Slightly improved lower bounds for homogeneous formulas of bounded depth and bounded individual degree

•A finer product decomposition for Homogeneous Formulas of small depth.•A finer product decomposition for small-depth homogeneous multi-r-ic formulas.•We give superpolynomial lower bounds for homogeneous multi-r-ic formulas of small-depth.•For all Δ∈[ω(1),o(log⁡nlog⁡r)] our lower bound is asymptotic...

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Vydáno v:Information processing letters Ročník 156; s. 105900
Hlavní autor: Chillara, Suryajith
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.04.2020
Témata:
Algebraic complexity theory
Computational complexity
Lower bounds
Theory of computation
Theory of computation
Algebraic complexity theory
Lower bounds
Computational complexity
ISSN:0020-0190, 1872-6119
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Shrnutí:•A finer product decomposition for Homogeneous Formulas of small depth.•A finer product decomposition for small-depth homogeneous multi-r-ic formulas.•We give superpolynomial lower bounds for homogeneous multi-r-ic formulas of small-depth.•For all Δ∈[ω(1),o(log⁡nlog⁡r)] our lower bound is asymptotically better than Kayal et al. [STOC 2014]. Kayal, Saha and Tavenas (Theory of Computing, 2018), showed that any bounded-depth homogeneous formula of bounded individual degree (bounded by r) that computes an explicit polynomial over n variables must have size exp⁡(Ω(1r(n4)1/Δ)) for all depths Δ≤O(log⁡nlog⁡r+log⁡log⁡n). In this article we show an improved size lower bound of exp⁡(Ω(Δr(nr2)1/Δ)) for all depths Δ≤O(log⁡nlog⁡r), and for the same explicit polynomial. In comparison to Kayal, Saha and Tavenas (Theory of Computing, 2018) (1) our results give superpolynomial lower bounds in a wider regime of depth Δ, and (2) for all Δ∈[ω(1),o(log⁡nlog⁡r)] our lower bound is asymptotically better. This improvement is due to a finer product decomposition of general homogeneous formulas of bounded-depth. This follows from an adaptation of a new product decomposition for bounded-depth multilinear formulas shown by Chillara, Limaye and Srinivasan (SIAM Journal of Computing, 2019).
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2019.105900