Some connections between nonuniform and uniform complexity classes

It is well known that every set in P has small circuits [13]. Adleman [1] has recently proved the stronger result that every set accepted in polynomial time by a randomized Turing machine has small circuits. Both these results are typical of the known relationships between uniform and nonuniform complexity bounds. They obtain a nonuniform upper bound as a consequence of a uniform upper bound. The central theme here is an attempt to explore the converse direction. That is, we wish to understand when nonuniform upper bounds can be used to obtain uniform upper bounds. In this section we will define our basic notion of nonuniform complexity. Then we will show how to relate it to more common notions.