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Barrington's theorem states that any Boolean circuit made up of gates of fan-in $2$ and with depth $d$ can be transformed into an equivalent Branching Program of constant width (in particular, of width $w=5$). The length of the resulting branching program is $4^d$.

I am wondering if there are any trade-off versions of this theorem. In particular, is it known if it is possible to decrease the length of the resulting program if we allow the width to become a larger constant?

For example, suppose I have a circuit of depth $d$ and total size $s$. If I allow the width of my branching program to become some constant $w>5$, could I hope to obtain a program of length $s\cdot 2^{d/f(w)}$, for some function $f$?

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Perhaps what you're looking for is theorem 2 in Cleve, R. Towards optimal simulations of formulas by bounded-width programs.

I don't think the precise statement that you're asking about is known (note the size $s$ could be much less than $2^d$).

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  • $\begingroup$ Thanks! This is essentially what I was looking for. I guess my precise statement was a little too ambitious. $\endgroup$ Dec 30, 2023 at 11:11

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