0–1 Laws for Fragments of Second-Order Logic: An Overview

  • Phokion G. Kolaitis
  • Moshe Y. Vardi
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 21)


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 any property expressible in first-order logic, i.e., the probability of any such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable.

We investigate here 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 ∑ 1 1 fragment are identical.


Satisfying Assignment Existential Quantifier Extension Axiom Asymptotic Probability Transfer Theorem 
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Copyright information

© Springer-Verlag New York, Inc 1992

Authors and Affiliations

  • Phokion G. Kolaitis
    • 1
  • Moshe Y. Vardi
    • 2
  1. 1.University of CaliforniaSanta CruzUSA
  2. 2.IBM Almaden Research CenterUSA

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