Π₀¹-classes and Rado's selection principle

Abstract

Introduction.There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded 77?-classes have arisen. For our purposes we may define a Hj-class to be a set Path(T) of all infinite paths through a recursive tree T. Here a recursive tree T is just a recursive subset of co <w , the set of all finite sequences of the natural numbers a> = {0,1,2,...}, which is closed under initial segments. If the tree T is finitely branching, then we say the /7?-class Path(T) is bounded. If T is highly recursive, i.e., if there exists a partial recursive function f:T-*co such that for each node n e T, f(n) equals the number of immediate successors of n, then we say that the 77?-class Path(T) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded /7?-class and the set of proper marriages for a highly recursive society was always an r.b. /7°-class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. 77?-class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. n l -class C there is a society S c such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of S c . Jockusch conjectured that the set of marriages of a recursive society can represent any bounded Z7?-class and the set of marriages of a highly recursive society can represent any r.b. /7?-class.These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. /7?-class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. /7°-class. Similarly, Bean [1] showed that the set of proper kcolorings of a highly recursive graph is always an r.b. 77?-class, while Remmel [13] showed that any r.b. 27j-class could be represented as the set of proper fc-colorings of a highly recursive graph G for any k > 3. In this paper we study an effective version of a combinatorial selection principle due to Rado. We shall prove that one can represent any bounded i7?-class as the set of "solutions" to the effective version of Rado's problem. What is actually interesting about this problem is that

The weakness of being cohesive, thin or free in reverse mathematics

“…Jockusch et al proved in[22] that for every Π 0,∅ ′1 class C ⊆ 2 ω , there exists a Π 0 1 class D ⊆ ω ω such that deg(C) = deg(D), where deg(C) is the class of degrees of members of C. For the reader who is familiar with Weihrauch degrees, what we actually prove here is that König's lemma is the jump of the cohesiveness principle under Weihrauch reducibility. Bienvenu [personal communication] suggested the use of Simpson's Embedding Lemma [42, Lemma 3.3] to prove the reducibility of some unsolvable instances of cohesiveness to various statements.…”

Index sets for ω‐languages

“…For the space ¼ ½ , this result is essentially due to Jockusch, Lewis and Remmel [7] who showed that for any finitely branching tree Ì which is highly computable in ¼ ¼ , there is a finitely branching recursive tree Ì ¼ with the same set of infinite paths.…”

The complexity of recursive constraint satisfaction problems

“…We also define a class of infinite recursive CSPs called recursively bounded CSPs where the problem of finding solutions is equivalent to the problem of finding an infinite path through a binary recursive tree. There is an extensive literature on the complexity of the problem of finding infinite paths through recursive trees of various types; see [11,12,10] for example. Our basic coding result will allow us to automatically transfer such results to give corresponding results on the complexity the problem of finding solutions to recursive CSPs.…”