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Optimal Models of Disjunctive Logic Programs: Semantics, Complexity, and Computation
April 2004 (vol. 16 no. 4)
pp. 487-503

Abstract—Almost all semantics for logic programs with negation identify a set, SEM(P), of models of program P, as the intended semantics of P, and any model M in this class is considered a possible meaning of P with regard to the semantics the user has in mind. Thus, for example, in the case of stable models [CHECK END OF SENTENCE], choice models [CHECK END OF SENTENCE], answer sets [CHECK END OF SENTENCE], etc., different possible models correspond to different ways of "completing” the incomplete information in the logic program. However, different end-users may have different ideas on which of these different models in SEM(P) is a reasonable one from their point of view. For instance, given SEM(P), user U_1 may prefer model M_1\in SEM(P) to model M_2\in SEM(P) based on some evaluation criterion that she has. In this paper, we develop a logic program semantics based on Optimal Models. This semantics does not add yet another semantics to the logic programming arena—it takes as input an existing semantics SEM(P)and a user-specified objective function Obj, and yields a new semantics \underline{{\rm{Opt}}}(P)\subseteq SEM(P) that realizes the objective function within the framework of preferred models identified already by SEM(P). Thus, the user who may or may not know anything about logic programming has considerable flexibility in making the system reflect her own objectives by building "on top” of existing semantics known to the system. In addition to the declarative semantics, we provide a complete complexity analysis and algorithms to compute optimal models under varied conditions when SEM(P) is the stable model semantics, the minimal models semantics, and the all-models semantics.

[1] P. Barth and A. Bockmayr, Modelling Discrete Optimization Problems in Constraint Logic Programming Annals of Operations Research, vol. 81, pp. 467-496, 1998.
[2] M. Cadoli, On the Complexity of Model Finding for Nonmonotonic Propositional Logics Proc. Fourth Italian Conf. Theoretical Computer Science, M. Venturini Zilli, A. Marchetti Spaccamela, and P. Mentrasti, eds. Singapore: World Scientific, pp. 125-139, 1992.
[3] J. Cohen, Constraint Logic Programming Comm. ACM, vol. 33, no. 7, 1990.
[4] P. Dasgupta, P.P. Chakrabarti, A. Dey, S. Ghose, and W. Bibel, Solving Constraint Optimization Problems from CLP-Style Specifications Using Heuristic Search Techniques IEEE Trans. Knowledge and Data Eng., vol. 14, no. 2, pp. 353-368, 2002.
[5] M. Dincbas, H. Simonis, and P. Van Hentenryck, Solving Large Combinatorial Optimization Problems in Logic Programming J. Logic Programming, vol. 8, nos. 1-2, pp. 75-93, 1990.
[6] J. Dix and F. Stolzenburg, Computation of Non-Ground Disjunctive Well-Founded Semantics with Constraint Logic Programming Proc. Workshop Nonmonotonic Extensions of Logic Programming, pp. 202-224, 1996.
[7] T. Eiter and G. Gottlob, On the Computational Cost of Disjunctive Logic Programming: Propositional Case Annals of Math. and Artificial Intelligence, vol. 15, nos. 3-4, pp. 289-323, 1995.
[8] T. Eiter, G. Gottlob, and H. Mannila, Disjunctive Datalog ACM Trans. Database Systems, vol. 22, no. 3, pp. 364-418, 1997.
[9] M.C. Fitting, Logic Programming on a Topological Bilattice Fundamenta Informatica, vol. 11, pp. 209-218, 1988.
[10] M. Gelfond and V. Lifschitz, The Stable Model Semantics for Logic Programming Logic Programming: Proc. Fifth Int'l Conf. Symp., pp. 1070-1080, 1988.
[11] M. Gelfond and V. Lifschitz, Classical Negation in Logic Programs and Disjunctive Databases New Generation Computing, vol. 9, pp. 365-385, 1991.
[12] J. Jaffar and J.-L. Lassez, Constraint Logic Programming Proc. 14th ACM Symp. Principles of Programming Languages, 1987.
[13] D.S. Johnson, A Catalog of Complexity Classes Handbook of Theoretical Computer Science, J. van Leeuwen, ed., vol. A, chapter 2, B.V. (North-Holland): Elsevier Science, pp. 67-161, 1990.
[14] J. Kadin, ${\rm{P}}^{({\rm NP}[O(\log n)])}$and Sparse Turing-Complete Sets for NP J. Computer and System Sciences, vol. 39, no. 3, pp. 282-298, 1989.
[15] M. Kifer and V.S. Subrahmanian, Theory of Generalized Annotated Logic Programming and Its Applications J. Logic Programming, vol. 12, no. 4, pp. 335-368, 1992.
[16] C. Koch and N. Leone, Stable Model Checking Made Easy Proc. 16th Int'l Joint Conf. Artificial Intelligence, Thomas Dean, ed., Morgan Kaufmann, pp. 70-75, Aug. 1999.
[17] V.S. Lakshmanan and F. Sadri, Modeling Uncertainty in Deductive Databases Proc. Int'l Conf. Database Expert Systems and Applications, pp. 724-733, Sept. 1994.
[18] V.S. Lakshmanan and F. Sadri, Probabilistic Deductive Databases Proc. Int'l Logic Programming Symp., Nov. 1994.
[19] L.V.S. Lakshmanan and N. Shiri, A Parametric Approach to Deductive Databases with Uncertainty IEEE Trans. Knowledge and Data Eng., vol. 13, no. 4, pp. 554-570, July/Aug. 2001.
[20] N. Leone, F. Scarcello, and V.S. Subrahmanian, Optimal Models of Disjunctive Logic Programs: Semantics, Complexity, and Computation Univ. of Maryland Technical Report CS-TR-4298,www.cs.umd.edu/Library/TRsCS-TR-4298.ps.Z , 2001.
[21] N. Leone, P. Rullo, and F. Scarcello, Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics and Computation Information and Computation, vol. 135, no. 2, pp. 69-112, June 1997.
[22] J.W. Lloyd, Foundations of Logic Programming. Springer-Verlag, 1987.
[23] V.W. Marek and M. Truszczynski, Logic Programming with Cost unpublished manuscript, 1999.
[24] K. Marriot and P.J. Stuckey, Programming with Constraints: An Introduction. MIT Press, 1998.
[25] J. Minker, On Indefinite Data Bases and the Closed World Assumption Proc. Sixth Conf. Automated Deduction, D.W. Loveland, ed., pp. 292-308, 1982.
[26] R. Ng and V.S. Subrahmanian, Probabilistic Logic Programming, Information and Computation Information and Computation, vol. 101, no. 2, pp. 150-201, Dec. 1992.
[27] I. Niemela, P. Simons, and T. Soininen, Extending and Implementing the Stable Model Semantics Artificial Intelligence, vol. 138, nos. 1-2, pp. 181-234, 2002.
[28] C.H. Papadimitriou, Computational Complexity. Addison-Wesley, 1994.
[29] F. Rossi, Constraint (Logic) Programming: A Survey on Research and Applications New Trends in Constraints, 1999.
[30] D. Saccà and C. Zaniolo, Stable Models and Non-Determinism in Logic Programs with Negation Proc. Ninth Symp. Principles of Database Systems, pp. 205-217, 1990.
[31] T. Sandholm, Algorithm for Optimal Winner Determination in Combinatorial Auctions Artificial Intelligence, vol. 135, nos. 1-2, 2002.
[32] E. Shapiro, Logic Programs with Uncertainties: A Tool for Implementing Expert Systems Proc. Int'l Joint Conf. Artificial Intelligence, pp. 529-532, 1983.
[33] N. Shiri, On a Generalized Theory of Deductive Databases PhD dissertation, Concordia Univ., Montreal, Canada, Aug. 1997.
[34] M.Y. Vardi, The Complexity of Relational Query Languages (extended abstract) Proc. 14th ACM Symp. Theory of Computing, pp. 137-146, 1982.
[35] P. Van Hentenryck, A Logic Language for Combinatorial Optimization Annals of Operations Research, vol. 21, pp. 247-274, 1990.
[36] M.H. van Emden, Quantitative Deduction and Its Fixpoint Theory J. Logic Programming, vol. 4, no. 1, pp. 37-53, 1986.

Index Terms:
Disjunctive logic programming, computational complexity, nonmonotonic reasoning, knowledge representation, optimization problems.
Citation:
Nicola Leone, Francesco Scarcello, V.S. Subrahmanian, "Optimal Models of Disjunctive Logic Programs: Semantics, Complexity, and Computation," IEEE Transactions on Knowledge and Data Engineering, vol. 16, no. 4, pp. 487-503, April 2004, doi:10.1109/TKDE.2004.1269672
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