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Martin-Löf randomness and other randomness notions arising from computable tests; as well as related concepts such as Kolmogorov complexity, K-triviality, and effective Hausdorff dimension.

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It is a result of Levin and (independently?) Kautz that if $X \in 2^\omega$ is 1-ML random relative to a computable measure $\mu$ the either $X$ is computable (it is an atom of $\mu$) or $X$ is Turing ...
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Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
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Let $K(x)$ denote the Kolmogorov complexity of a finite binary string $x$. A finite binary string $x$ is called Kolmogorov random if $K(x) \geq |x|$. And an infinite binary sequence is called Martin-...
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Suppose we are given a set $U$, and a black-box objective function $f: U \mapsto [0, 1]$. The job is to maximise $f(\cdot)$. Now, for a given $\delta \in (0,1)$, consider the following randomised ...
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Let $A$ be a bi-immune set, that is, an immune set whose complement is also immune. An immune set (let's say $A$) is a set of natural numbers (the natural numbers include 0) such that: i. $A$ is ...
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This is a reference request. There is a large body of work, I'm familiar with, that describes the existence of bi-Hölder embeddings of finite metric spaces into Euclidean space (such as this ...
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Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$. I would ...
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We are given a vector $\mathbf{b}$ of size $h$. Initially we have $\mathbf{b}_i=1$ for all $i\in \{1, 2, \ldots, h\}$. In a sequential fashion, at each time step $t=1, \ldots, n$, an index $j(t)$ is (...
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Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\...
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I am implementing an algorithm to generate free unlabelled trees uniformly at random (uar). For this I found this paper by Herbert S. Wilf (The uniform selection of trees. 1981. In Journal of ...
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$P$ means polynomial complexity. $S_p$ is the class of all $P$_random set, and $S_{pc}$ is the class of all $P$ noncomputable sets, is $S_p \bigcap S_{pc}$ empty? If not empty, any example? what is ...
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Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate $P=BPP$ in $1986$. They do this without getting into circuit lower bounds and from a different view ...
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Consider some finite string $x=(x_1,x_2,...,x_{n-1},x_n)$ that is drawn from a non-stationary process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as, $$ P_M(...
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Consider an algebraic irrational number in $(0,1)$ with binary expansion $x = \sum_{n\ge 1} \frac{a_n}{2^n}$. Is it possible that the number $\sum_{n\ge 1}\frac{a_{2n}}{2^n}$ is again algebraic ...
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Actually, in many works of probability theory/stochastic process, there is no explicit definition of randomness. Maybe because we think we can deduce the definition easily. But in Kolmogorov ...
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Is there an algorithm which, given a string $s$, generates a sequence of $|s|$ strings, such that it can be proven in some axiomatic system $S$, that the Kolmogorov complexity of each successive ...
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Chaitin's famous incompleteness theorem states that for every r.e. theory $T\supseteq Q$ in the language of arithmetic, there is a constant $d_T$ such that for any $m\geq d_T$ and any $x$, $T$ can ...
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2 weeks ago, Samir Siksek https://arxiv.org/abs/1505.00647 proved the more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, ...
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Let's call an infinite sequence of bits $f:N\rightarrow \{0,1\}$ absolutely random if any computably constructed subsequence is not computable, i.e. there aren't monotonic computable function $g:N \...
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Roughly speaking, the Kolmogorov Complexity proof of Lovasz local lemma states that for any $k$-CNF $S$ on $n$ variables and $m$ clauses, where the dependency of every clause is bounded by $2^{k-c}$, ...
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Has anyone studied an equivalent to algorithmic complexity for probability distributions? This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...
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A set A is cohesive if $A\subseteq ^* W_e$ or $A\subseteq^* \bar{W_e}$ for each $e\in \omega$ (standard enumeration of r.e. sets). By Jockusch and Stephan's 1993 paper 'A cohesive set which is not ...
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