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A super-polynomial lower bound for regular arithmetic formulas

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Ramprasad SaptharishiAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 146 - 153

https://doi.org/10.1145/2591796.2591847

Published: 31 May 2014 Publication History

Abstract

We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n² + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least n^{Ω(log n)}.

Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as ΣΠ^[b]ΣΠ^[a]-formulas. We show that there exists an n²-variate polynomial of degree n in VNP such that any ΣΠ^[O(√n)]ΣΠ^[√n]-formula computing it must have top fan-in at least 2^{Ω(√n·log n)}. In comparison, Tavenas [Tav13] has recently shown that every n^O(1)-variate polynomial of degree n in VP admits a ΣΠ^[O(√n)]ΣΠ^[√n]-formula of top fan-in 2^{O(√n·log n)}. This means that any further asymptotic improvement in our lower bound for such formulas (to say 2^{ω(√n log n)}) will imply that VP is different from VNP.

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A super-polynomial lower bound for regular arithmetic formulas

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

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Published: 31 May 2014

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Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
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