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

Abstract

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).