Skip to main content
MathOverflow

Questions tagged [algorithmic-randomness]

Ask Question

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.

43 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
6 votes
0 answers
130 views

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 ...
Peter Gerdes's user avatar
  • 4,039
1 vote
0 answers
54 views

I'm sure this idea isn't new. Suppose a computer program has to predict a binary time series by calculating the most probable continuation (this is a widely known "joke" if you want to show ...
Hauke Reddmann's user avatar
  • 4,859
0 votes
0 answers
139 views

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 ...
XM73's user avatar
  • 1
8 votes
0 answers
231 views

Computing generators of a Mordell curve $$y^2 = x^3 - 44275089430000,$$ can be done in Magma by running the following code: ...
Max Alekseyev's user avatar
  • 38.5k
4 votes
1 answer
202 views

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-...
Keshav Srinivasan's user avatar
  • 4,593
6 votes
1 answer
245 views

There is a plethora of notions of randomness for Cantor space ($\phantom{}^\omega 2$) (Schnorr randomness, Martin-Löf randomness, weak 2-randomness, the various forms of higher randomness such as $\...
Beau Madison Mount's user avatar
  • 1,165
2 votes
0 answers
112 views

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 ...
aroyc's user avatar
  • 221
1 vote
3 answers
379 views

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 &...
Rubi Shnol's user avatar
  • 801
2 votes
2 answers
160 views

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 ...
Thomas Benjamin's user avatar
  • 6,268
0 votes
0 answers
153 views

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 ...
shenyuantao's user avatar
  • 11
4 votes
1 answer
330 views

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 ...
AB_IM's user avatar
  • 4,842
3 votes
0 answers
155 views

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 ...
Florian Ente's user avatar
  • 183
0 votes
1 answer
106 views

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 ...
AB_IM's user avatar
  • 4,842
4 votes
1 answer
246 views

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 ...
Mathman's user avatar
  • 153
1 vote
0 answers
166 views

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 (...
Penelope Benenati's user avatar
  • 1,707
1 vote
1 answer
115 views

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 ...
Peter Gerdes's user avatar
  • 4,039
0 votes
1 answer
130 views

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^\...
Peter Gerdes's user avatar
  • 4,039
1 vote
0 answers
141 views

Is relationship between c.e.set, productive set, immune set, ML-random set similar to relationship between polynomial complexity set, polynomial complexity-productive set, P-immune set, P-random set?
XL _At_Here_There's user avatar
  • 3,316
4 votes
1 answer
761 views

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 ...
Lluís Alemany-Puig's user avatar
  • 275
0 votes
1 answer
155 views

$P$ means polynomial complexity. $S_p$ is class of all $P$_random sets, and $S_{pc}$ is class of all $P$ incomputable sets, is $S_{pc} \setminus S_p$ empty? If not empty, any example? what is the ...
XL _At_Here_There's user avatar
  • 3,316
3 votes
1 answer
341 views

$P$ 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 ...
XL _At_Here_There's user avatar
  • 3,316
1 vote
0 answers
130 views

Let $A$ be an invertible $N$-by-$N$ matrix, for some large $N$ (say $N = 10^6$). Suppose the only thing we know how to do is apply $A$ to a vector, i.e compute matrix-vector products $Az$. Question. ...
dohmatob's user avatar
  • 7,043
6 votes
0 answers
135 views

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 ...
Turbo's user avatar
  • 1
0 votes
0 answers
105 views

Take two integers $n$ and $m$ with $0<\log_2m<n<m$ and let $r_1=f_1(n)\bmod m$ and $r_2=f_2(n)\bmod m$ for functions $f_1,f_2:\mathbb Z\rightarrow\mathbb Z$. Denote the $\ell$ roots of $(f_i(...
Turbo's user avatar
  • 1
1 vote
1 answer
179 views

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(...
litmus's user avatar
  • 91
5 votes
3 answers
431 views

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 ...
Bogdan's user avatar
  • 841
5 votes
0 answers
165 views

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 ...
orangeskid's user avatar
  • 723
4 votes
1 answer
444 views

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 ...
XL _At_Here_There's user avatar
  • 3,316
0 votes
1 answer
163 views

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 ...
XL _At_Here_There's user avatar
  • 3,316
8 votes
1 answer
2k views

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 ...
XL _At_Here_There's user avatar
  • 3,316
2 votes
1 answer
203 views

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 ...
ARi's user avatar
  • 851
2 votes
0 answers
114 views

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 ...
Andrew Penland's user avatar
  • 413
3 votes
0 answers
509 views

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 ...
Payam Seraji's user avatar
  • 890
3 votes
0 answers
299 views

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 ...
Robin Saunders's user avatar
  • 3,818
9 votes
0 answers
301 views

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, ...
Bogdan's user avatar
  • 91
2 votes
1 answer
182 views

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 ...
Peva Blanchard's user avatar
  • 370
2 votes
2 answers
469 views

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 \...
Dan's user avatar
  • 1,358
2 votes
0 answers
59 views

I am not so expert in theoretical computer science, so sorry if the question is trivial, i just could not find it in literature. Suppose we have a source $X$ with min-entropy $\ell$, the randomness ...
math-Student's user avatar
  • 1,139
2 votes
0 answers
193 views

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 $2^{k-c}$, ...
tap1cse's user avatar
  • 69
3 votes
1 answer
146 views

An additive cost function is defined as $c: \omega\times \omega \to \mathbb{Q}_2$ such that it is recursive, monotonic (i.e. $c(x+1,y)\leq c(x,y)\leq c(x,y+1)$ and $c(x,y)=0$ whenever $x\geq y$, the ...
Jing Zhang's user avatar
  • 3,078
5 votes
3 answers
1k views

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 ...
Joseph Soulbringer's user avatar
  • 171
4 votes
0 answers
140 views

Is there a nice example of a sequence that looks random to any predictor whose predictions use a finite-state machine? More precisely, consider a finite-state machine $M$ with input alphabet {0,1} ...
James Propp's user avatar
  • 20.1k
9 votes
1 answer
444 views

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 ...
Jing Zhang's user avatar
  • 3,078