Testing formula satisfaction

Fischer, E. and Goldhirsh, Y. and Lachish, Oded (2012) Testing formula satisfaction. In: Fomin, F.V. and Kaski, P. (eds.) Algorithm Theory – SWAT 2012. Lecture Notes in Computer Science 7357. Berlin, Germany: Springer, pp. 376-387. ISBN 9783642311550.

Abstract

We study the query complexity of testing for properties defined by read once formulae, as instances of 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 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 formulae only involving And/Or gates, we provide a more efficient test whose query complexity is only quasi-polynomial in \epsilon. On the other hand we show that such testability results do not hold in general for formulae over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over alphabets of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size.

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