# Which research fields deal with this variant definition of Boolean circuit depth?

Disclaimer: I admit that the question is not very clear. I think it cannot be helped because the question is very open-ended.

First of all, I present the interested type of circuits. We only consider this structure of Boolean / Arithmetic circuits: a DAG consisting of multiple inputs, 1 output and multiple gates. Each gate has fan-in $$\le 2$$ and unlimited fan-out (in principle). Also, we only consider Boolean gates (NOT, AND, OR), and Arithmetic gates (ADD, MULTIPLY).

Let $$f: \mathbb{B}^n \rightarrow \mathbb{B}$$ be a Boolean function and $$G(f)$$ be the set of its corresponding Boolean circuits with size poly$$(n)$$. For each circuit $$C \in G(f)$$, we can transform it into $$C' \in G(f)$$ only consisting of NOT and AND gates, by using De Morgan's laws.

Then, we place weights on every edge of $$C'$$ as follows (called Rule 0): place indeterminate $$X$$ on every input edge of AND gates, and place $$1$$ on every input edge of NOT gates. Now we can construct $$f_0(C) \in \mathbb{Z}[X]$$ as follows:

1. At each input node of $$C'$$, assign $$1 \in \mathbb{Z}[X]$$ as its value.
2. At each of other nodes of $$C'$$, say node $$u$$, compute its value $$\in \mathbb{Z}[X]$$: $$\text{val}(u) = \sum_{\text{edge } (v,u)} \text{val}(v)\cdot\text{weight}(v, u)$$
3. $$f_0(C)$$ is the value of the output node of $$C'$$.

With $$f_0(C)$$, we have an alternative definition of circuit depth: the depth of $$C$$ is $$\deg f_0(C)$$. Therefore, in a way, the best circuits to compute $$f$$ are the minimum circuits in $$G(f)$$ w.r.t $$\deg f_0(\cdot)$$.

In a similar manner, we can change the weighting rule into Rule 1 to get $$f_1(C)$$. Particularly, Rule 1 is as follows: place indeterminate $$X$$ on only one of the input edges of each AND gate, and place $$1$$ on each of other edges. Now $$f_1(C)$$ is the minimum-degree value of the output node of $$C'$$, out of all weighting scenarios. So we have a variant definition of circuit depth: the depth of $$C$$ is $$\deg f_1(C)$$.

Naturally, the research question has risen: what is the best circuits to compute $$f$$ w.r.t $$\deg f_1(\cdot)$$. In other words, given $$f$$, what are the minimum circuits in $$G(f)$$ w.r.t $$\deg f_1(\cdot)$$?

Lastly, I quickly present why there is an Arithmetic aspect of all this. One can see easily that we can transform $$C'$$ into Arithmetic poly$$(n)$$-size circuits $$D_0, D_1$$, both of which have 1 input (indeterminate $$X$$), 1 output, and $$f_0(C), f_1(C)$$ as their outputs, respectively. In the transformations, the input nodes of $$C'$$ become the constant nodes of 1.