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Fischer, E. and Yonatan, G. and Oded, Lachish (2016) Testing read-once formula satisfaction. ACM Transactions on Computation Theory (TOCT) 8 (2), ISSN 1942-3454.

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## Abstract

We study the query complexity of testing for properties defined by read once formulas, as instances of {\em massively parametrized properties}, and prove several testability and non-testability 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 $\epsilon$, 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 {\em 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 $\epsilon$. 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 additionally satisfies a strong monotonicity condition.

Item Type: Article © ACM, 2016. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published at the link above. Massively parametrized properties, property testing, read once formula School of Business, Economics & Informatics > Computer Science and Information Systems Oded Lachish 12 May 2016 10:43 12 Jun 2021 12:29 https://eprints.bbk.ac.uk/id/eprint/14832

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