This paper studies the Turing degrees of various properties defined for universal numberings, that is, for numberings which list all partial-recursive functions. In particular properties relating to the domain of the corresponding functions are investigated like the set DEQ of all pairs of indices of functions with the same domain, the set DMIN of all minimal indices of sets and DMIN∗ of all in...

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholak and the first author by proving the fundamental index sets associated with a computable set A, {e : We ≤q A} for q ∈ {m, T } are 0 4 complete. We also show that FPT (≤q), that is {e : We computable and ≡q ∅},...

The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Q-vector spaces, Archimedean real closed ordered fields, reduced Abelian p-groups of length less than ω, and models of the original Ehrenfeucht theory. The index sets for...

Index-sets are an integral and fundamental part of every mathematical modeling language. They assist the modeler in grouping various objects and entities. Indexsets are also used extensively in the mathematical notation in order to write an expression in a concise way. An example is the sigma notation for formulating the summation of an unknown positive number n of terms. In this paper, the con...