Encyclopedia of Database Systems

2018 Edition
| Editors: Ling Liu, M. Tamer Özsu

Certain (and Possible) Answers

  • Gösta Grahne
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_1254

Synonyms

True answer (Maybe answer); Validity (Satisfiability)

Definition

Let T be a finite theory expressed in a language L, and φ an L-sentence. Then T finitely entails φ, in notation Tφ, 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 Tφ or T ⊧ ¬ φ. In particular, if T is incomplete (not complete in the finite), then there is an L-sentence φ, such that Tφ 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
$$ {T}_1\,{=}\!\left\{\!\begin{array}{c}\hfill R(a,b){\wedge} R(a,c),\hfill \\ {}\hfill \forall x,y{:}R(x,y){\to} (x,y){=}(a,b){\vee} (a,c),\hfill \\ {}\hfill a\ne b,a\ne c,b\ne c.\hfill \end{array}\right. $$
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Abiteboul S, Duschka OM. Complexity of answering queries using materialized views. In: Proceedings of the 17th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems; 1998. p. 254–63.Google Scholar
  2. 2.
    Abiteboul S, Kanellakis PC, Grahne G. On the representation and querying of sets of possible worlds. Theor Comput Sci. 1991;78(1):158–87.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Di Paola RA. The recursive unsolvability of the decision problem for the class of definite formulas. J ACM. 1969;16(2):324–7.zbMATHCrossRefGoogle Scholar
  4. 4.
    Eiter T, Gottlob G, Gurevich Y. Curb your theory! a circumspective approach for inclusive interpretation of disjunctive information. In: Proceedings of the 13th International Joint Conference on AI; 1993. pp. 634–39.Google Scholar
  5. 5.
    Green TJ, Tannen V. Models for incomplete and probabilistic information. In: Advances in Database Technology, Proceedings of the 10th International Conference on Extending Database Technology; 2006.Google Scholar
  6. 6.
    Imielinski T, Lipski W. Incomplete information in relational databases. J ACM. 1984;31(4):761–91.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Kolaitis PG. Schema mappings, data exchange, and metadata management. In: Proceedings of the 24th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems; 2005. p. 61–75.Google Scholar
  8. 8.
    Libkin L. Data exchange and incomplete information. In: Proceedings of the 25th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems; 2006. p. 60–9.Google Scholar
  9. 9.
    Reiter R. On closed world data bases. In Logic and Data Bases. 1977. p. 55–76.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Concordia UniversityMontrealCanada

Section editors and affiliations

  • Leonid Libkin
    • 1
  1. 1.School of InformaticsUniversity of EdinburghEdinburghUK