Questions tagged [kolmogorov-complexity]
The Kolmogorov complexity of a string s is equal to the length of the shortest program computing s and halting. Measures the lack of structure in a string.
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On proving the uncomputability of Kolmogorov complexity by contradiction
I have seen a proof by contradiction for the uncomputability of Kolmogorov complexity. The idea the basically the same as in the proof for halting problem (i.e., there are cases that lead to Berry ...
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Time bounded Kolmogorov complexity and one way functions
I recently read the following article https://www.quantamagazine.org/researchers-identify-master-problem-underlying-all-cryptography-20220406/ which links to https://arxiv.org/abs/2009.11514 that ...
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Kolmogorov Complexity: Is there a Program P which outputs Kolmogorov string K, P doesn't contain K and P is longer than K?
Given a binary string, K, with length N, and Kolmogorov Complexity N, is there a program P, of length M, and with Kolmogorov Complexity M such that:
P outputs K.
M >= N.
P does not contain K.
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Chaitin’s version of Gödel’s theorem and pseudorandomness
Chaitin’s version of Gödel’s theorem roughly states that there exists a constant c such that for each string of one’s and zeroes x, the sentence “the algorithmic information complexity (Kolmogorov ...
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In Kolmogorov's $ K(s) \leq |s| + c $, why is $c$ +ve?
Kolmogorov said "It seems natural to call a chain random if it cannot be written down in a more condensed form, i.e., if the shortest program for generating it is as long as the chain itself. &...
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Storing N bits on the smallest possible space in a real computer
Update. Since my original question was misunderstood by many, and lead to a lot of debate about various issues, let me try to pose this modified and rephrased question:
Assume that I have a computer ...
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Solomonoff induction generates sequences of unbounded Kolmogorov complexity?
I was reading this paper, and the author (in private correspondence, though I'm sure it's also in the paper) explains that a bit string generated through Solomonoff induction will have growing (and ...
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is Kolmogorov complexity computable on a finite domain?
The proof in the wikipedia article for the uncomputability of Kolmogorov complexity uses the fact that there are strings of arbitrarily large Kolmogorov complexity. What if we restrict to a finite ...
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Is there a known relationship between Kolmogorov Complexity of a binary string and the logic optimization of the corresponding Boolean function?
I haven't thought about how to go about proving it or finding a counterexample (I probably don't have the right background), but it seems intuitive to me that, given some representation of a Boolean ...
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Small Turing machine accepting single complicated input?
This imprecise question is about a simple example for the following problem.
I would like a Turing machine with few states that accepts only inputs which look complicated to the naked eye.
Of course ...
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kolmogorov complexity for finite Language?
In lectures my professor proved that there is no Turing machine that for every x it calculates k(x).
On the other hand, I saw a claim online that for finite language L there is a Turing machine that ...
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What is the most random Markov martingale?
Consider a boxing match between players A and B. If A wins, then $1$ point is gained; otherwise $1$ point is lost. Assume that $A$ is as same good as B. Denote by $X_t$ the expected point of A at time ...
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Kolmogorov complexity of probability distributions, with and without time bound
Kolmogorov complexity of a string is the length of the shortest algorithm that generates it. Here I'm focusing instead on randomly distributed strings $x$, with length $n$, with a probability ...
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Algorithmic information theory with stochastic algorithms?
Suppose we define a class of algorithms that is allowed to sample i.i.d. Bernoulli bitstrings of arbitrary length, and use these to generate outputs. If we are allowed to use algorithms like this, ...
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Turing machine that checks whether a given string is an output of a given machine and input
Is there a Turing machine such that, given a description $\langle M \rangle$ of a Turing machine $M$, an input $x$ and a string $y$, computes whether or not $y$ is the output of $M$ input $x$?
My ...
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How many strings of length |w| are unrelated?
For sufficiently large |w|, how many of the 2^|w| strings of length |w| are entirely unrelated?
A way to define this: two strings are unrelated if their joint Kolomogorov complexity is practically ...
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A property of Kolmogorov random strings
I am working on the following problem:
Prove that, for all $k\in\mathbb N$, there exists $n\in\mathbb N$ so that every binary string $x\in\{0,1\}^{kn}$ with Kolmogorov complexity $K(x)$ at least $kn$ ...
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Expressivity of neural networks, how much information can be stored
I want to know whether a given neural network (with a finite number of nodes) is able to store all injective maps f: D -> C, where D has cardinality k and C has cardinality N (so the number of maps ...
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How to define enumeration of the set of finite state machines?
I want to write a function that takes N (maximum number of states) as a parameter, enumerates all possible finite state machines up to N states, and returns random FSM with a probability in proportion ...
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Computability of Kolmogorov Complexity of Turing-Incomplete language
I am trying to determine whether Kolmogorov complexity is computable for a specific language. I am certain this given language is not Turing-Complete. The language is defined as follows:
$A;B \text{ - ...
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Kolmogorov complexity of a product of two numbers
In his book "Theoretical Computer Science", Juraj Hromkovic informally defines the Kolmogorov complexity $K(x)$ of a word $x$ consisting of zeros and ones as the binary length of the shortest Pascal ...
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Why don't prefix-free Turing machines suffer from complexity dips?
It's claimed in several texts on algorithmic complexity that prefix-free Turing machines are better for understanding randomness, at least in infinite sequences. In Nies' Computability and Randomness,...
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Kolmogorov Complexity, struggle with equation
I am working on understanding Kolmogorov Complexity. I've been struggling with the following exercise for quite some time now and would appreciate any inputs.
Show that there exists a constant $c$, ...
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Kolmogorov Complexity of String Concatenation
For all bit strings $x$, $y$ and Kolmogorov complexity $K$, is $K(xy) > K(x)$?
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How many "compressible" strings are there?
Let's say that a string of length $N$ is "compressible" iff its Kolmogorov complexity is less than $N$. To keep it simple, we can assume binary strings for this.
It is easy to see that almost all ...
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How connected are information theory and algorithmic information theory?
In the book by Cover and Thomas on information theory, there is a chapter on algorithmic information theory (kolmogorov complexity and so forth).
As far as I understand, algorithmic information ...
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On teaching Kolmogorov complexity with Python and the complexity of composed strings
The setting of this question is a bit long-winded, but please bear with me.
This fall I will be lecturing a course on mathematical information theory, and on a few lectures we will be discussing ...
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How can I prove the languages of incompressible words is undecidable?
I have hard time understanding the proof by contradiction for the claim "$L=\{x : K(x) \ge |x| \}$" is undecidable ".
The proof is as follows :
M' = " On input $n$
Enumerate over all $n$-...
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Examples of exact computation of Kolmogorov complexity?
First question: It is known that Kolmogorov Complexity (KC) is not computable (systematically). I would like to know if there are any "real-world" examples-applications where the KC has been computed ...
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Range of values for Kolmogorov complexity
Let $n$ be a positive integer. Is it true that for all $1\leq i \leq n$ there exists a length $n$ binary string $w$ such that $K(w) = i$. Where $K(w)$ is the Kolmogorov complexity of $w$.
For each ...
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Kolmogorov randomness for Pseudo random number generator
I am working on pseudo random number generation for one of my projects. My goal is to prove that the output is almost Kolmogorov Random since Kolmogorov complexity is uncomputable.
So would appreciate ...
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What doest it mean: "computer is an intelligence amplifier"?
There is one example in Kolmogorov complexity books and related articles:
Consider we have a monkey at a typewriter and a monkey at a computer keyboard.
If the monkey types at random on a typewriter,...
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Computability of Kolmogorov complexity in total languages
It is well known that the Kolmogorov complexity is uncomputable in Turing-complete programming languages. However, what about total programming languages?
For example, is the Kolmogorov complexity of ...
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How do you resolve this paradox with the invariance theorem?
The invariance theorem of kolmogorov complexity states that for two different languages with complexity functions $K_1$ and $K_2$, we have
$$\exists c.\forall s. K_1(s) \le K_2(s) + c$$
Here is an ...
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Is there a hierarchy theory for time-bounded Kolmogorov complexity?
We know that there are languages in $DTIME(n^t)$ and not in $DTIME(n^s)$ for all $t>s$ due to simple diagonal arguments (i.e., the Time Hierarchy Theorem), but I'm wondering if there is any ...
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How can Kolmogorov complexity help me practically with measuring entropy?
A comment was made to me saying the following in relation to Kolmogorov complexity:-
You're not the first to think non-computability = impractical or even useless. But it can be useful. In particular ...
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VC Dimension of A Set of Hypothesis
I am confused about what does a VC dimension of a set of hypothesis means.
I have two hypothesis, say $H_1$ with VC dimension of $x$, and $H_2$ of VC dimension of $y$. Does this automatically mean ...
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Perturbative Kolmogorov Complexity Bounds
Are there any known bounds on the impact of changing (for example) one bit in a string on the resulting string's Kolmogorov Complexity? In mathematical terms, does the equation $|K(x) - K(x')|$ (with $...
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Solomonoff's theory of induction, Kolmogorov complexity and Bayesian Inference
My motivations for asking this question are philosophical in nature. I'm by no means a computer scientist though, and I feel as though this question should be answered by someone who is since it's one ...
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Kolmogorov complexity of a sequence of n bits with k ones
Let Program P be :"Generate, in lexicographic order, all sequences with k ones and n bit length;
Of these sequences, print the ith sequence."
Apparently the length of this program is $\log(n) + \log(\...
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Is there a universal metric of "size of a program"?
There is a universal metric of information: amount of bits. It's universal in the sense that if we write a piece of information in DNA (4-ary digits), we can simply multiply by 2-log-4 to get the ...
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Prove that A is non-regular using K-Complexity Non regularity theorem
Given $Y^A_{x,n}$= the nth string $y∈Σ^∗$ (in lex order) such that $xy∈A$ (if n such y exits). So what completes $x$ if adding $n$ such $y$'s brings us to an element in the set $A$
Given $A \subseteq ...
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Non Regularity proof using Kolmogorov Complexity (Li - Vitanyi Theorem)
When proving a language is non regular we can use Kolmogorov complexity.
As far I know to do this we just have to use this satisfy the following conditions
Given $Y^A_{x,n}$= the nth string $y∈Σ^∗$ (...
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Kolmogorov complexity of prefixes of computable sequences
Let the characteristic sequence of a set $A ⊆ \mathbb{Z^+}$ be the following infinite binary sequence:
$$χ_A = b_1b_2b_3\ldots,$$
whose $n$th bit is 1 if $n ∈ A$ And 0 otherwise.
Write $χ_{A,n}$ for ...
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Kolmogorov Complexity proving there exists a constant for when if two strings are equal length
When talking about kolmogorov complexity, I understand that it describes true randomness of given (for now) a string $x$, if we can describe x in less than the $|x|$ then its complexity is said to be
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Proof: Kolmogorov complexity of string concatenation
Does there exist a universal constant $c$ such that for any strings $x, y$, we have: $$K(xy)\leq K(x) + K(y) + c$$
where $K(\cdot)$ denotes the Kolmogorov Complexity of a binary string and $xy$ means ...
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United space-time complexity of finite strings
Let's consider bit string as a program for some computational model. If after $k$ steps program represented by number $n$ halts and outputs bit string $s$, then complexity of s is (n+1)*k. For example ...
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What is static complexity?
Definition : Kolmogorov complexity is a static complexity measure that captures the difficulty of describing a string.
For example, the string consisting of three million zeroes can be described with ...
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How to calculate Kolmogorov Complexity if we have access to an Oracle for the HALT Problem
I try to solve the following exercise:
We know that K (x), the complexity of Kolmogorov, is incomputable. Show how
calculate it, if we have an oracle for the membership problem (or for the HALT ...
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What Good Is Kolmogorov Complexity Since It Is Relative?
Kolmogorov complexity is relative to a choice of Universal Turing Machine. Because of the Invariance Theorem, the difference in complexity assigned by two Universal Turing Machines is bounded by a ...
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