Questions: Are there references which mention the relation between pebble games and conversions to bounded width circuits?
Here, "conversions to bounded width circuits" means that circuits of size $s$ are converted to circuits of size $f(s)$ and width $g(s)$.
I have noticed that a pebble game is closely related to the conversions (even for nondeterministic circuits). The relation is quite natural, and I'm wondering whether the relation has been already mentioned somewhere, although I think there will be probably no reference. (Since the definition of the pebble game is a little bit long, I write a reference. The pebble game is Definition 2.10 of Nordstrom's arXiv manuscript (page 14 of the 3rd version) https://arxiv.org/abs/1307.3913 .)