Related papers: On the computability of ordered fields
We prove that there exists a $\Sigma^0_1$ closed subset of $[0,1]$ that is not homeomorphic to any computably compact space. We show that the index set of c.e. subspaces of $[0,1]$ that admit a computably compact presentation is not…
We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for…
Implicit computational complexity is a lively area of theoretical computer science, which aims to provide machine-independent characterizations of relevant complexity classes. % for uniformity with subsequent uses >> 1960s (but feel free to…
We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the…
We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we…
We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of…
The class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games,…
The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these…
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…
The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for…
We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures…
Building on our prior work on axiomatization of exact real computation by formalizing nondeterministic first-order partial computations over real and complex numbers in a constructive dependent type theory, we present a framework for…
A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…
It is well known that there is no basis of the field for real numbers regarded as a vector space over any proper subfield that is closed under multiplication. Mabry has extended this result to bases of arbitrary proper field extensions. The…
The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper we consider the related problems of computing all minimal overorders…