On the relative complexity of hard problems for complexity classes without complete problems

AbstractWe show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound. As a consequence, any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems. A further corollary is that, assuming P ≠ NP, the partial ordering of the polynomial time degrees of NP-sets is not complete, and that there are no degree invariant approximations to NP-complete problems

Diagonalizations over polynomial time computable sets

AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP

Inductive inference and computable numberings

AbstractIt has been previously observed that for many TxtEx-learnable computable families of computably enumerable (c.e. for short) sets all their computable numberings are evidently 0′-equivalent, i.e. are equivalent with respect to reductions computable in the halting problem. We show that this holds for all TxtEx-learnable computable families of c.e. sets, and prove that, in general, the converse is not true. In fact there is a computable family A of c.e. sets such that all computable numberings of A are computably equivalent and A is not TxtEx-learnable. Moreover, we construct a computable family of c.e. sets which is not TxtBC-learnable though all of its computable numberings are 0′-equivalent. We also give a natural example of a computable TxtBC-learnable family of c.e. sets which possesses non-0′-equivalent computable numberings. So, for the computable families of c.e. sets, the properties of TxtBC-learnability and 0′-equivalence of all computable numberings are independent