I've seen poly-time and logspace uniformity for circuit families, typically defined as the existence of a poly-time/logspace Turing machine "generator" that outputs the correctly sized circuit $C_n$ on input $n$. These restrictions upper bound the power of circuit families by requiring that they are computationally generated.
I was wondering what happens when you change the generator model. Here are two (vague) questions:
- Are there other useful types of uniformity for circuit families?
- Are there interesting "generators" besides time/space bounded TMs? For example, consider families that progress from $C_i$ to $C_{i+1}$ by "small" modifications of the circuit. Can these be broad and have interesting complexity properties?
The idea in #2 is far too inclusive, since it allows things like simple recursive definitions. I am more interested in severe restrictions on the range of possible families by limiting the types of modifications.