Related papers: Universal computably enumerable sets and initial s…
In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free…
Given a countable structure $\mathcal{A}$, the degree spectrum of $\mathcal{A}$ is the set of all Turing degrees which can compute an isomorphic copy of $\mathcal{A}$. One of the major programs in computable structure theory is to determine…
We show that every countable ideal of degrees that are low for isomorphism is contained in a principal ideal of degrees that are low for isomorphism by adapting an exact pair construction. We further show that within the hyperimmune-free…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
The purpose of this paper is to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47--58]. Namely, we consider the…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $\omega$-power of the Sierpi\'nski space).…
We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that…
In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.
We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of…
The halting probabilities of universal prefix-free machines are universal for the class of reals with computably enumerable left cut (also known as left-c.e. reals), and coincide with the Martin-Loef random elements of this class. We study…
It is well known that the R, the set of real numbers, is an abstract set, where almost all its elements cannot be described in any finite language. We investigate possible approaches to what might be called an epi-constructionist approach…
We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show…
Schnorr showed that a real is Martin-Loef random if and only if all of its initial segments are incompressible with respect to prefix-free complexity. Fortnow and independently Nies, Stephan and Terwijn noticed that this statement remains…
We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this…
Every K-trivial set is computable from an incomplete Martin-L\"of random set, i.e., a Martin-L\"of random set that does not compute 0'.
Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct…
We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2log n upper bound on the Kolmogorov complexity of initial segments of r.e.\ sets is optimal and characterize…
The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…
It is easy to see that no n-REA set can form a (non-trivial) minimal pair with 0' and only slightly more difficult to observe that no {\omega}-REA set can form a (non-trivial) minimal pair with 0". Shore has asked whether this can be…