Fixed-point Selection Functions

Let similar or equal to 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 is an element of omega), there are R-complete sets, i.e. such sets D that for any A is an element of A, A <=(R) D Earlier we considered completeness criteria for such reducibilities roughly of the following type: For any A is an element of A, A is R-complete if and only if there is a function f, defined on omega such that f <=(R) D and Omega(f(i)) not similar or equal to Omega(i) for all i is an element of omega. This means that for any set A is an element of A, if it is non-complete, then any function f <=(R) A has a fixed-point e: Omega(f(e)) similar or equal to Omega(e). In this paper we introduce a notion of fixed-point selection function for sequences of such sets and study their complexity characteristics.