholar
  20. GL91.
    M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing 9:365–385, 1991. 108CrossRefGoogle Scholar
  21. JM94.
    J. Jaffar and M. J. Maher. Constraint logic programming: A survey. Journal of Logic Programming, 19(20):503–581, 1994. 107CrossRefMathSciNetGoogle Scholar
  22. Lif98.
    V. Lifschitz. Action languages, answer sets and planning. The Logic Programming Paradigm, pp. 357–373. Series Artificial Intelligence, Springer-Verlag, 1999. 107Google Scholar
  23. MNR94.
    W. Marek, A. Nerode, and J. B. Remmel. The stable models of predicate logic programs. Journal of Logic Programming, 21(3):129–154, 1994. 109zbMATHMathSciNetGoogle Scholar
  24. MT91.
    W. Marek and M. Truszczyński. Autoepistemic logic. Journal of the ACM, 38:588–619, 1991. 110zbMATHCrossRefGoogle Scholar
  25. MT93.
    V. W. Marek and M. Truszczyński. Nonmonotonic Logic-Context-Dependent Reasoning. Series Artificial Intelligence, Springer-Verlag, 1993. 110, 116Google Scholar
  26. MT99.
    V. W. Marek and M. Truszczyński. Stable Models and an Alternative Logic Programming Paradigm. The Logic Programming Paradigm, pp. 375–398. Series Artificial Intelligence, Springer-Verlag, 1999. 107, 110Google Scholar
  27. MS99.
    K. Marriott and P. J. Stuckey. Programming with Constraints: An Introduction. MIT Press, Cambridge, MA, 1998. 107zbMATHGoogle Scholar
  28. Nie98.
    I. Niemelä. Logic programs with stable model semantics as a constraint programming paradigm. In Proceedings of the Workshop on Computational Aspects of Nonmonotonic Reasoning, pages 72–79, 1998. 107Google Scholar
  29. NS96.
    I. Niemelä and P. Simons. Efficient implementation of the well-founded and stable model semantics. In Proceedings of JICSLP-96. MIT Press, 1996. 107, 108, 110Google Scholar
  30. Prz88.
    T. Przymusiński. On the declarative semantics of deductive databases and logic programs. In Foundations of deductive databases and logic programming, pages 193–216, Los Altos, CA, 1988. Morgan Kaufmann. 109Google Scholar
  31. RRS+97.
    P. Rao, I. V. Ramskrishnan, K. Sagonas, T. Swift, D. S. Warren, and J. Freire. XSB: A system for efficiently computing well-founded semantics. In Proceedings of LPNMR’97, pages 430–440, Lecture Notes in Computer Science, 1265, Springer-Verlag, 1997. 109Google Scholar
  32. Sch95.
    J. Schlipf. The expressive powers of the logic programming semantics. Journal of the Computer Systems and Science, 51:64–86, 1995. 107, 111zbMATHCrossRefMathSciNetGoogle Scholar
  33. Ull88.
    J. D. Ullman. Principles of Database and Knowledge-Base Systems. Computer Science Press, Rockville, MD, 1988. 110Google Scholar
  34. VRS91.
    A. Van Gelder, K. A. Ross, and J. S. Schlipf. Unfounded sets and wellfounded semantics for general logic programs. Journal of the ACM, 38:620–650, 1991. 110zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • V. W. Marek
    • 1
  • J. B. Remmel
    • 2
  1. 1.Department of Computer ScienceUniversity of KentuckyUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan Diego

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