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How many circuits are of size $n$ are there? In general for a size $n$ circuit, I know there are $O(2^{poly(n)})$ circuits$^1$, but surely this is reduced by the succinctness condition?

$^1$ https://cs.stackexchange.com/questions/89488/proving-that-exp-doesnt-have-polynomial-size-circuits

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  • $\begingroup$ What exactly do you mean by “succinctly generated circuits”, and how do you measure their size? Note that in general, only $2^{O(n)}$ objects can be described (using whatever fixed encoding scheme) by strings of length $n$ in a finite alphabet. $\endgroup$ – Emil Jeřábek Feb 8 at 19:02
  • $\begingroup$ Size here meaning the number of gates in the circuit. I appreciate that in general there are only $2^{O(n)}$ circuits for an $n$ bit entry. But does that necessarily tell me anything about the size? Does it straightforwardly imply there are only $log(n)$ circuits with $n$ gates that can be described in any succinct encoding scheme? $\endgroup$ – user138901 Feb 8 at 20:45
  • $\begingroup$ No, a more likely bound is $n^{O(1)}$ if the encoding produces exponentially larger circuits, but this entirely depends on the encoding scheme, of which you still haven’t specified one iota. $\endgroup$ – Emil Jeřábek Feb 9 at 7:51

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