Skip to main content
SpringerLink
Log in

The expressiveness of locally stratified programs

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence, to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of logically stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene's hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets. In particular, a set is definable in the well-founded semantics by a programP whose well-founded partial model is total iff it is hyperarithmetic.

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

    Google Scholar 

  2. K.R. Apt, Logic programming,Handbook of Theoretical Computer Science, ed. J. van Leeuven (Elsevier, 1990) pp. 495–574.

  3. K.R. Apt and M. Bezem, Acyclic programs, Technical Report CS-R9010, Centre for Mathematics and Computer Science, Amsterdam (1990).

    Google Scholar 

  4. K.R. Apt, H.A. Blair and A. Walker, Towards a theory of declarative knowledge, in:Foundations of Deductive Databases and Logic Programming, ed. J. Minker (Morgan Kaufmann, Los Altos, CA, 1988) pp. 89–148.

    Google Scholar 

  5. K.R. Apt and H.A. Blair, Arithmetic classification of perfect models of stratified programs, Fundamenta Inf. 13(1990)1–17.

    Google Scholar 

  6. K.R. Apt and H.A. Blair, Arithmetic classification of perfect models of stratified programs — addendum, Fundamenta Inf. 14(1991)339–344.

    Google Scholar 

  7. C. Baral and V.S. Subrahmanian, Dualities between alternative semantics for logic programming and non-monotonic reasoning, in:Logic Programming and Non-Monotonic Reasoning: Proc. 1st Int. Workshop, eds. A. Nerode, W. Marek and V.S. Subrahmanian (MIT Press, Cambridge, MA, 1991) pp. 69–87.

    Google Scholar 

  8. N. Bidoit and Ch. Froixdevaux, Negation by default and non stratifiable logic programs, Internal Report 437, Université Paris-Sud (1988).

  9. H.A. Blair, The recursion-theoretic complexity of the semantics of predicate logic as a programming language, Inf. Contr. (July–August, 1982)25–47.

  10. C. Cholak and H. Blair, The complexity of local stratification, Manuscript (1991), submitted.

  11. M.H. van Emden and R.A. Kowalski, The semantics of predicate logic as a programming language, J. ACM 23(1976)733–742.

    Google Scholar 

  12. M. Gelfond, Autoepistemic logic and formalization of commonsense reasoning,Non-Monotonic Reasoning, eds. M. Reinfrank, J. de Kleer, M.L. Ginsberg and E. Sandewall, Lecture Notes in Artificial Intelligence 346 (Springer, 1989) pp. 176–186.

  13. M. Gelfond and V. Lifschitz, The stable model semantics for logic programming,Proc. ICLP/SLP-5 (1988) pp. 1070–1080.

  14. P.G. Hinman, Recursion-theoretic hierarchies,Persepctives in Mathematical Logic (Springer, 1978).

  15. P.G. Kolaitis, The expressive power of stratified logic programs, Manuscript (1987).

  16. Private communication.

  17. R.A. Kowalski, Predicate logic as a programming language,Proc. IFIP Congress 1974 (North-Holland, Amsterdam, 1974) pp. 569–574.

    Google Scholar 

  18. V. Lifschitz, On the declarative semantics of logic programs with negation,Foundations of Deductive Databases and Logic Programming, ed. J. Minker (Morgan Kaufmann, Los Altos, CA, 1988) pp. 172–192.

    Google Scholar 

  19. J.W. Lloyd,Foundations of Logic Programming, 2nd Ed. (Springer, 1987).

  20. W. Marek, A. Nerode and J. Remmel, A theory of nonmonotonic rule systems, MSI Technical Report 90-31, Mathematical Sciences Institute, Cornell University (1990).

  21. W. Marek, A. Nerode and J. Remmel, How complicated is the set of stable models of a general logic program?, Technical Report, Mathematical Sciences Institute, Cornell University (1991).

  22. W. Marek and V.S. Subrahmanian, The relationship between logic program semantics and nonmonotonic reasoning,Proc. 6th Int. Conf. on Logic Programming, eds. G. Levi and M. Martelli (MIT Press, 1989) pp. 600–617.

  23. W. Marek and M. Truszczyński, Stable semantics for logic programs and default theories,Proc. North American Conf. on Logic Programming (MIT Press, Cambridge, MA, 1989) pp. 243–256.

    Google Scholar 

  24. A. Nerode and R. Shore,Logic for Applications (Springer, 1993).

  25. T. Przymusinski, On the declarative semantics of deductive databases and logic programs, in:Foundations of Deductive Databases and Logic Programming, ed. J. Minker (Morgan Kaufmann, Los Altos, CA, 1988).

    Google Scholar 

  26. T. Przymusinski, Every logic program has a natural stratification and an iterated fixed point model,8th ACM Symp. on Principles of Database Systems (1989) pp. 11–21

  27. H. Rogers,Theory of Recursive Functions and Effective Computability (McGraw-Hill, 1967).

  28. J. Schlipf, The expressive powers of the logic programming semantics, to appear in J. CSS. A preliminary version appeared in the9th ACM Symp. on Principles of Database Systems (1990).

  29. J.C. Shepherdson, Unsolvable problems for SLDNF-resolution, J. Logic Progr. 10(1991)19–22.

    Google Scholar 

  30. J.C. Shepherdson and H.E. Sturgis, Computability of recursive functions, J. ACM 10(1963)217–255.

    Google Scholar 

  31. R.M. Smullyan,Theory of Formal Systems (Princeton University Press, Princeton, NJ, 1961).

    Google Scholar 

  32. A. Van Gelder, Negation as failure using tight derivations for general logic programs, in:Foundations of Deductive Databases and Logic Programming, ed. J. Minker (Morgan Kaufmann, Los Altos, CA, 1988) pp. 149–176.

    Google Scholar 

  33. A. Van Gelder, The alternating fixpoint of logic programs with negation,8th ACM Symp. on Principles of Database Systems (1989) pp. 1–10.

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer and Information Science, Syracuse University, 13244-4100, Syracuse, NY, USA

    Howard A. Blair

  2. Department of Computer Science, University of Kentucky, 40506, Lexington, KY, USA

    V. Wiktor Marek

  3. Department of Computer Science, University of Cincinnati, 45221-0008, Cincinnati, OH, USA

    John S. Schlipf

Authors
  1. Howard A. Blair
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Wiktor Marek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John S. Schlipf
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.

Research partially supported by NSF Grant IRI-9012902 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.

Research partially supported by NSF Grant IRI-8905166 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blair, H.A., Marek, V.W. & Schlipf, J.S. The expressiveness of locally stratified programs. Ann Math Artif Intell 15, 209–229 (1995). https://doi.org/10.1007/BF01534455

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01534455

Keywords

Search

Navigation