Testing Formula Satisfaction

TL;DR

This paper investigates the query complexity of testing properties defined by read-once formulas, establishing testability results for Boolean formulas and demonstrating non-testability in certain non-Boolean cases.

Contribution

It proves the testability of Boolean read-once formulas with various gate types and introduces efficient algorithms, while also showing limitations for non-Boolean formulas.

Findings

Boolean formulas are testable with exponential or quasipolynomial queries.

Monotone formulas admit an approximation algorithm for distance estimation.

Certain non-Boolean formulas require queries dependent on formula size for testing.

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.