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Testable and untestable classes of first-order formulae

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Title: Testable and untestable classes of first-order formulae
Authors: Jordan, Charles Browse this author
Zeugmann, Thomas Browse this author
Keywords: Property testing
Randomized algorithms
Ackermann's class
Ramsey's class
Issue Date: Sep-2012
Publisher: Elsevier
Journal Title: Journal of Computer and System Sciences
Volume: 78
Issue: 5
Start Page: 1557
End Page: 1578
Publisher DOI: 10.1016/j.jcss.2012.01.007
Abstract: In property testing, the goal is to distinguish structures that have some desired property from those that are far from having the property, based on only a small, random sample of the structure. We focus on the classification of first-order sentences according to their testability. This classification was initiated by Alon et al. [2], who showed that graph properties expressible with prefix there exists*for all* are testable but that there is an untestable graph property expressible with quantifier prefix for all*there exists*. The main results of the present paper are as follows. We prove that all (relational) properties expressible with quantifier prefix there exists*for all there exists* (Ackermann's class with equality) are testable and also extend the positive result of Alon et al. [2] to relational structures using a recent result by Austin and Tao [8]. Finally, we simplify the untestable property of Alon et al. [2] and show that prefixes for all(3)there exists, for all(2)there exists for all, for all there exists for all(2) and for all there exists V there exists can express untestable graph properties when equality is allowed.
Type: article (author version)
Appears in Collections:情報科学院・情報科学研究院 (Graduate School of Information Science and Technology / Faculty of Information Science and Technology) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: Charles Jordan

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