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

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Mathematical Foundations of Computer Science 2000 (MFCS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1893))

Abstract

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.

Work partially supported by NSF grants CCR-9610257 and CCR-9732041.

Work partially supported by NSF grant CCR-9700061. Work partly done at LIFO, University of Orléans

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Authors and Affiliations

  1. University of California, Santa Cruz, USA

    Phokion G. Kolaitis

  2. Rice University, USA

    Moshe Y. Vardi

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  1. Phokion G. Kolaitis
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  2. Moshe Y. Vardi
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Editors and Affiliations

  1. Department of Computer Science, University of Aarhus, Ny Munkegade, Bldg. 540, 8000, Aarhus C, Denmark

    Mogens Nielsen

  2. Department of Computer Sciene, Comenius University, 84248, Bratislava, Slovakia

    Branislav Rovan

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Kolaitis, P.G., Vardi, M.Y. (2000). 0–1 Laws for Fragments of Existential Second-Order Logic: A Survey. In: Nielsen, M., Rovan, B. (eds) Mathematical Foundations of Computer Science 2000. MFCS 2000. Lecture Notes in Computer Science, vol 1893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44612-5_6

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  • DOI: https://doi.org/10.1007/3-540-44612-5_6

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  • Print ISBN: 978-3-540-67901-1

  • Online ISBN: 978-3-540-44612-5

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