A complexity theory based on Boolean algebra
Journal of the ACM (JACM)
Finding irreducible polynomials over finite fields
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Diagonal Circuit Identity Testing and Lower Bounds
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Deterministically testing sparse polynomial identities of unbounded degree
Information Processing Letters
Tensor-rank and lower bounds for arithmetic formulas
Proceedings of the forty-second ACM symposium on Theory of computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
On the arithmetic complexity of euler function
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
On identity testing of tensors, low-rank recovery and compressed sensing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Arithmetic circuits: The chasm at depth four gets wider
Theoretical Computer Science
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
A Case of Depth-3 Identity Testing, Sparse Factorization and Duality
Computational Complexity
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We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition X1⊔⋅⋅⋅⊔ Xd of the variable indices [n] that the top product layer respects, i.e. C(term{x})=∑i=1k ∏j=1d fi,j(term{x}Xj), where fi,j is a sparse polynomial in F[term{x}Xj]. Extending this definition to any depth - we call a depth-D formula C (consisting of alternating layers of Σ and Π gates, with a Σ-gate on top) a set-depth-D formula if every Π-layer in C respects a (unknown) partition on the variables; if D is even then the product gates of the bottom-most Π-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-D formulas (over any field) with running time polynomial in exp((D2log s) Δ - 1), where s is the size bound on the input set-depth-D formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of D=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995) and recently by Forbes & Shpilka (STOC 2012 & ECCC TR12-115). Our work settles this question, not only for depth-3 but, up to depth εlog s / log log s, for a fixed constant ε Hadamard algebra, after applying a 'shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-D formulas.