Kolmogorov complexity concerns the size of the shortest program that generates a given string.

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Kolmogorov complexity of a computer?

Warning: Vague, unclear question ahead. Proceed at your own risk. The Shannon entropy and Kolmogorov complexity give you in broad informal terms how unpredictable a string is and to what degree the ...
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Does a random binary sequence almost always have a finite number of prime prefixes?

Does a random binary sequence almost always have a finite number of prime prefixes? Specifically, let $x = \sum_{1 \le i}{2^{-i} \cdot x_i}$ with $x_i \in \{0,1\}$ be a random real in $[0,1)$, $X_i = ...
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Relationship between compression, shannon entropy and kolmogorov complexity

I have read that the Shannon Entropy is used as a bound for the compressibility of a message, for example here 1 it says "In other words, the best possible lossless compression rate is the entropy ...
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Kolmogorov complexity of an algorithm?

I've read that Kolmogorov comlexity is about calculating the least number of bits needed to describe a string or other mathematical objects. Does 'other mathematical objects' include algorithms too? ...
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Prove that bitstrings with 1/0-ratio different from 50/50 are compressable

I'm looking for a proof, that $$ \sum_{i=0}^{\lambda n} \binom{n}{i} \le 2^{nH(\lambda)} $$ with $n>0$, $0 \le \lambda \le 1/2$ and $ H(\lambda)=-[\lambda log \lambda + (1-\lambda) log (1-\lambda)] ...
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Prove that these Sets Containing Infinitely Many Incompressible Strings Exist

We define a set $A$ to be special if: $$\liminf_{n \to \infty} \frac{|A^{\leq n}|}{n} = 0$$ I want to prove that there are special recursive sets that contain infinitely many incompressible strings. ...
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Solomonoff induction , Shannon Entropy, Kolmogorov Complexity.

If Expected Kolmogorov Complexity equals Shannon Entropy why can't Shannon Entropy be used as an approximation of Kolmogorov Complexity in Solomonoff Induction? Regarding Kolmogorov Complexity and ...
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How much information is in the question “How much information is in this question?”?

I'm actually not sure where to pose this question, but we do have an Information Theory tag so this must be the place. The "simple" question is in the title: how do I know how many bits of information ...
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Kolmogorov complexity for infinite strings

I'm struggling with a problem that I believe I've managed to reduce to a question of Kolmogorov complexity for infinite strings, but since I'm not an expert in this field, I'm not sure about the ...
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A question on Kolmogorov Complexity

Is it true that for all strings of a given length at least one has its Kolmogorov complexity equal to its length ? Is there a proof if the answer is in affirmative? (For any alphabet with more than ...
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Kolmogorov (Kolmogoroff- ) Complexity of infinite sequences, Request for Proof

Let $\xi \in X^{\omega}$ be an infinite sequence and denote by $\xi[1\ldots n]$ its length $n$ initial segment. Then (due to Martin-Löf) the following holds: For every $\xi \in X^{\omega}$ there ...
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Kolmogorov (Kolmogoroff-) Complexity, Contradiction with Invariance Theorem.

Fix some programming languages $S$ which is rich enough such that one can write interpreters for $S$ in $S$. Define $$ K(w) := \mbox{length of a shortest program producing $w$}. $$ Now fix some ...
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Is it possible to create a string with known Kolmogorov Complexity?

I wish to compare compressors using strings with known Kolmogorov Complexity, but I haven't got the theoretical background and tools to understand how to do that. I'm just starting in this area and ...
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Kolmogorov complexity proof of Lovasz local lemma

Roughly speaking, Kolmogorv 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 $2^{k-c}$, for ...
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Definability of Kolmogorov Complexity?

This paper claims to have a proof of Godel's Second Incompleteness Theorem using Kolmogorov Complexity: http://www.ams.org/notices/201011/rtx101101454p.pdf As far as I can tell, it seems to assume ...
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Regularity of balanced binary strings

How can one tell which number of propositional variables is necessary to express a Boolean function given as a sequence of 0s and 1s (a binary string) of length $2^n$ as a Boolean formula? ...
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Measure of how much information is lost in an implication

In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$: ...
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Respective complexities of a string and its substring

If $s$ is a substring so that $s \subset S$, can Kolmogorov complexity of the whole string $S$ be lower than that of the given substring, $K(S) < K(s)$?
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Kolmogorov complexity of sequence and its fragment

Is it possible that part of sequence is more complex than all sequence because the best way to encode it is to use the complete sequence and starting and ending positions of the fragment. Maybe, for ...
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Is it really true that $K(x|y) = K(x,y) - K(y)$?

Denote by $y^*$ the shortest program computing the string $y$. In the main textbook and various papers of Li & Vitanyi, I have seen the following statements. The first is well established: the ...
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Solovay Randomness

Say that an $x\in 2^{\omega}$ is Solovay random if for all computably enumerable collections of intervals $\{I_n\}$ such that $\sum_n\mu(I_n)<\infty$, then $x\in I_n$ for at most finitely many $n$. ...
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Does the $k$th forward difference of Radó's $\Sigma$ eventually dominate every computable function?

Let $\Sigma$ be Radó's Busy Beaver function, and let $\Delta[\Sigma]$ denote the forward difference of $\Sigma$, such that $\Delta[\Sigma] \ (n) = \Sigma(n+1) - \Sigma(n)$ for all $n \in \mathbb{N}$. ...
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The minimal number of states required to run Goldbach's Conjecture

It is well known that being able to compute Busy Beaver numbers would allow one to solve (in theory) such open problems as Goldbach's conjecture. Simply run a Turing machine with $n$ states to check ...
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Proof that the set of incompressible strings is undecidable

I would like to see a proof or a sketch of a proof that the set of incompressible strings is undecidable. Definition: Let x be a string, we say that x is c-compressible if K(x) $\leq$ |x|-c. If x is ...
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Questions about the proof that minimal Turing machines are not recursively enumerable & proof that Kolmogorov complexity is uncomputable

This thread can be broken up into two questions. First I am trying to understand the proof that $MIN_{TM}$ is not recursively enumerable. If M is a Turing machine, then we say that the length of ...
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How to assess the complexity of a string

Q How do I quantify complexity of a given string? Consider a sample: ...
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Is there an infinite set of strings whose Kolmogorov complexities are computable?

Is there an infinite set of strings whose Kolmogorov complexities are computable?