Kolmogorov Complexity and Solovay Functions

Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality

Every recursive boolean algebra is isomorphic to one with incomplete atoms

AbstractThe theorem of the title is proven, solving an old question of Remmel. The method of proof uses an algebraic technique of Remmel-Vaught combined with a complex tree of strategies argument where the true path is needed to figure out the final isomorphism

The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

Lattice nonembeddings and intervals of the recursively enumerable degrees

AbstractLet b and c be r.e. Turing degrees such that b>c. We show that there is an r.e. degree a such that b>a>c and all lattices containing a critical triple, including the lattice M5, cannot be embedded into the interval [c, a]

Tabular degrees in \Ga-recursion theory

AbstractBailey, C. and R. Downey, Tabular degrees in \Ga-recursion theory, Annals of Pure and Applied Logic 55 (1992) 205–236.We introduce several generalizations of the truth-table and weak-truth-table reducibilities to \Ga-recursion theory. A number of examples are given of theorems that lift from \Gw-recursion theory, and of theorems that do not. In particular it is shown that the regular sets theorem fails and that not all natural generalizations of wtt are the same

Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a difference of left-$\Pi_2^0$ reals. Further, we show that the densities of the $\omega$-c.e.\ sets coincide with the densities of the $\Delta^0_2$ sets, and there are $\omega$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in \omega$.Comment: To appear in Mathematical Logic Quarterl

05301 Abstracts Collection -- Exact Algorithms and Fixed-Parameter Tractability

From 24.07.05 to 29.07.05, the Dagstuhl Seminar 05301 ``Exact Algorithms and Fixed-Parameter Tractability\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. This is a collection of abstracts of the presentations given during the seminar