The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, $\mathbb{Q}$-vector spaces, Archimedean real closed ordered fields, reduced Abelian $p$-groups of length less than $ω^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $Π_{n}^{0}$, $d$-$Σ_{n}^{0}$, or $Σ_{n}^{0}$, for various $n$. In each case, the calculation involves finding an \textquotedblleft optimal\textquotedblright% \ sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable $Π_{n}$, $d$-$Σ_{n}$, or $Σ_{n}$) yields a bound on the complexity of the index set. When we show $m$% -completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.
The notion of generic reducibility was introduced by A.Rybalov in his CiE 2018 paper: a set A is generically reducible to set B if there exists a total computable function f that m-reduces A to B such that the f-preimage of every set that has density 0 has density 0. It may be considered as the ``generic version'' of the notion of m-reducibility. In this note we improve one of his results and show that every two computable sets that do not have density 0 or 1 are equivalent with respect to generic reducibility, and that every computable set is reducible to every computable set that does not ha…
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 to mathematics. While most constructive mathematics is concerned with constructive proofs, the agenda here is that the objects that we study, specifically the class of numbers that we study, should be an enumerable set of finite symbol strings. These might also be called decidable constructive real numbers, that is our class of numbers should be a computable …
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is computable if its boundary $\partial M$ is computable. We also show how this result combined with certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.
A computable structure $\mathcal{A}$ is decidable if, given a formula $φ(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $φ$ holds of $\bar{a}$. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is $Σ^1_1$-complete. This result holds even if we restrict out attention to groups, graphs, or fields. We also show that the index sets of the computable structures with $n$-decidable pre…
In a previous paper, the author introduced the idea of intrinsic density --- a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high ($\mathbf{a}'\ge_{\rm T}\emptyset"$) or compute a diagonally non-computable function. By contrast, a classic construction of an immune set in every non-computable degree actually yields a set with intrinsic lower density 0 in every non-computable degree. We also show that the f…
Martin-Löf (ML)-reducibility compares $K$-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every $K$-trivial set is computable from a c.e.\ set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e.\ set ML-above the sets obeying it (called an ML-complete set for the cost function). We characterise the $K$-trivial se…
Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly.
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 finite rank. For both kinds of groups, the computable ones all have computable $Σ_3$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable $d$-$Σ_2$" (the conjunction of a computable $Σ_2$ sentence and a computable $Π_2$ sentence). This was already shown fo…
We study two complexity notions of groups - a computable Scott sentence and the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not the case sometimes. J. Knight et al. determined the complexity of index sets of various structures. In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups, generalizing methods that was previously used by J. Knight et al. We found computable Scott sentences for various different groups or class of groups, including nilpotent…
Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Π_{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $Π_{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear …
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number …
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this paper, what the influence of this extra data and structure is on the logical and computational properties of basic theorems pertaining to open sets. To answer this question, we study various basic theorems of analysis, like the Baire category, Heine, Heine-Borel, Urysohn, and Tietze theorems, all for open sets given by their (third-order) characteristic funct…
We study the sets that are computable from both halves of some (Martin-Löf) random sequence, which we call \emph{$1/2$-bases}. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e.\ elements. It is a proper subideal of the $K$-trivial sets. We characterise $1/2$-bases as the sets computable from both halves of Chaitin's $Ω$, and as the sets that obey the cost function $\mathbf c(x,s) = \sqrt{Ω_s - Ω_x}$. Generalising these results yields a dense hierarchy of subideals in the $K$-trivial degrees: For $k< n$, let $B_{k/n}$ be the collection o…
Intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate the intrinsic complexity of unary total recursive functions on nonnegative integers with standard order. According to existing results, possible spectra of such functions include three sets consisting of precisely: the computable degree, all c.e. degrees and all $Δ_2$ degrees. These results, however, fall far short of the full classification. In this paper, …
Kolmogorov complexities Kmax, Kmin on computable partially ordered sets
We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes Max^{X \to D}_{PR} and Max^{X \to D}_{Rec} of functions X \to D which are pointwise maximum of partial or total computable sequences of functions where D = (D,<) is some computable partially ordered set . The enumeration theorem and the invariance theorem always hold for Max^{X \to D}_{PR}, leading to a variant KD;max of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem …
We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $Δ^0_α$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $Δ^0_α$ bi-embeddable categoricity and relative $Δ^0_α$ bi-embeddable categoricity coincide for equivalence structures for $α=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddabl…
In this paper we develop a technique for proving determinacy of classes of the form $ω^2-Π^1_1+Γ$ (a refinement of the difference hierarchy on the co-analytic sets lying between $ω^2-Π^1_1$ and $(ω^2+1)-Π^1_1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes $ω^2-Π^1_1+Σ^0_α$ for all computable $α$ and of $ω^2-Π^1_1+Δ^1_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.
We examine sets of codes such that certain properties are invariant under the choice of oracle from a range of possible oracles and establish a connection between such codes and Medvedev reductions. In examing the complexity of such sets of \emph{universal codes}, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining $Π_1^1$-completeness for sets of universal codes. Among other corollaries, we show that the set of codes for Medvedev reductions of bi-immune sets to DNC functions is $Π_1^1$-complete.
A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computabi…
We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.
A set $I$ is said to be a universal indifferent set for $1$-genericity if for every $1$-generic $G$ and for all $X \subseteq I$, $G ΔX$ is also $1$-generic. Miller showed that there is no infinite universal indifferent set for $1$-genericity. We introduce two variants (union and subsequence sets for $1$-genericity) of the notion of universal indifference and prove that there are no non-trivial universal sets for $1$-genericity with respect to these notions. In contrast, we show that there is a non-computable subsequence set for weak-$1$-genericity.
We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.
