Abstract
The partial stable models of a logic program form a class of models that include the (unique) well-founded model, total stable models and other two interesting subclasses: maximal stable models and least undefined stable models. As stable models different from the well-founded are not unique, DATALOG queries do not in general correspond to functions. The question is: what are the expressive powers of the various types of stable models when they are restricted to the class of all functional queries? The paper shows that this power does not go in practice beyond the one of stratified queries, except for least undefined stable models which, instead, capture the whole Boolean hierarchy BH. Finally, it is illustrated how the latter result can be used to design a ``functional'' language which, by means of a disciplined usage of negation, allows to achieve the desired level of expressiveness up to BH so that exponential time resolution is eventually enabled only for hard problems.
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References
S. Abiteboul and V. Vianu, DATALOG extensions for databases queries and updates, Journal of Computer and System Science 43 (1991) 62–124.
S. Abiteboul, R. Hull and V. Vianu, Foundations of Databases (Addison-Wesley, 1994).
S. Abiteboul, E. Simon and V. Vianu, Non-deterministic languages to express deterministic transformations, in: Proc. ACM PODS Symp. (1990) pp. 218–229.
C. Baral and V. Subrahmanian, Stable and extension class theory for logic programs and default logic, Journal of Automated Reasoning (1992) 345–366.
A. Chandra and D. Harel, Structure and complexity of relational queries, Journal of Computer and System Sciences 25(1) (1982) 99–128.
P. Dung, Negation as hypotheses: an abductive foundation for logic programming, in: Proc. 8th Conf. on Logic Programming (1991) pp. 3–17.
T. Eiter, G. Gottlob and H. Mannila, Expressive power and complexity of disjunctive DATALOG, in: Proc. ACM PODS Symp., Minneapolis, USA (May 1994).
T. Eiter, N. Leone and D. Saccà, The expressive power of partial models for disjunctive deductive databases, in: Proc. Workshop on Logic in Databases (LID '96), San Miniato (July, 1996).
R. Fagin, Generalized first-order spectra and polynomial-time recognizable sets, in: Complexity of Computation, ed. R. Karp, SIAM-AMS Proc., Vol. 7 (1974) pp. 43–73.
M. Gelfond and V. Lifschitz, The stable model semantics for logic programming, in: Proc. 5th Int. Conf. and Symp. on Logic Programming, MIT Press, Cambridge (1988) pp. 1070–1080.
F. Giannotti, S. Greco, D. Saccà and C. Zaniolo, Programming with non-determinism in deductive databases, Ann. of Math. and AI (this issue).
F. Giannotti, D. Pedreschi, D. Saccà and C. Zaniolo, Non-determinism in deductive databases, in: Proc. 2nd Conf. on Deductive and Object-Oriented Databases (1991) pp. 129–146.
S. Greco, D. Saccà and C. Zaniolo, DATALOG queries with stratified negation and choice: from p to Dp, in: Proc. of the Fifth Int. Conf. on Database Theory (1995) pp. 82–96.
S. Greco and D. Saccà, A query language that captures the query hierarchy, unpublished manuscript (1995).
S. Grumbach, Z. Lacroix and S. Lindell, Implicit definitions on finite structures, in: Proc. of the Conf. on Computer Science Logic (1995).
Y. Gurevich, Logic and the challenge of computer science, in: Trends in Theoretical Computer Science, ed. E. Borger (Computer Science Press, 1988).
N. Immerman, Languages which capture complexity classes, SIAM Journal on Computing 16(4) (1987) 760–778.
D.S. Johnson, A catalog of complexity classes, in: Handbook of Theoretical Computer Science, Vol. 1 (North-Holland, 1990).
P.C. Kanellakis, Elements of relational database theory, in: Handbook of Theoretical Computer Science, Vol. 2 (North-Holland, 1991).
P.G. Kolaitis and C.H. Papadimitriou, Why not negation by fixpoint?, Journal of Computer and System Sciences 43 (1991) 125–144.
R. Krishnamurthy and S.A. Naqvi, Non-deterministic choice in DATALOG, in: Proc. 3rd Int. Conf. on Data and Knowledge Bases (Morgan-Kaufmann, Los Altos, 1988) pp. 416–424.
J.W. Lloyd, Foundations of Logic Programming (Springer-Verlag, Berlin, 1987).
W. Marek and M. Truszcynski, Autoepistemic logic, J. of the ACM 38(3) (1991) 588–619.
C.H. Papadimitriou, Computational Complexity (Addison-Wesley, 1994).
T.C. Przymusinski, Well-founded semantics coincides with three-valued stable semantics, Foundamenta Informaticae 13 (1990) 445–463.
D. Saccà, The expressive powers of stable models for bound and unbound DATALOG queries, Journal of Computer and System Science (to appear).
D. Saccà, Multiple total stable models are definitely needed to solve unique solution programs, Information Processing Letters 58(5) (1996) 249–254.
D. Saccà and C. Zaniolo, Stable models and non-determinism in logic programs with negation, in: Proc. ACM PODS Symp. (1990) pp. 205–218.
D. Saccà and C. Zaniolo, Deterministic and non-deterministic stable models, submitted to Journal of Logic and Computation.
J.S. Schlipf, The expressive powers of the logic programming semantics, Journal of Computer and System Science 51 (1995) 64–86.
J.D. Ullman, Principles of Database and Knowledge Base Systems, Vols. 1–2 (Computer Science Press, 1989).
M.Y. Vardi, The complexity of relational query languages, in: Proc. ACM Symp. on Theory of Computing (1982) pp. 137–146.
A. Van Gelder, K. Ross and J.S. Schlipf, The well-founded semantics for general logic programs, Journal of the ACM 38(3) (1991) 620–650.
A. Van Gelder, The alternating fixpoint of logic programming with negation, Journal of Computer and System Sciences 43 (1992) 125–144.
J. You and L.Y. Yuan, On the equivalence of semantics for normal logic programs, Journal of Logic Programming (1995) 211–222.
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Greco, S., Saccà, D. ``Possible is certain'' is desirable and can be expressive. Annals of Mathematics and Artificial Intelligence 19, 147–168 (1997). https://doi.org/10.1023/A:1018903705269
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DOI: https://doi.org/10.1023/A:1018903705269