Related papers: On some computational properties of open sets

We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural…

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly…

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of…

We start by an introduction to the basic concepts of computability theory and the introduction of the concept of Turing machine and computation universality. Then se turn to the exploration of trade-offs between different measures of…

Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…

In this article we call a sequence $(a_n)_n$ of elements of a metric space nearly computably Cauchy if for every strictly increasing computable function $r:\mathbb{N}\to\mathbb{N}$ the sequence $(d(a_{r(n+1)},a_{r(n)}))_n$ converges…

We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types. This is achieved through G\"odel's functional interpretation, and…

We consider concept generalization at a large scale in the diverse and natural visual spectrum. Established computational modes (i.e., rule-based or similarity-based) are primarily studied isolated and focus on confined and abstract problem…

This talk describes how a combination of symbolic computation techniques with first-order theorem proving can be used for solving some challenges of automating program analysis, in particular for generating and proving properties about the…

Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning…

This work considers special types of interval linear systems - overdetermined systems. Simply said these systems have more equations than variables. The solution set of an interval linear system is a collection of all solutions of all…

We study the sets that are computable from both halves of some (Martin-L\"of) 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.\…

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first…

The aim of this paper is to define and study $\mathcal{B}$-open sets and related properties. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is in turn a generalization of a pre-open set and a…

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of…

In several domains, data objects can be decomposed into sets of simpler objects. It is then natural to represent each object as the set of its components or parts. Many conventional machine learning algorithms are unable to process this…

This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…

Algebraic characterizations of the computational aspects of functions defined over the real numbers provide very effective tool to understand what computability and complexity over the reals, and generally over continuous spaces, mean. This…

It is common practice to compare the computational power of different models of computation. For example, the recursive functions are strictly more powerful than the primitive recursive functions, because the latter are a proper subset of…