Related papers: On some computational properties of open sets
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
A method of representation of a solution as segments of the series in powers of the step of the independent variable is expanded for solving complex systems of ordinary differential equations (ODE): the Lorenz system and other systems. A…
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…
A major open problem in computational complexity is the existence of a one-way function, namely a function from strings to strings which is computationally easy to compute but hard to invert. Levin (2023) formulated the notion of one-way…
We describe arithmetic computations in terms of operations on some well known free algebras (S1S, S2S and ordered rooted binary trees) while emphasizing the common structure present in all them when seen as isomorphic with the set of…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of…
A common approach in neuroscience is to study neural representations as a means to understand a system -- increasingly, by relating the neural representations to the internal representations learned by computational models. However, a…
Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results…
We study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, G\"{o}del's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their…
Sequential data such as time series, video, or text can be challenging to analyse as the ordered structure gives rise to complex dependencies. At the heart of this is non-commutativity, in the sense that reordering the elements of 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 construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter.…
In this paper we consider two hierarchies of hereditarily total and continuous functionals over the reals based on one extensional and one intensional representation of real numbers, and we discuss under which asumptions these hierarchies…
The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological…
A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of…
Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2],…
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…
The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(\alpha^n x)^{}_{n\in\mathbb{N}}$, where $\alpha$ is a fixed real number with $| \alpha | > 1$ and…
We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We analyze numerically various ensembles of linear programming problems and obtain, for each of these…