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Volume 10 (2014) Article 18 pp. 465-514
On Reconstruction and Testing of Read-Once Formulas
Received: June 18, 2012
Revised: July 18, 2014
Published: December 26, 2014
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Keywords: read-once formulas, identity testing, reconstruction, arithmetic circuits, bounded-depth circuits
ACM Classification: F.1.3, F.2.2
AMS Classification: 68Q15, 68Q25

Abstract: [Plain Text Version]

$ \newcommand{\BigO}{\mathcal{O}} $

An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are $\{+,\times\}$ and each input variable labels at most one leaf. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable $x_i$ with a univariate polynomial $T_i(x_i)$. We obtain a deterministic non-adaptive reconstruction algorithm for PROFs, that is, an algorithm that, given black-box access to a PROF, constructs a PROF computing the same polynomial. The running time of the algorithm is $(nd)^{\BigO(\log n)}$ for PROFs of individual degrees at most $d$. To the best of our knowledge our results give the first subexponential-time deterministic reconstruction algorithms for ROFs.

Another question that we study is the following generalization of the polynomial identity testing (PIT) problem. Given an arithmetic circuit computing a polynomial $P(\bar{x})$, decide whether there is a PROF computing $P(\bar{x})$, and find one if one exists. We call this question the read-once testing problem (ROT for short). Previous (randomized) algorithms for reconstruction of ROFs imply that there exists a randomized algorithm for the ROT problem. In this work we show that most previous PIT algorithms be strengthened to yield algorithms for the ROT problem. In particular we give ROT algorithms for the following circuit classes: depth-$2$ circuits (circuits computing sparse polynomials), depth-$3$ circuits with bounded top fan-in, and sum of $k$ ROFs. The running time of the ROT algorithm is essentially the same as the running time of the corresponding PIT algorithm for the class.

The main tool in most of our results is a new connection between PIT and reconstruction of ROFs. Namely, we show that in any model that satisfies some “nice” closure properties (for example, a partial derivative of a polynomial computed by a circuit in the model, can also be computed by a circuit in the model) and that has an efficient deterministic polynomial identity testing algorithm, we can also answer the read-once testing problem.

An extended abstract of this paper appeared in the Proceedings of the 40th Annual Symposium on Theory of Computing (STOC 2008), pages 507-516.