Encyclopedia of Database Systems

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

Zero-One Laws

  • Nicole Schweikardt
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_1272

Definition

A query language is said to have the 01 law if every Boolean query that contains no constants (i.e., the query does not mention any particular element from the domain of potential data values) is almost surely true or almost surely false. The notions of being “almost surely true,” respectively, “almost surely false” are defined as follows: Let σ be a fixed database schema. For each natural number n, let DBn (σ) be the set of all database instances of schema σ whose active domain is a subset of {1,…,n} (i.e., all database entries belong to {1,…,n}). For a Boolean query q of schema σ let μn (q) be the probability that a database D chosen uniformly at random from DBn (σ) is a “yes”‐instance of query q. In other words, μn (q) is the number of databases in DBn (σ) on which q evaluates to “yes,” divided by the number of all databases in DBn (σ). Query q is said to be almost surely true (respectively, almost surely false), if the limit μ(q) := limn→∞μn (q) exists and is equal to...

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Recommended Reading

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    Ebbinghaus H-D, Flum J. Finite model theory. 2nd ed. Berlin: Springer; 1999.zbMATHGoogle Scholar
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    Hodges W. Model theory. Cambridge: Cambridge University Press; 1993.CrossRefGoogle Scholar
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    Kolaitis P, Vardi MY. 0–1 laws for fragments of existential second-order logic: a survey. In: Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science; 2000. p. 84–98.Google Scholar
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    Libkin L. Elements of finite model theory. New York: Springer; 2004.CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Johann Wolfgang Goethe-University, Frankfurt am MainFrankfurtGermany

Section editors and affiliations

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