Related papers: Coarse Reducibility and Algorithmic Randomness
This paper is an extended version of our work in \cite{Ca2025}. We extend the concept of effective reducibility between statements of set theory with ordinal Turing machines (OTMs) explored in \cite{Ca2018} for $\Pi_{2}$-statements to…
When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…
We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{M}^{[0,1)}$ has a…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space…
We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment…
Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is…
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…
We define $\Psi$-autoreducible sets given an autoreduction procedure $\Psi$. Then, we show that for any $\Psi$, a measurable class of $\Psi$-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly…
A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness…
In contrast with the notion of complexity, a set $A$ is called anti-complex if the Kolmogorov complexity of the initial segments of $A$ chosen by a recursive function is always bounded by the identity function. We show that, as for…
An open question in reverse mathematics is whether the cohesive principle, $\COH$, is implied by the stable form of Ramsey's theorem for pairs, $\SRT^2_2$, in $\omega$-models of $\RCA$. One typical way of establishing this implication would…
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse…
Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable \emph{trivial} measures, where a measure $\mu$ on…
We introduce and study several notions of computability-theoretic reducibility between subsets of $\omega$ that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be…
One of the common obstacles for learning causal models from data is that high-order conditional independence (CI) relationships between random variables are difficult to estimate. Since CI tests with conditioning sets of low order can be…
Kobayashi introduced a uniform notion of compressibility of infinite binary sequences in terms of relative Turing computations with sub-identity use of the oracle. Kobayashi compressibility has remained a relatively obscure notion, with the…
Given partial information about a set, we are interested in fully recovering the original set from what is given. If a set encodes itself robustly, any partial information about the set suffices to fully recover the information about the…
We introduce several notions of reduction in distributed computing, and investigate reduction properties of two fundamental agreement tasks, namely Consensus and Atomic Commitment. We first propose the notion of reduction "a la Karp'', an…
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…