New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree

摘要

Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers n and d such that d ω(log n), any syntactic depth four circuit of bounded individual degree δ = o(d) that computes the Iterated Matrix Multiplication polynomial (IMMn,d) must have size n ? √ d/δ . Unfortunately, this bound deteriorates as the value of δ increases. Further, the bound is superpolynomial only when δ is o(d). It is natural to ask if the dependence on δ in the bound could be weakened. Towards this, in an earlier result [STACS, 2020], we showed that for all large enough integers n and d such that d = Θ(log2 n), any syntactic depth four circuit of bounded individual degree δ 6 n 0.2 that computes IMMn,d must have size n?(log n) . In this paper, we make further progress by proving that for all large enough integers n and d, and absolute constants a and b such that ω(log2 n) 6 d 6 n a, any syntactic depth four circuit of bounded individual degree δ 6 n b that computes IMMn,d must have size n?( √ d) . Our bound is obtained by carefully adapting the proof of Kumar and Saraf [SIAM J. Computing, 2017] to the complexity measure introduced in our earlier work [STACS, 2020].