Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem. Therefore, our construction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computably enumerable set. Finally, we relate the H-trivials to other classes of "highly nonrandom " reals that have been previously studied.

. We give a decision procedure for the 89-theory of the weak truthtable (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: A finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theor...

Downey and Lempp [8] have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5element lattice N5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with local nonmodularity in the computably enumerable degrees. 1 Introduction Lachlan [13] has shown that the two five-element nondistributive lattices, the modular lattice M 3 and the nonmodular lattice N 5 (see Figure 1 below) can be embedded into the upper semilattice (E; ) of the computably enumerable degrees. These lattices capture nondistributivity and nonmodularity in the sense that every nondistributive lattice contains one of these lattices as a sublattice and that every nonmo...

We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from Ambos-Spies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm [14], demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via grant 95-VIC-MIS-0698 under contract VIC-509. Some of these results were obtained whilst Downey was a Visiting Professor at Cornell University in fall 1995. decomposed into a pair of incomparible nonzero computa...

This paper will appear in the Proceedings of the Conference on Recursion Theory held at Oberwolfach, March 20--24, 1989.

this paper, we show that they can be, by proving that for every nonzero a 2 R, every countable distributive lattice can be embedded into R( a) preserving 1 (Theorem 18). The long gap between the appearance of [16] and our paper is perhaps explained by the fact that the techniques of Lachlan's paper have not been wellunderstood. Our first step in proving our result was to reprove the Lachlan Splitting Theorem using techniques which have been developed since Lachlan's paper appeared. In Section 2 of this paper, we give this reproof of the Splitting Theorem. In Section 3, we extend the techniques of Section 2 to show our result, and in Section 4, we discuss related results and open questions. In particular, in this section, we show how our results, together with some other recent and older results, provide a complete answer to all questions of the form Which finite distributive lattices can be embedded into A preserving B? where A can be:

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We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees. 1 Introduction One of the most basic and important questions concerning the structure of the upper semilattice R of recursively enumerable degrees is the embedding question: what (finite) lattices can be embedded as lattices into R? This question has a long and rich history. After the proof of the density theorem by Sacks [31], Shoenfield [32] made a conjecture, one consequence of which would be that no lattice embeddings into R were possible. Lachlan [21] and Yates [40] independently refuted Shoenfield's conjecture by proving that the 4 element boolean algebra could be embedded into R (even preserving 0). Using a little lattice representation theory, this result was subsequently extended by Lachlan-Lerman-Thomason [38], [36] who proved that all countable distributive lattices could be embedded (pre...

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