Beginnings of a theory of general database completions
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Ordinary logical implication is not enough for answering queries in a logic database, since especially negative information is only implicitly represented in the database state. Many database completions have been proposed to remedy this problem, but a clear understanding of their properties and differences is still needed.
In this paper, a general framework is proposed for studying database completions, which reveals an interesting and fruitful relation to social choice theory. We apply our framework to parameterized forms of the minimal model approach and the closed world assumption, as well as to two versions of the default logic: simple defaults with constraints and normal defaults. Thereby the relationship between these important classes of database completions is clarified. In particular, it is shown that the GCWA cannot be simulated by any of the other three approaches. We also give a characterization of those completions which can be represented as a form of minimal implication. The whole discussion is based on various properties of database completions which have proven useful as a means for classifying the proposed approaches.
Finally, we illustrate the applicability of general database completions to conventional deductive databases by means of transformation.
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- [BH86]N. Bidoit, R. Hull: Positivism vs. minimalism in deductive databases. In Proc. of the Fifth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS'86), 123–132, 1986.Google Scholar
- [BL89]S. Brass, U. W. Lipeck: Specifying closed world assumptions for logic databases. In Second Symposium on Mathematical Fundamentals of Database Systems (MFDBS'89), 68–84, Lecture Notes in Computer Science 364, Springer-Verlag, Berlin, 1989.Google Scholar
- [Bra88]S. Brass: Vervollständigungen für Logikdatenbanken (completions of logic databases). Diploma thesis, Informatics, Techn. Univ. Braunschweig, 1988. In German. Revised Version available as technical report 315/1989, Informatics, Univ. Dortmund.Google Scholar
- [Bra90]S. Brass: Remarks on first order circumscription. Submitted for publication, 1990.Google Scholar
- [Dav80]M. Davis: The mathematics of non-monotonic reasoning. Artificial Intelligence 13 (1980), 73–80.Google Scholar
- [Gab85]D. M. Gabbay: Theoretical foundations for non-monotonic reasoning in expert systems. In K. R. Apt (ed.), Logics and Models of Concurrent Systems, 439–457, Springer, Berlin, 1985.Google Scholar
- [GL89]M. Gelfond, V. Lifschitz: Compiling circumscriptive theories into logic programs. In Non-Monotonic Reasoning (2nd International Workshop), 74–99, Lecture Notes in Artificial Intelligence 346, Springer-Verlag, Berlin, 1989.Google Scholar
- [GPP86]M. Gelfond, H. Przymusinska, T. Przymusinski: The extended closed world assumption and its relationship to parallel circumscription. In Proc. of the Fifth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS'86), 153–185, 1986.Google Scholar
- [KLM90]S. Kraus, D. Lehmann, M. Magidor: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44 (1990), 167–207.Google Scholar
- [Lif88]V. Lifschitz: On the declarative semantics of logic programs with negation. In J. Minker (ed.), Foundations of Deductive Databases and Logic Programming, 177–192, Morgan Kaufmann Publishers, Los-Altos (Calif.), 1988.Google Scholar
- [Llo87]J. W. Lloyd: Foundations of Logic Programming, second edition. Springer-Verlag, Berlin, 1987.Google Scholar
- [Luk85]W. Lukaszewics: Two results on default logic. In Proc. 9th International Joint Conference on Artificial Intelligence (IJCAI), 459–461, Los Angeles, 1985.Google Scholar
- [Mak89]D. Makinson: General theory of cumulative inference. In Non-Monotonic Reasoning (2nd International Workshop), 1–18, Lecture Notes in Artificial Intelligence 346, Springer-Verlag, Berlin, 1989.Google Scholar
- [Min82]J. Minker: On indefinite databases and the closed world assumption. In D. W. Loveland (ed.), 6th Conference on Automated Deduction, 292–308, Lecture Notes in Computer Science 138, Springer-Verlag, Berlin, 1982.Google Scholar
- [Mou85]H. Moulin: Choice functions over a finite set: A summary. Social Choice and Welfare 2 (1985), 147–160.Google Scholar
- [Prz88]T. C. Przymusinski: On the declarative semantics of deductive databases and logic programs. In J. Minker (ed.), Foundations of Deductive Databases and Logic Programming, 193–216, Morgan Kaufmann Publishers, Los-Altos (Calif.), 1988.Google Scholar
- [Rei78]R. Reiter: On closed world data bases. In H. Gallaire, J. Minker (eds.), Logic and Data Bases, 55–76, Plenum, New York, 1978.Google Scholar
- [RT88]K. A. Ross, R. W. Topor: Inferring negative information from disjunctive databases. Journal of Automated Reasoning 4 (1988), 397–424.Google Scholar
- [Sch76]T. Schwartz: Choice functions, rationality conditions and variations on the weak axiom of revealed preferences. J. Econom. Theory 13 (1976), 414–427.Google Scholar
- [Sho87]Y. Shoham: Nonmonotonic logics: Meaning and utility. In Proc. 10th International Joint Conference on Artificial Intelligence (IJCAI), 388–393, Milan, 1987.Google Scholar
- [YH85]A. Yahya, L. J. Henschen: Deduction in non-horn databases. Journal of Automated Reasoning 1 (1985), 141–160.Google Scholar