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Circuit Depth Lower Bound for Iterated Majority Function

Let $k \geq 3$ be a fixed integer. The function computed by a balanced $k$-ary tree of depth $d(n) = \Theta(\log n)$, where each node computes the majority of its $k$ inputs, cannot be computed by circuits of depth $o(\log \log n)$, even when using unbounded fan-in threshold gates.

Can someone please help me find this theorem (proof of it)? I tried to look at old paper by Yao and others but could not find it.

I am new to this areas so any helpful resources will be great to read more about it.

CuriousScientist
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1 Answers1

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The theorem you're referring to is related to classic results in circuit complexity. As explored in relation to unbounded fan-in threshold circuits, majority circuits, and their computational limitations.

As Stated, the key result states that the function computed by a balanced $k$-ary tree of depth $d(n) = \Theta(\log n)$, where each node computes the majority of its $k$ inputs, cannot be computed by circuits of depth $o(\log \log n)$, even when using unbounded fan-in threshold gates.

The foundational results in this area are attributed to Yao's work on lower bounds for Boolean circuit complexity. more recently Kulikov and Podolskii explore the complexity of majority circuits in depth. You can find a detailed analysis of why certain computational tasks, require specific depth in circuits, even with access to more powerful computational models like threshold gates. References in these articles can help you through the underlying combinatorial probabilistic arguments used to establish lower bounds

Refer to the Springer article [Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates] (https://link.springer.com/article/10.1007/s00224-018-9900-3) for a more detailed treatment of these topics.

DrJay
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