What is the name for a topological space whose Kolmogorov quotient is discrete?
Are these the almost discrete topological spaces?
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What is the name for a topological space whose Kolmogorov quotient is discrete?
Are these the almost discrete topological spaces?
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)
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}.
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 $2|M|+|x|+1$.
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)
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:
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.)
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.)
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:
There exists a constant $c > 0$ such that for all palindromes $x in {0, 1}^*$ we have $K(x) leq lfloor x / 2 rfloor + c$.
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?
For all bit strings x, y and complexity of Kolmogorov K, is
K (xy)> K (x)?
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.
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?