Related papers: The Rice-Shapiro theorem in Computable Topology
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and…
In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems:…
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…
In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
We provide a detailed study of two properties of spaces and pairs of spaces, the surjection property and the epsilon-surjection property, that were recently introduced to characterize the notion of computable type arising from computability…
A computable structure $\mathcal{A}$ is decidable if, given a formula $\varphi(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $\varphi$ holds of $\bar{a}$. We…
Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented…
Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.
The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a…
We investigate conditions under which a co-computably enumerable set in a computable metric space is computable. Using higher-dimensional chains and spherical chains we prove that in each computable metric space which is locally computable…
We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the…
There are a variety of results in the literature proving forms of computability for topological entropy and pressure on subshifts. In this work, we prove two quite general results, showing that topological pressure is always computable from…
When can a model of a physical system be regarded as computable? We provide the definition of a computable physical model to answer this question. The connection between our definition and Kreisel's notion of a mechanistic theory is…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction…
The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, then any…
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…