# Questions tagged [algorithmic-randomness]

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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### Is identifying the best in randomly chosen $n$ elements, equivalent to identifying one from the best half of randomly chosen $2n$ elements?

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 ...
1 vote
149 views

### Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem. I am curious if someone developed kind of &...
86 views

### Are finite subsets of bi-immune sets random?

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 ...
134 views

### Use $n$ different integers with $+$ or $-$ to make equations, find out the unused numbers

I am just a normal Chinese student and I can't communicate well with English. The question is there are $n$ different integers, we can use $+$ or $-$ to make equations to let the result be zero and in ...
172 views

### Bi-Hölder embeddings of finite metric spaces

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 ...
96 views

### Applications of products of random matrices

I'm studying the paper "Matrix concentration for products" and I'm trying to find simple applications of the inequalities for the expected value of the spectral norm of products of random ...
65 views

### Algorithm for economically sampling method for Gaussian matrix product

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 ...
198 views

### Expectation of multiplied random variables given their individual expectations

Suppose that I have two non-negative real valued random variables $x, y \in Z_+$ that always satisfy $$x+y \leq 1.$$ Also suppose that $E[x] = 1/2$ and $E[y] = 1/4$. What is the maximum possible value ...
1 vote
138 views

### Algorithmic combinatorial discrete problem (randomized lazy update?)

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 (...
1 vote
73 views

### Non-relativized, Computable and Schnor randomness w.r.t a measure

Riemann and Slaman have some great work classifying what reals are 1-random with respect to a measure $\mu$ relative to $\mu$. In that paper they cite Levin and Kautz (but not to refs I can find) for ...
70 views

1 vote
148 views

### Find probability of non-stationary inputs into Turing machine?

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(...
299 views

### Example of concrete statement which requires probabilistic algorithm

In my paper I would like to include an example of easy concrete finitary statement which can be easily verified by probabilistic algorithm to any reasonable confidence level, but which looks ...
121 views

### Subsequence of a sequence of digits of an algebraic irrational number

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 ...
364 views

### Existence of normal number except random numbers

For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence. Now, is there any number that is normal ...
107 views

### Randomness defined in Kolmogorov complexity is identified with one in probability theory/stochastic process?

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 ...
1k views

### What is the precise relation between Kolmogorov complexity and Shannon's entropy?

Consider the discrete case: Shannon's entropy is $H(x)=-\sum\limits_i^n p(x_i) log\space p(x_i)$. Probability based on prefix-free Kolmogorov complexity is $R(x_i)=2^{-K(x_i)}$ where $K(x_i)$ is ...
164 views

### Generating an arbitrarily long sequence with decreasing Kolmogorov complexity of terms

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 ...
98 views

### Non-uniform distribution in digits of chaotic orbits?

For a class I was teaching, I was demonstrating time-series analysis with the famous discrete-time dynamical system $x_{n+1} = ax_n(1-x_n)$, where $a = 4 - \epsilon$ ($\epsilon$ small'') and ...
444 views

### A question about the Chaitin constant of a theory

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 ...
244 views

### "Kolmogorov complexity" of models of computation

This question was partly inspired by my learning about John Tromp's binary lambda calculus and similar minimal languages such as Jot. A more detailed discussion of some of these ideas is in Michael ...
263 views

### Complete list of exceptions and efficient algorithm for Waring's problem

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, ...
As presented in Oxtoby's book ( http://link.springer.com/book/10.1007%2F978-1-4615-9964-7 ), there are two notions of largeness for subspace $Y$ of a given space $X$: Topology: $X$ is a topological ...