Has anyone explored what is the circuit complexity of classic decision problems such as Primes or Graph-Isomorphism for small input size $N$?
While most people are interested in the how the scaling goes as $N \to \infty$, I think it would also be interesting to see how this grows for small N. Sure, we now know Primes is in P, but it would be neat to see how it grows, and maybe even sharp changes in the growth rate of the graph as the inputs get large enough that a different algorithm becomes more efficient.
There is even the (unlikely) possibility that someone could extract from a sequence of circuits a general algorithm.
It seems like this approach could answer different questions than are usually asked about $N \to \infty$. With the advances of algorithms knowledge (SAT solvers, etc.) and super computing power, concrete answers could be obtained for small $N$.
Are there any references or lists of results for people explicitly computing circuit complexity of decision problems for small $N$?
If there are people working on this, what algorithms do they currently use to solve the minimal circuit problem (given a boolean function and set of gates, output a circuit using the minimal number of gates necessary)?