Relationships between computability-theoretic properties of problems

Abstract

A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computabilitytheoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set A and a computable instance of a problem P, to find a solution relative to which A is still non-computable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of ω cones, of 1 hyperimmunity or of 1 non-Σ 0 1 definition. We also prove that the hierarchies of preservation of hyperimmunity and non-Σ 0 1 definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.Definition 1.1. A Turing ideal is a collection of reals S ⊆ 2 ω which is closed under the effective join and downward-closed under the Turing reduction. In other wordsMany statements studied in reverse mathematics can be formulated as mathematical problems, with instances and solutions. For example, weak König's lemma (WKL) asserts that every infinite, finitely branching subtree of 2 <ω has an infinite path. Here, an instance is such a tree T , and a solution to T is an infinite path 1 The authors are thankful to Mariya Soskova for interesting comments and discussions about cototal degrees.through it. An ω-structure M with second-order part S is a model of a problem P (written M ⊧ P) if every instance in S has a solution in it. In this case we also say that P holds in S. In order to separate a problem P from another problem Q in reverse mathematics, one usually constructs a Turing ideal S in which P holds, but not Q. However, when closing the Turing ideal with solution to instances of P, one must be careful not to make it a model of Q. This motivates the use of preservation properties. Definition 1.2. Fix a collection of sets W ⊆ 2 ω downward-closed under Turing reduction. A problem P admits preservation of W if for every set Z ∈ W and every Z-computable instance X of P, there is a solutionThe following basic lemma is at the core of separations in reverse mathematics.Lemma 1.3. Suppose a problem P admits preservation of some collection W, but another problem Q does not. Then there is a Turing ideal S ⊆ W in which P holds, but not Q.Proof. Since Q does not admit preservation of W, there is some Z ∈ W, and a Q-instance X Q ⩽ T Z such that for every solution Y to X Q , Z ⊕ Y ∈ W. We will build a Turing ideal S ⊆ W containing Z and in which P holds. In particular, Q cannot hold in any such Turing ideal. We build a countable sequence of sets Z 0 , Z 1 , . . . such that for every n ∈ ω, ⊕ s<n Z s ∈ W, and for every P-instance X ⩽ T ⊕ s<n Z s , ther...