Diagonalizing over Deterministic Polynomial Time

Though some of our concepts will be defined for arbitrary complexity classes, we are only interested in diagonalizing over deterministic polynomial time. (In particular, we do not consider the polynomially bounded classes for other complexity measures like nondeterministic time (NP), (bounded) alternating time (PH), or space (PSPACE).) We say that a diagonalization concept D pertains to deterministic polynomial time (or P) if the following holds. First, the concept is sufficiently strong to diagonalize over P, i.e., to construct a recursive set D~ P. Second, we obtain diagonals of (essentially) arbitrarily low hyperpolynomial deterministic time complexity, i.e., any property Q which can be enforced by a diagionalization of type D is shared by some set in DTIME(f(n)), f any "natural" hyperpolynomial function. Our interest was in finding as powerful as possible diagonalization concepts which still pertain to P. From the concepts discussed here, only some will pertain to P. Other ones will be too powerful, that is they will also allow certain diagonalizations over deterministic exponential time or nondeterministic polynomial time.