Bounded Relativization

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization . For a complexity class C , we say that a statement is C -relativizing if the statement holds relative to every oracle O ∈ C . It is easy to see that every result that relativizes also C -relativizes for every complexity class C . On the other hand, we observe that many non-relativizing results, such as IP = PSPACE , are in fact PSPACE -relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0,