Relations Between Diagonalization, Proof Systems, and Complexity Gaps

In this paper we study diagonal processes over time-bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional "clock" in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. Furthermore, these diagonalization methods show that the Gap Theorem for resource bounded computations does not hold for complexity classes consisting only of languages accepted by Turing machines for which it can be formally proven that they run in the required time bound