We want to show that given any daemon, there is a smaller circuit that solves the problem. The most natural approach is showing how to construct a smaller circuit, given a daemon. But if the daemon is obfuscated, there is no efficient procedure which takes the daemon circuit as input and produces a smaller circuit that still solves the problem.
Is there a non-obfuscated circuit corresponding to every obfuscated one? And would the non-obfuscated circuit be at least as small as the obfuscated one?
If so it seems like you could just show how to construct the smaller circuit for any non-obfuscated daemon, and then even though you don’t technically have a construction for efficiently converting all daemons into smaller circuits, you’d still have a solid argument that daemons aren’t minimal.
“Obfuscated circuit” implies there is some circuit that get obfuscated, obfuscation is a map from circuits to circuits that increases the size and makes them inscrutable.
So obfuscation per se is not a problem for the statement. But it is an obstruction to a proof. You can’t just handle obfuscation as a special case, it’s not a natural kind, just an example that shows what kind of thing is possible.
Is there a non-obfuscated circuit corresponding to every obfuscated one? And would the non-obfuscated circuit be at least as small as the obfuscated one?
If so it seems like you could just show how to construct the smaller circuit for any non-obfuscated daemon, and then even though you don’t technically have a construction for efficiently converting all daemons into smaller circuits, you’d still have a solid argument that daemons aren’t minimal.
Yes, this is why I think the statement is true.
“Obfuscated circuit” implies there is some circuit that get obfuscated, obfuscation is a map from circuits to circuits that increases the size and makes them inscrutable.
So obfuscation per se is not a problem for the statement. But it is an obstruction to a proof. You can’t just handle obfuscation as a special case, it’s not a natural kind, just an example that shows what kind of thing is possible.