Related papers: Kobayashi compressibility
We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther,…
The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional…
A proposition that connects randomness and compression is put forward via Gibbs entropy over set of measurement vectors associated with a compression process. The proposition states that a lossy compression process is equivalent to {\it…
We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and H\"olderian error bounds and includes…
Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them…
In this paper, we present the Monte-Carlo Compressive Optimization algorithm, a new method to solve a combinatorial optimization problem that is assumed compressible. The method relies on random queries to the objective function in order to…
We introduce and study a generalization of majorization called relative submajorization and show that it has many applications to the resource theories of thermodynamics, bipartite entanglement, and quantum coherence. In particular, we show…
We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time…
The randomness rate of an infinite binary sequence is characterized by the sequence of ratios between the Kolmogorov complexity and the length of the initial segments of the sequence. It is known that there is no uniform effective procedure…
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
Incomputability as a mathematical notion arose from work of Alan Turing and Alonzo Church in the 1930s. Like Turing himself, it attracted less attention than it deserved beyond the confines of mathematics. Today our experiences in computer…
The unwavering success of deep learning in the past decade led to the increasing prevalence of deep learning methods in various application fields. However, the downsides of deep learning, most prominently its lack of trustworthiness, may…
Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation…
We define several notions of a limit point on sequences with domain a barrier in $[\omega]^{<\omega}$ focusing on the two dimensional case $[\omega]^2$. By exploring some natural candidates, we show that countable compactness has a number…
The paper introduces a new lossless, highly robust compression algorithm that similar with LZW algorithm, yet the algorithm discards dictionary processing and uses irregular sequences with massive, random information instead. Then the paper…
Recent oracle separations [Kretschmer, TQC'21, Kretschmer et. al., STOC'23] have raised the tantalizing possibility of building quantum cryptography from sources of hardness that persist even if the polynomial hierarchy collapses. We…
Coherence is a fundamental notion in quantum mechanics, defined relative to a reference basis. As such, it does not necessarily reveal the locality of interactions nor takes into account the accessible operations in a composite quantum…
In computable analysis typically topological spaces with countable bases are considered. The Theorem of Kreitz-Weihrauch implies that the subbase representation of a second-countable $T_0$ space is admissible with respect to the topology…
Randomness plays a central rol in the quantum mechanical description of our interactions. We review the relationship between the violation of Bell inequalities, non signaling and randomness. We discuss the challenge in defining a random…
Solomonoff's uncomputable universal prediction scheme $\xi$ allows to predict the next symbol $x_k$ of a sequence $x_1...x_{k-1}$ for any Turing computable, but otherwise unknown, probabilistic environment $\mu$. This scheme will be…