# 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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### 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 ...
128 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 (...
35 views

### Computational complexity of optimization algorithms using random algorithm theory

A fundamental and undoubtedly much-studied problem is that of determining not only whether or not an optimization algorithm converges to its optimum but also how fast it converges (see a discussion on ...
32 views

### Is every productive set not Martin-Lof random？

every productive set is not Martin-Lof random，and so are some immune set.Therefore,do we have to have a definition for productive sets and such immune set which is different from Martin-Lof ...
63 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 ...
61 views

135 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(...
266 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 ...
111 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 ...
341 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 ...
94 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 ...
583 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 ...
150 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 ...
97 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 ...
413 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 ...
236 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 ...
206 views

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

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...
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 ...