المفاهيم الأساسية
There exists a function that is not computable by any circuit of depth (4-o(1)) log n with the restriction that the top (2-o(1)) log n layers consist only of AND (or OR) gates.
الملخص
The paper presents an improved XOR composition theorem for formulas with restrictions on the top layers, and uses this result to establish a nearly tight depth lower bound for a modified version of the Andreev function.
The key insights are:
The authors show that the well-mixed set of functions, a crucial component in the previous work by Mihajlin and Sofronova, can be significantly simplified and improved using a simple counting argument.
With this improved well-mixed set, the authors are able to obtain a nearly tight XOR composition theorem, which implies a depth lower bound of (4-o(1)) log n for formulas with the restriction that the top (2-o(1)) log n layers consist only of AND (or OR) gates.
The depth lower bound is established by first choosing a hard function f with large formula complexity, and then showing that its XOR composition with a carefully chosen function g cannot be computed by small formulas with the given restriction on the top layers.
The authors also discuss the challenges in extending their techniques to prove depth lower bounds against depth-2 formulas on top, and suggest that proving a general composition theorem for two depth-2 formulas may be a necessary first step.
الإحصائيات
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