Let us say that any (Turing) degree d > 0 satisfies the minimal complementation property (MCP) if for every degree 0 < a < d there exists a minimal degree b < d such that a ∨ b = d (and therefore a ∧ b = 0). We show that every degree d ≥ 0′ satisfies MCP. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

An incomplete degree is cuppable if it can be joined by an incomplete degree to a complete degree. For sets fulfilling some type of simplicity property one can now ask whether these sets are cuppable with respect to a certain type of reducibilities. Several such results are known. In this paper we settle all the remaining cases for the standard notions of simplicity and all the main strong reducibilities.

We give two new characterizations of K-triviality. We show that if for all Y such that Ω is Y-random, Ω is -random, then A is K-trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of K-triviality and answering a question of Yu. We also prove that if A is K-trivial, then for all Y such that Ω is Y-random, ≡LRY. This answers a question of Merkle and Yu. The other direction is immediate, so (...) we have the second characterization of K-triviality.The proof of the first characterization uses a new cupping result. We prove that if A≰LRB, then for every set X there is a B-random set Y such that X is computable from Y⊕A. (shrink)

Let ω be the set of natural numbers. For functions f, g: ω → ω, we say f is dominated by g if f < g for all but finitely many n ∈ ω. We consider the standard “fair coin” probability measure on the space 2ω of in-finite sequences of 0's and 1's. A Turing oracle B is said to be almost everywhere dominating if, for measure 1 many X ∈ 2ω, each function which is Turing computable from X is (...) dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i. e., 0″ is truth-table computable from B ′, the Turing jump of B. (shrink)

We show that given any (Turing) degree 0

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In this paper we initiate the study of the ω-Turing reducibility between sequences of sets of natural numbers. We shall prove that the induced degree structure is an extension of the structure of the Turing degrees and that the two structures are closely connected, but different enough. Further we shall prove some definability results for the local theory of the newly defined structure.

We give natural examples of factors of the Muchnik lattice which capture intuitionistic propositional logic , arising from the concepts of lowness, 1-genericity, hyperimmune-freeness and computable traceability. This provides a purely computational semantics for IPC.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

Let $\mathcal{P}_w$ be the lattice of Muchnik degrees of nonempty $\Pi^0_1$ subsets of $2^\omega$. The lattice $\mathcal{P}$ has been studied extensively in previous publications. In this note we prove that the lattice $\mathcal{P}$ is not Brouwerian.

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see (...) text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman. (shrink)

Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a non-recursive oracle. This paper combines these two types — it considers three basic models of queries to a teacher (QEX[Succ], QEX[ The results for each of these three models of query-inference are the same: If an oracle is omniscient for query-inference then it is already omniscient for EX. There is an oracle of trivial EX-degree, which allows nontrivial query-inference. Furthermore, (...) queries to a teacher cannot overcome differences between oracles and the query-inference degrees are a proper refinement of the EX-degrees. In the case of finite learning, the query-inference degrees coincide with the Turing degrees. Furthermore oracles can not close the gap between the different types of queries to a teacher. (shrink)

Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that the theory of every non-trivial factor of the Medvedev lattice is contained in Jankov’s logic, the deductive closure of IPC plus the weak law of the excluded middle $${\neg p \vee \neg \neg (...) p}$$. This answers a question by Sorbi and Terwijn. (shrink)

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ${\Delta^0_2}$ enumeration degree cannot have a ${\Delta^0_2}$ enumeration. We also prove that no subclass of the ${\Sigma^0_2}$ enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to ${\mathbf{0}_e'}$ by a single incomplete ${\Sigma^0_2}$ enumeration degree.

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.