Definition
Let T be a finite theory expressed in a language L, and φ an L-sentence. Then T finitely entails φ, in notation T ⊧ φ, if all finite models of T also are models of φ. A theory T is said to be complete in the finite if for each L-sentence φ either T ⊧ φ or T ⊧ ¬φ. In particular, if T is incomplete (not complete in the finite), then there is an L-sentence φ, such that T ⊭ φ and T ⊭ ¬φ. It follows from classical logic that a first order theory is complete in the finite if and only if all its finite models are isomorphic. Consider now a theory
where a, b, and c are constants. This theory is complete, and clearly for instance T ⊧ R(a, b), T ⊧ R(a, c), and T ⊭ R(d, c), for all constants d different from a and b. Consider then the theory
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsRecommended Reading
Abiteboul S. and Duschka O.M. Complexity of answering queries using materialized views. In Proc. 17th ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, 1998, pp. 254–263.
Abiteboul S., Kanellakis P.C., and Grahne G. On the representation and querying of sets of possible worlds. Theor. Comput. Sci., 78(1):158–187, 1991.
Di Paola R.A. The recursive unsolvability of the decision problem for the class of definite formulas. J. ACM, 16(2):324–327, 1969.
Eiter T., Gottlob G., Gurevich Y. Curb your theory! a circumspective approach for inclusive interpretation of disjunctive information. In Proc. 13th Int. Joint Conf. on AI, 1993, pp. 634–639.
Green T.J. and Tannen V. Models for incomplete and probabilistic information. In Proc. EDBT 2006 workshops LNCS Vol. 4251, 2006.
Imielinski T. and Lipski W. Incomplete information in relational databases. J. ACM, 31(4):761–791, 1984.
Kolaitis P.G. Schema mappings, data exchange, and metadata management. In Proc. 24th ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, 2005, pp. 61–75.
Libkin L. Data exchange and incomplete information. In Proc. 25th ACM SIGACT-SIGMOD-SIGART Symp. on Principles of Database Systems, 2006, pp. 60–69.
Reiter R. On closed world data bases. In Logic and Data Bases, 1977, pp. 55–76.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this entry
Cite this entry
Grahne, G. (2009). Certain (and Possible) Answers. In: LIU, L., ÖZSU, M.T. (eds) Encyclopedia of Database Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-39940-9_1254
Download citation
DOI: https://doi.org/10.1007/978-0-387-39940-9_1254
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-35544-3
Online ISBN: 978-0-387-39940-9
eBook Packages: Computer ScienceReference Module Computer Science and Engineering