Fixed-point Selection Functions

Abstract

Abstract: Let $$\simeq$$ be a binary relation between sets of integers, and ≤R be a Post reducibility, i.e. a reflexive and transitive relation between sets of integers such that if A ≤R B then the computational complexity of recognition of elements of A is easier than (or equal to) the recognition of elements of B. Suppose that for a class A of arithmetical sets, which have an effective enumeration $$\{\Omega_{e}\}_{e\in\omega}$$, there are R-complete sets, i.e. such sets D that for any A ∈ A, A ≤R D. Earlier we considered completeness criteria for such reducibilities roughly of the following type: For any $$A\in{\mathcal{A}}$$, A is R-complete if and only if there is a function f, defined on ω such that f ≤R D and $$\Omega_{f(i)}\not\simeq\Omega_{i}$$ for all $$i\in\omega$$. This means that for any set A ∈ A, if it is non-complete, then any function $$f\leq_{R}A$$ has a fixed-point $$e$$: $$\Omega_{f(e)}\simeq\Omega_{e}$$. In this paper we introduce a notion of fixed-point selection function for sequences of such sets and study their complexity characteristics.