Characterizing Arithmetic Read-Once Formulae

TL;DR

This paper provides a simple characterization of arithmetic read-once formulas and introduces the first property testing algorithm for functions computable by these formulas, advancing understanding in this area.

Contribution

It offers the first known characterization and property testing algorithm for arithmetic read-once formulas, filling a gap in the theoretical understanding of these structures.

Findings

Characterization of arithmetic read-once formulas

Development of a property testing algorithm for these formulas

No prior characterization or testing algorithms existed for this class

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.