0–1 Laws for Fragments of Existential Second-Order Logic: A Survey

  • Phokion G. Kolaitis
  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)


The probability of a property on the collection of all finite relational structures is the limit as n → ∞ of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that the 0-1 law holds for every property expressible in first-order logic, i.e., the probability of every such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable.

In this survey, we consider fragments of existential second-order logic in which we restrict the patterns of first-order quantifiers. We focus on fragments in which the first-order part belongs to a prefix class. We show that the classifications of prefix classes of first-order logic with equality according to the solvability of the finite satisfiability problem and according to the 0-1 law for the corresponding Σ1 1 fragments are identical, but the classifications are different without equality.


Decision Problem Random Graph Relation Symbol Extension Axiom Countable Structure 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Phokion G. Kolaitis
    • 1
  • Moshe Y. Vardi
    • 2
  1. 1.University of CaliforniaSanta CruzUSA
  2. 2.Rice UniversityUSA

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