Lower Bounds for Syntactically Multilinear Algebraic Branching Programs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Arithmetic Circuits, Syntactic Multilinearity, and the Limitations of Skew Formulae
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Short proofs for the determinant identities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Separating multilinear branching programs and formulas
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
Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits
Computational Complexity
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In their seminal paper, Valiant, Skyum, Berkowitz and Rackoff proved that arithmetic circuits can be balanced. That is, they showed that for every arithmetic circuit Ф of size s and degree r, there exists an arithmetic circuit Ψ of size poly (r, s) and depth O (log(r) log(s)) computing the same polynomial. In the first part of this paper, we follow the proof of Valiant el al. and show that syntactically multilinear arithmetic circuits can be balanced. That is, we show that if Ф is syntactically multilinear, then so is Ψ. Recently, a super-polynomial separation between multilinear arithmetic formula and circuit size was shown. In the second part of this paper, we use the result of the first part to simplify the proof of this separation. That is, we construct a (simpler) polynomial f (x 1, ... , x n ) such that Every multilinear arithmetic formula computing f is of size n Ω(log(n)). There exists a syntactically multilinear arithmetic circuit of size poly(n) and depth O(log2(n)) computing f.