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Testing Read-Once Formula Satisfaction

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Published:25 April 2016Publication History
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Abstract

We study the query complexity of testing for properties defined by read-once formulas, as instances of massively parametrized properties, and prove several testability and nontestability results. First, we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in ϵ, doubly exponential in the arity, and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm that outputs an approximation of the distance of the input from satisfying the property. For formulas only involving And/Or gates, we provide a more efficient test whose query complexity is only quasipolynomial in ϵ. On the other hand, we show that such testability results do not hold in general for formulas over non-Boolean alphabets. Specifically, we construct a property defined by a read-once arity 2 (non-Boolean) formula over an alphabet of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size. We also present such a formula over an alphabet of size 5 that also satisfies a strong monotonicity condition.

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            cover image ACM Transactions on Computation Theory
            ACM Transactions on Computation Theory  Volume 8, Issue 2
            May 2016
            92 pages
            ISSN:1942-3454
            EISSN:1942-3462
            DOI:10.1145/2930059
            Issue’s Table of Contents

            Copyright © 2016 ACM

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            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 25 April 2016
            • Accepted: 1 February 2016
            • Revised: 1 October 2015
            • Received: 1 July 2015
            Published in toct Volume 8, Issue 2

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