## What is the name for a topological space whose Kolmogorov quotient is discrete?

What is the name for a topological space whose Kolmogorov quotient is discrete?

Are these the almost discrete topological spaces?

## Can a Kolmogorov complexity oracle solve halting problem?

I can find https://www.nearly42.org/cstheory/halting_to_kolmogorov/ but the halting problem part is unrelated to proof that kolmogorov can’t be solved(still that assume it solvable and output a longer uncompressable string)

## Is the Kolmogorov Complexity of 11…1 with even length L less than for the string 1010…10 of the same length?

We define the Kolmogorov Complexity to be independent of any

particular programming language for bit string x as the length of the shortest

string <M,w> where TM M on input w halts with x on its tape

and <M,w> is some specified fixed encoding of the pair {M,w}.

## turing machines – A property of Kolmogorov random strings

I am working on the following problem:

Prove that, for all $$kinmathbb N$$, there exists $$ninmathbb N$$ so that every binary string $$xin{0,1}^{kn}$$ with Kolmogorov complexity $$K(x)$$ at least $$kn$$ satisfies the following property:

By interpreting $$x$$ as $$x_1cdots x_n$$, with $$|x_i|=k$$, for any $$zin{0,1}^k$$ there is an index $$i$$ for which $$x_i=z$$.

To show this, I want to suppose that if there exists a string $$x$$ of length $$kn$$ so that, after writing $$x=x_1cdots x_n$$, not all $$k$$-bit binary strings appear in $$x_1cdots x_n$$. Then ideally I want to show that $$x$$ can be described in less than $$kn$$ bits, which means $$x$$ have Kolmogorov complexity less than $$kn$$, which establishes the contradiction I want.

Any hints on how to do this?

For completeness, the Kolmogorov complexity of a string $$x$$ is defined as the length of the shortest description of $$x$$. And by the description of a string $$x$$, I refer to a pair $$(M,w)$$, where $$M$$ is a Turing Machine and $$w$$ is some string, so that $$M$$ halts on $$w$$ as input, leaving behind $$x$$ on the tape. I encode the pair $$(M,w)$$ as $$0^{|M|}1Mw$$. Then, for any string $$x$$, if $$M$$ is a Turing Machine that halts immediately upon execution, then $$(M,x)$$ is a description for $$x$$, of length $$2|M|+|x|+1$$. Hence the Kolmogorov complexity of $$x$$ does not exceed $$2|M|+|x|+1$$.

## lo.logic – The connections between Kolmogorov complexity and mathematical logic

We know that Kolmogorov Cmplexity (KC) has connections to mathematical logic since it can be used to prove the Gödel incompleteness results (Chaitin’s Theorem and Kritchman-Raz). Are there any other striking application of Kolmogorov complexity to mathematical logic (outside KC itself of course)?

Relatively simple examples like the ones I mentioned are preferred, but more complicated illustrations are also very welcome!

(This is cross posted from MSE, because I wasn’t getting any useful replies there)

## reference request – Equivalence relation induced by Kolmogorov quotients

Recall: given a (possibly non-$$T_0$$) topological space $$X$$, its Kolmogorov quotient $$KX$$ is the $$T_0$$ topological space formed by $$X/sim$$ where $$xsim y$$ if they are topologically indistinguishable. Denote the mapping $$Xto KX$$ of $$x$$ to its equivalence class $$pi$$.

I have two loosely related terminology questions:

1. Is it okay to use the word “section” to refer to a mapping (and/or the image of such a mapping) $$gamma: KX to X$$ such that $$picirc gamma = id$$? (This would be the word from category theory, just wondering if there is another established terminology that is used.)

2. Is there a word for the equivalence relationship where two topological spaces $$X$$ and $$Y$$ are said to be equivalent if their quotients $$KX$$ and $$KY$$ are homeomorphic?

(I am particularly interested in the case where the non-$$T_0$$ topology comes from a semi-norm or a pseudo-metric; so if there is an answer to 2 when restricted to semi-normed spaces or pseudometric spaces, I’d be happy too.)

## turing machines – What are the three points of view in Kolmogorov Complexity?

I was reviewing for my finals and find this question that I have totally no clue.

Compare the following to statements from three points of view:

1. There exists a constant $$c > 0$$ such that for all palindromes $$x in {0, 1}^*$$ we have $$K(x) leq lfloor x / 2 rfloor + c$$.

2. There exists a constant $$c > 0$$ such that for all $$x in {0, 1}^*$$ we have $$K(overline{x}) leq K(x) + c$$ where $$overline{x}$$ is the complement of $$x$$.

So what are the three points of view am I suppose to use and where should I start?

## Kolmogorov Kolmogorov complexity of string concatenation

For all bit strings x, y and complexity of Kolmogorov K, is

K (xy)> K (x)?

## Kolmogorov complexity of \$ y \$ given \$ x = yz \$ with \$ K (x) geq ell (x) – O (1) \$

I am trying to solve Exercise 2.2.2 of "An Introduction to the Kolmogorov Complexity and Its Applications" (Li & Vitányi, Vol 3). The exercise is the following (paraphrased):

Let $$x$$ satisfied $$K (x) geq n – O (1)$$, or $$n = ell (x)$$ is the length of $$x$$ in a binary encoding. CA watch $$K (y) geq frac {n} {2} – O (1)$$ for $$x = yz$$ with $$ell (y) = ell (z)$$.

I have tried but I have failed to apply the method of incompressibility, exploiting the fact that there are many $$x$$ in a way. More direct approaches to trying to find intelligent recursive functions of $$y$$ and $$z$$ also did not work.

## real analysis – Kolmogorov overlay on the Hilbert cube

A result of Kolmogorov and Arnold says that continuous functions on $$mathbb {R} ^ n$$ can be represented as sums of the form

$$f (x_1, dots, x_n) = sum_ {q = 0} ^ {2n} Phi_q left ( sum_ {p = 1} ^ n phi_ {p, q} (x_p) right) ,$$

or $$Phi_p$$ and $$phi_ {p, q}$$ are unary continuous functions.

I am curious to know the similar results obtained for the functions of the Hilbert cube.

Suppose I have a continuous function $$f:[0,1]^ Omega rightarrow [0,1]$$. Is it still possible to write this in the form

$$f (x_0, x_1, dots) = sum_ {q < omega} Phi_q left ( sum_ {p < omega} phi_ {p, q} (x_p) right),$$

or $$Phi_p$$ and $$phi_ {p, q}$$ the continuous functions and all the sums converge uniformly? If this fails, are there any known similar results?