Characterizing Arithmetic Read-Once Formulae

Abstract: An \emph{arithmetic read-once formula} (ROF for short) is a formula (i.e. a tree of computation) in which the operations are ${+,\times}$ and such that every input variable labels at most one leaf. We give a simple characterization of such formulae. Other than being interesting in its own right, our characterization gives rise to a property testing algorithm for functions computable by such formulae. To the best of our knowledge, prior to our work no characterization and/or property testing algorithm was known for this kind of formulae.