Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically zero. Klivans & Spielman (STOC 2001) observed that the problem is open even when k is a constant. This case has been subjected to a serious study over the past few years, starting from the work of Dvir & Shpilka (STOC 2005). We give the first polynomial time blackbox algorithm for this problem. Our algorithm runs in time poly(nd^k), regardless of the base field. The only field for which polynomial time algorithms were previously known is F=Q (Kayal & Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first blackbox algorithm for depth-3 circuits that does not use the rank based approaches of Karnin & Shpilka (CCC 2008). We prove an important tool for the study of depth-3 identities. We design a blackbox polynomial time transformation that reduces the number of variables in a sps(k,d,n) circuit to k variables, but preserves the identity structure.Comment: 14 pages, 1 figure, preliminary versio

Efficient and Error-Correcting Data Structures for Membership and Polynomial Evaluation

We construct efficient data structures that are resilient against a constant fraction of adversarial noise. Our model requires that the decoder answers most queries correctly with high probability and for the remaining queries, the decoder with high probability either answers correctly or declares "don't know." Furthermore, if there is no noise on the data structure, it answers all queries correctly with high probability. Our model is the common generalization of a model proposed recently by de Wolf and the notion of "relaxed locally decodable codes" developed in the PCP literature. We measure the efficiency of a data structure in terms of its length, measured by the number of bits in its representation, and query-answering time, measured by the number of bit-probes to the (possibly corrupted) representation. In this work, we study two data structure problems: membership and polynomial evaluation. We show that these two problems have constructions that are simultaneously efficient and error-correcting.Comment: An abridged version of this paper appears in STACS 201

Randomness in completeness and space-bounded computations

The study of computational complexity investigates the role of various computational resources such as processing time, memory requirements, nondeterminism, randomness, nonuniformity, etc. to solve different types of computational problems. In this dissertation, we study the role of randomness in two fundamental areas of computational complexity: NP-completeness and space-bounded computations. The concept of completeness plays an important role in defining the notion of \u27hard\u27 problems in Computer Science. Intuitively, an NP-complete problem captures the difficulty of solving any problem in NP. Polynomial-time reductions are at the heart of defining completeness. However, there is no single notion of reduction; researchers identified various polynomial-time reductions such as many-one reduction, truth-table reduction, Turing reduction, etc. Each such notion of reduction induces a notion of completeness. Finding the relationships among various NP-completeness notions is a significant open problem. Our first result is about the separation of two such polynomial-time completeness notions for NP, namely, Turing completeness and many-one completeness. This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis. Our next result involves a conjecture by Even, Selman, and Yacobi [ESY84,SY82] which states that there do not exist disjoint NP-pairs all of whose separators are NP-hard via Turing reductions. If true, this conjecture implies that a certain kind of probabilistic public-key cryptosystems is not secure. The conjecture is open for 30 years. We provide evidence in support of a variant of this conjecture. We show that if there exist certain secure one-way functions, then the ESY conjecture for the bounded-truth-table reduction holds. Now we turn our attention to space-bounded computations. We investigate probabilistic space-bounded machines that are allowed to access their random bits {\em multiple times}. Our main conceptual contribution here is to establish an interesting connection between derandomization of such probabilistic space-bounded machines and the derandomization of probabilistic time-bounded machines. In particular, we show that if we can derandomize a multipass machine even with a small number of passes over random tape and only O(log^2 n) random bits to deterministic polynomial-time, then BPTIME(n) ⊆ DTIME(2^{o(n)}). Note that if we restrict the number of random bits to O(log n), then we can trivially derandomize the machine to polynomial time. Furthermore, it can be shown that if we restrict the number of passes to O(1), we can still derandomize the machine to polynomial time. Thus our result implies that any extension beyond these trivialities will lead to an unknown derandomization of BPTIME(n). Our final contribution is about the derandomization of probabilistic time-bounded machines under branching program lower bounds. The standard method of derandomizing time-bounded probabilistic machines depends on various circuit lower bounds, which are notoriously hard to prove. We show that the derandomization of low-degree polynomial identity testing, a well-known problem in co-RP, can be obtained under certain branching program lower bounds. Note that branching programs are considered weaker model of computation than the Boolean circuits

Sparse Polynomial Interpolation and Testing

Interpolation is the process of learning an unknown polynomial f from some set of its evaluations. We consider the interpolation of a sparse polynomial, i.e., where f is comprised of a small, bounded number of terms. Sparse interpolation dates back to work in the late 18th century by the French mathematician Gaspard de Prony, and was revitalized in the 1980s due to advancements by Ben-Or and Tiwari, Blahut, and Zippel, amongst others. Sparse interpolation has applications to learning theory, signal processing, error-correcting codes, and symbolic computation. Closely related to sparse interpolation are two decision problems. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial f is zero from its evaluations. Sparsity testing is the problem of testing whether f is in fact sparse. We present effective probabilistic algebraic algorithms for the interpolation and testing of sparse polynomials. These algorithms assume black-box evaluation access, whereby the algorithm may specify the evaluation points. We measure algorithmic costs with respect to the number and types of queries to a black-box oracle. Building on previous work by Garg–Schost and Giesbrecht–Roche, we present two methods for the interpolation of a sparse polynomial modelled by a straight-line program (SLP): a sequence of arithmetic instructions. We present probabilistic algorithms for the sparse interpolation of an SLP, with cost softly-linear in the sparsity of the interpolant: its number of nonzero terms. As an application of these techniques, we give a multiplication algorithm for sparse polynomials, with cost that is sensitive to the size of the output. Multivariate interpolation reduces to univariate interpolation by way of Kronecker substitu- tion, which maps an n-variate polynomial f to a univariate image with degree exponential in n. We present an alternative method of randomized Kronecker substitutions, whereby one can more efficiently reconstruct a sparse interpolant f from multiple univariate images of considerably reduced degree. In error-correcting interpolation, we suppose that some bounded number of evaluations may be erroneous. We present an algorithm for error-correcting interpolation of polynomials that are sparse under the Chebyshev basis. In addition we give a method which reduces sparse Chebyshev-basis interpolation to monomial-basis interpolation. Lastly, we study the class of Boolean functions that admit a sparse Fourier representation. We give an analysis of Levin’s Sparse Fourier Transform algorithm for such functions. Moreover, we give a new algorithm for testing whether a Boolean function is Fourier-sparse. This method reduces sparsity testing to homomorphism testing, which in turn may be solved by the Blum–Luby–Rubinfeld linearity test