Efficient parallel evaluation of straight-line code and arithmetic circuits
Proc. of the Aegean workshop on computing on VLSI algorithms and architectures
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Properties that characterize LOGCFL
Journal of Computer and System Sciences
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Non-commutative arithmetic circuits: depth reduction and size lower bounds
Theoretical Computer Science
On arithmetic branching programs
Journal of Computer and System Sciences
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Exponential lower bounds for restricted monotone circuits
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
A survey of lower bounds for satisfiability and related problems
Foundations and Trends® in Theoretical Computer Science
Characterizing Valiant's algebraic complexity classes
Journal of Complexity
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Minimizing Disjunctive Normal Form Formulas and $AC^0$ Circuits Given a Truth Table
SIAM Journal on Computing
Lower Bounds for Syntactically Multilinear Algebraic Branching Programs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Balancing Syntactically Multilinear Arithmetic Circuits
Computational Complexity
On the Power of Small-Depth Computation
Foundations and Trends® in Theoretical Computer Science
Quasi-polynomial hitting-set for set-depth-Δ formulas
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Arithmetic circuit lower bounds via maxrank
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of nxn matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O^(^n^l^o^g^n^). If the original circuit uses only integer constants of polynomial size, then the same is true for the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: -the existence of nontrivial boolean circuits of constant depth for languages in LOGCFL; -reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree.