Abstract We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa’s result that there are no minimal pairs for relative generic computability and answers a basic structural question mentioned in several papers in the area.

Abstract A coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

We study the number of changes of the initial segment Zs ↾n for computable approximations of a Martin-Löf random Δ20 set Z. We establish connections between this number of changes and various notions of computability theoretic lowness, as well as the fundamental thesis that, among random sets, randomness is antithetical to computational power. We introduce a new randomness notion, called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2n times. We establish various connections with ω-c.e. tracing and omega;-c.e. jump domination, a new lowness property. We also examine some relationships to randomness theoretic notions of highness, and give applications to the study of (weak) Demuth cuppability.

Abstract We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

Abstract A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model , i.e., the elementary diagram De ( ) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree.

Abstract We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs ( ) splits into a stable version ( ) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

Abstract A real is called properly n-generic if it is n-generic but not n + 1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

Abstract. A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X

Abstract Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

Abstract We give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings. We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that $\alpha$ dominates $\beta$ if there are a constant c and a partial computable function $\varphi$ such that for all positive rationals $q < \alpha$ we have $\varphi(q)\!\downarrow < \beta$ and $\beta- \varphi(q) \leqslant c(\alpha- q)$. The intuition is that an approximating sequence for $\alpha$generates one for $\beta$ whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if $\alpha$ dominates $\beta$, then the initial segment complexity of $\alpha$ is at least that of $\beta$. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random computably enumerable reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.

Abstract We construct the set of the title, answering a question of Cholak, Jockusch. and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of must contain a nonlow set.

Abstract We show that for every c.e. degreea>0there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of setsSthere exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements ofS. Then we show that ifα∈ω∪ {ω} then for anyα-c.e. degreea>0there exists an intrinsicallyα-c.e. relation on the domain of a computable structure whose degree spectrum {0, a}. All of these results also hold for m-degree spectra of relations.