Abstract
We present a probabilistic data model for complex values. More precisely, we introduce probabilistic complex value relations, which combine the concept of probabilistic relations with the idea of complex values in a uniform framework. We elaborate a model-theoretic definition of probabilistic combination strategies, which has a rigorous foundation on probability theory. We then define an algebra for querying database instances, which comprises the operations of selection, projection, renaming, join, Cartesian product, union, intersection, and difference. We prove that our data model and algebra for probabilistic complex values generalizes the classical relational data model and algebra. Moreover, we show that under certain assumptions, all our algebraic operations are tractable. We finally show that most of the query equivalences of classical relational algebra carry over to our algebra on probabilistic complex value relations. Hence, query optimization techniques for classical relational algebra can easily be applied to optimize queries on probabilistic complex value relations.
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References
S. Abiteboul, R. Hull and V. Vianu, Foundations of Databases (Addison-Wesley, Reading, MA, 1995).
J.F. Baldwin, Evidential support logic programming, Fuzzy Sets and Systems 24 (1987) 1-26.
D. Barbara, H. Garcia-Molina and D. Porter, The management of probabilistic data, IEEE Transactions on Knowledge and Data Engineering 4(5) (1992) 387-502.
R. Carnap, Logical Foundations of Probability (University of Chicago Press, Chicago, 1950).
R. Cavallo and M. Pittarelli, The theory of probabilistic databases, in: Proceedings of the 13th International Conference on Very Large Databases (Morgan Kaufmann, 1987) pp. 71-81.
P. Ciaccia, D. Montesi, W. Penzo and A. Trombetta, Imprecision and user preferences in multimedia queries: A generic algebraic approach, in: Proceedings of the International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2000), Lecture Notes in Computer Science, Vol. 1762 (Springer, New York, 2000) pp. 50-71.
D. Dey and S. Sarkar, A probabilistic relational model and algebra, ACM Transactions on Database Systems 21(3) (1996) 339-369.
D. Dey and S. Sarkar, PSQL: A query language for probabilistic relational data, Data & Knowledge Engineering 28 (1998) 107-120.
T. Eiter, J.J. Lu, T. Lukasiewicz and V.S. Subrahmanian, Probabilistic object bases, Technical Report INFSYS RR-1843-99-11, Institut für Informationssysteme, Technische Universität Wien (1999).
T. Eiter, T. Lukasiewicz and M. Walter, Extension of the relational algebra to probabilistic complex values, in: Proceedings of the International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2000), Lecture Notes in Computer Science, Vol. 1762 (Springer, New York, 2000) pp. 94-115.
R. Fagin, Fuzzy queries in multimedia database systems, in: Proceedings of the 17th ACM Symposium on Principles of Database Systems (ACM Press, New York, 1998) pp. 1-10.
N. Fuhr and T. Rölleke, A probabilistic NF2 relational algebra for integrated information retrieval and database systems, in: Proceedings of the 2nd World Conference on Integrated Design and Process Technology (Society for Design and Process Science, 1996) pp. 17-30.
N. Fuhr and T. Rölleke, A probabilistic relational algebra for the integration of information retrieval and database systems, ACM Transactions on Information Systems 15(1) (1997) 32-66.
H. Gaifman, Concerning measures in first order calculi, Israel Journal of Mathematics 2 (1964) 1-18.
M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).
J.Y. Halpern, An analysis of first-order logics of probability, Artificial Intelligence 46(3) (1990) 311-350.
Y. Kornatzky and S.E. Shimony, A probabilistic object-oriented data model, Data & Knowledge Engineering 12 (1994) 143-166.
Y. Kornatzky and S.E. Shimony, A probabilistic spatial data model, Information Sciences 90 (1996) 51-74.
H.E. Kyburg, Jr., Interval-valued probabilities, in: Imprecise Probabilities Project, eds. G. de Cooman, P.Walley and F.G. Cozman (1998). Available from http://ippserv.rug.ac.be/.
L.V.S. Lakshmanan, N. Leone, R. Ross and V.S. Subrahmanian, ProbView: A flexible probabilistic database system, ACM Transactions on Database Systems 22(3) (1997) 419-469.
L.V.S. Lakshmanan and F. Sadri, Probabilistic deductive databases, in: Proceedings of the International Logic Programming Symposium (1994) pp. 254-268.
T. Lukasiewicz, Local probabilistic deduction from taxonomic and probabilistic knowledge-bases over conjunctive events, International Journal of Approximate Reasoning 21(1) (1999) 23-61.
T. Lukasiewicz, Probabilistic deduction with conditional constraints over basic events, Journal of Artificial Intelligence Research 10 (1999) 199-241.
M. Pittarelli, An algebra for probabilistic databases, IEEE Transactions on Knowledge and Data Engineering 6(2) (1994) 293-303.
K.V.S.V.N. Raju and A.K.Majumdar, Fuzzy functional dependencies and lossless join decomposition of fuzzy relational database systems, ACM Transactions on Database Systems 13(2) (1988) 129-166.
H.-J. Schek and P. Pistor, Data structures for an integrated data base management and information retrieval system, in: Proceedings of the 8th International Conference on Very Large Data Bases (Morgan Kaufmann, 1982) pp. 197-207.
D. Scott and P. Krauss, Assigning probabilities to logical formulas, in: Aspects of Inductive Logic, eds. J. Hintikka and P. Suppes (North-Holland, Amsterdam, 1966) pp. 219-264.
M. Walter, An extension of relational algebra to probabilistic complex values, Master's thesis, Universität Gießen (1999).
E. Zimányi, Query evaluation in probabilistic relational databases, Theoretical Computer Science 171(1/2) (1997) 179-219.
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Eiter, T., Lukasiewicz, T. & Walter, M. A data model and algebra for probabilistic complex values. Annals of Mathematics and Artificial Intelligence 33, 205–252 (2001). https://doi.org/10.1023/A:1013121110704
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DOI: https://doi.org/10.1023/A:1013121110704