Testing Read-Once Formula Satisfaction

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.