An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are {+, ×} and such that every 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 ). In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for preprocessed ROFs. In particular we obtain the following results. We give an (nd)O(log n) black-box polynomial identity-testing algorithm for PROFs in n variables of individual degrees at most d (i.e. each T i (x i ) is of degree at most d). This improves and generalizes the previous n2. Given k PROFs in n variables of individual degrees at most d we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k) . This result improves and extends the previous results of [SV08].3. Combining the two results above we obtain an (nd) O(k+log n) time deterministic algorithm for checking whether a black box holding the sum of k n-variate PROFs computes the zero polynomial. In other words, we provide a hitting set of size (nd) O(k+log n) for the sum of k PROFs. This result greatly improves and extends [SV08] where an n O(k 2 + √ n) algorithm was given for identity testing the sum of k ROFs. We give an (nd)O(D+k) time deterministic black-box identity testing algorithm for the sum of k PROFs of depth D and individual degrees at most d, which improves the n O(D+k 2 ) algorithm of [SV08] for depth D ROFs.As a corollary of the results above we obtain an n O(k) time deterministic identity testing algorithm for multilinear depth-3 ΣΠΣ(k) circuits. This matches the non black-box algorithm of [KS07] for the multilinear case. In fact the same result also holds for preprocessed-multilinear-ΣΠΣ(k) circuits which are depth-4 circuits of a restricted form.In addition to the above results we obtain a deterministic reconstruction algorithms for preprocessed read-once formulas. The running time of the algorithm is (nd) O(log n) for general preprocessed read-once formulas, of individual degrees at most d, and (nd) O(D) for depth D preprocessed read-once formulas of individual degrees at most d.Most of our results are obtained using hardness of representation approach which was first used in [SV08]. This technique can be thought of as a very explicit way of transforming (mild) hardness of a very structured polynomial to an identity testing algorithm.

An arithmetic read-once formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of read-once formulas. the following are some of the results that we obtain.1. Given k ROFs in n variables, over a field F, we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n2. We give an n O(d+k 2 ) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k 2 ) for the sum of k depth d ROFs. If |F| is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs.3. We give a hitting set of size exp(Õ( √ n + k 2 )) for the sum of k ROFs (without depth restrictions). In particular this implies a sub-exponential time deterministic algorithm for black-box identity testing and reconstructing of ROFs. As before, if |F| is too small then we make queries from a polynomial size extension field.To the best of our knowledge our results give the first sub-exponential time black-box identity testing algorithm for the sum of (a constant number of) ROFs and the first polynomial time identity testing algorithm for the sum of (a constant number of) ROFs, in the non black-box setting.Another question that we study is the following generalization of the polynomial identity testing problem. Given an arithmetic circuit computing a polynomial P (x), decide whether there is a ROF computing P (x). If there is such a formula then output it. Otherwise output "No". 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 read-once testing problem. In this work we show that most previous algorithms for polynomial identity testing can be strengthen to yield algorithms for the read-once testing 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 (both in the black-box and non black-box settings, where the running time depends on the model), non-commutative formulas and sum of k ROFs. The running time of the ROT algorithm is essentially the same running time as the corresponding PIT algorithm for the class.The main tool in most of our results is a new connection between polynomial identity testing and reconstruction of read-once formulas. Namely, we show that in any model that is closed under partial derivatives (that is, 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 det...

We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth-4 circuits with fan-in k at the top + gate. We give the first polynomial-time deterministic identity testing algorithm for such circuits. Our results also hold in the black-box setting.The running time of our algorithm is (ns), where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. The importance of this model arises from [AV08], where it was shown that derandomizing black-box polynomial identity testing for general depth-4 circuits implies a derandomization of polynomial identity testing (PIT) for general arithmetic circuits. Prior to our work, the best PIT algorithm for multilinear ΣΠΣΠ(k) circuits [KMSV10] ran in quasi-polynomial-time, with the running time being n O(k 6 log(k) log 2 s) .We obtain our results by showing a strong structural result for multilinear ΣΠΣΠ(k) circuits that compute the zero polynomial. We show that under some mild technical conditions, any gate of such a circuit must compute a sparse polynomial. We then show how to combine the structure theorem with a result by Klivans and Spielman [KS01], on the identity testing for sparse polynomials, to yield the full result. * CSAIL, MIT.

An arithmetic read-once formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of read-once formulas. the following are some of the results that we obtain.1. Given k ROFs in n variables, over a field F, we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k 2 ) .2. We give an n O(d+k 2 ) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k 2 ) for the sum of k depth d ROFs. If |F| is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs.3. We give a hitting set of size exp(Õ( √ n + k 2 )) for the sum of k ROFs (without depth restrictions). In particular this implies a sub-exponential time deterministic algorithm for black-box identity testing and reconstructing of ROFs. As before, if |F| is too small then we make queries from a polynomial size extension field.To the best of our knowledge our results give the first sub-exponential time black-box identity testing algorithm for the sum of (a constant number of) ROFs and the first polynomial time identity testing algorithm for the sum of (a constant number of) ROFs, in the non black-box setting.Another question that we study is the following generalization of the polynomial identity testing problem. Given an arithmetic circuit computing a polynomial P (x), decide whether there is a ROF computing P (x). If there is such a formula then output it. Otherwise output "No". 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 read-once testing problem. In this work we show that most previous algorithms for polynomial identity testing can be strengthen to yield algorithms for the read-once testing 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 (both in the black-box and non black-box settings, where the running time depends on the model), non-commutative formulas and sum of k ROFs. The running time of the ROT algorithm is essentially the same running time as the corresponding PIT algorithm for the class.The main tool in most of our results is a new connection between polynomial identity testing and reconstruction of read-once formulas. Namely, we show that in any model that is closed under partial derivatives (that is, 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 eff...

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Before our work no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in quasi-polynomial time in general, and polynomial time for constant depth. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four circuits.