An Example of a Theorem that has Contradictory Relativization and Diagonalization Proof

The central questions in complexity theory (e.g. the P =?NP question) can only be solved with proof techniques that do not relativize [BGS75]. There had been some debate about whether such techniques are within reach of the “current state of mathematics” [Hop84]. Recent advances in the area of interactive protocols have produced techniques that do not relativize [Sha90, LFKN90, HCRR90]. However, these advances do not resolve the debate on whether diagonalization can be used to solve problems that have contradictory relativizations [BGS75, Koz80]. In the following, we give an example of a theorem that has contradictory relativizations and a diagonalization proof.