Questions tagged [kolmogorov-complexity]
Kolmogorov complexity concerns the size of the shortest program that generates a given string.
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Intuition of Kolmogorov dimension in metric space
What is the intuition of Kolmogorov dimension as defined below?
Let $N(\mathcal{F}, \alpha, \|\cdot\|_2)$ be the $\alpha$-covering number of $\mathcal{F}$ with respect to the $\|\cdot\|_2$-norm. ...
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Formalization in PA in the Kritchman-Raz proof
In their paper Kritchman and Raz present a proof of Gödel's second theorem using Kolmogorov complexity. To make it work, they operate in some (weak) formal theory $T$ that incorporates some arithmetic,...
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Recursion theoretic definition of Kolmogorov complexity
In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem the Kolmogorov Complexity (KC) of $x$ is defined s
$$ K(x) = \mu e (\varphi_e(0) \simeq x) \, . $$
This seems to give ...
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How to universalize $\text{Prov}(\ulcorner y < K(x)\urcorner) \to y < K(x)$ in a paper of Kikuchi
In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem he defines for $\Sigma_1$ binary predicates $R(x, y)$ the condition
$$ \Gamma_{1}(R) \Leftrightarrow \forall x\forall y(R(...
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The philosophical significance of Chaitin's Theorem
In a book review of Torkel Franzén's "Gödel’s Theorem:
An Incomplete Guide to Its Use and Abuse" in the Notices the reviewer (Raatikainen) writes:
Franzén also devotes a brief chapter to ...
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The connection between Kolmogorov complexity and mathematical logic
We know that Kolmogorov complexity 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 ...
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question related to Kolmorogov Smirnov statistic.
Question related to Kolmogorov Smirnov statistics:
If $F_n$ is the empirical distribution function for $n$ IID random variables with an unknown distribution function $F$, what does the random function ...
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The use of soundness in the Kritchman-Raz proof and Berry's paradox
In the Kritchman-Raz paper the authors recall Chaitin's proof of a version of the first incompleteness theorem (italics are mine):
Chaitin’s incompleteness theorem states that for any rich enough
...
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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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A consequence of Chaitin's incompleteness theorem
According to Wikipedia due to Chaitin's incompleteness theorem, the output of any program computing a lower bound of the Kolmogorov complexity cannot exceed some fixed limit, which is independent of ...
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Proving two different universal machine types give equivalent results in original Solomonoff induction paper
Solomonoff's original paper about Solomonoff induction contains the following (p. 18):
Suppose $M$ to be a universal machine with binary input alphabet, and an output alphabet that is the same as ...
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Martin-Löf randomness tests relative to conditional probability?
Background:
Martin-Löf's way of defining randomness of finite strings (over a finite alphabet such as $\{0,1\}$) and infinite sequences uses a generalized notion of a statistical test.
Often, when ...
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Is there an algorithmic complexity measure that strikes a balance between regularity and randomness of a string?
If my understanding is correct, Kolmogorov complexity would assign the highest value (description length) to a totally random string, such as:
abewdwflkweoasfksalsfnlka
the lowest value to a ...
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Any examples of exact calculation 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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Bound on the Kolmogorov complexity of integers
I am reading Elements of Information Theory (Thomas M. Cover, Joy A. Thomas, 2nd edition) in which the following theorems are given (page 475--476):
For any integer $n$
$$ K(n) \leq \log^* n + c. $$...
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Probability distributions based on Kolmogorov complexity?
Suppose a human being randomly chooses a real number $x$ with $0<x<1$. It seems the probability of choosing $x$ is closely related to the Kolmogorov complexity of $x$. That is, a number like $0....
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Application of (Solomonoff) Algorithmic Probability formula?
Ray Solomonoff gives the Algorithmic Probability formula as,
$$
P_M(x)=\sum_{i=1}^{\infty}2^{-|s_{i}(x)|} \tag{1}
$$
If I understand the formula correctly, $M$ is a Turing machine ...
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Proving a binary string of length i is less than or equal to $2^i$?
This is a problem I've gotten on my Graph Theory Homework, and I'm I'm not quite sure how to start off with proving it.
The question is as follows:
Let $\mathcal S$ be a finite collection of binary ...
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Motivation for Algorithmic Randomness Definition
Wikipedia gives this definition for algorithmic randomness in terms of Kolmogorov complexity:
"Given a natural number c and a sequence w, we say that w is c-incompressible if $K(w) \geq |w|-c$.
An ...
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Uncomputability of Kolmogorov complexity
I have read different proofs of Kolmogorov Complexity Uncomputability but I fail to understand why the example below does not work.
Certainly there is something important that I don't get.
Could you ...
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Is there an analogue of Kolmogorov Complexity for Strongly Normalizing Languages?
The definition of Kolmogorov Complexity relies upon the definition of Turing Complete description languages. Famously, Kolmogorov Complexity is uncomputable and akin to the halting problem. I have two ...
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Deriving Probability Theory from Information Theory
In the paper "A Philosophical Treatise of Universal Induction" section 3 on Probability describes three different interpretations of probability theory: frequentist, objectivist, and subjectivist.
I ...
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Average Kolmogorov complexity of an integer factorization
The other day I discovered the Hardy-Ramanujan theorem, which suggested to me that the Kolmogorov complexity of any factorization of some $n$, given $n$, is negligible, a claim I am looking to verify. ...
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What if the random variable are not IID, and we sum them?
Let $a_n > 0$ be decreasing with $\sum a_n^2=1$. Starting at the origin we do drunken random walk on the real line, with step size $a_1,a_2...$, so the total deviation is $\sum a_n^2=1$. If our ...
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Normal number and Kolmogorov complexity
For real number $r$, infinite sequence of its digits in base 10, (I mean 1/9=>1,1,1,1,1,1,1,1,1,1,1,1.....)
I heard that if this sequence is the random sequence in the sense of kolmogorov complexity ...
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Showing that $3/4$ of all words over $\{0,1\}^n$ have $K(w) \geq n-2$
$K(n)$ is the kolmogorov-complexity of a word n. I know that for every $n$, there's at least one word $w_{n}$ of length $n$, such that $k(w_{n}) \geq n$. There's $2^n$ words in $\{0,1\}^n$, how can I ...
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Proving that at least half of all words with length bigger than $n$ are not compressible.
We say that a word is not compressible, if it's Kolmogorov-Complexity is bigger than the length of the word. We know that for every natural number $n$, we have at least one word which is not ...
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Kolmogorov complexity vs. intuitive notion of simplicity
Consider the infinite sequence $1,2,4,8,16,...$ If one were asked how the sequence continues the answer would likely be $32,64,..,2^k,..$ and one would implicitly assume the question to be about the "...
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Is it impossible to prove one very large integer is bigger than another?
We have two integers $i$ and $j$, and the difference between them is large enough such that
$(i >j) \rightarrow (K(i) > K(j))$.
$K$ is the Kolmogorov complexity function.
We have a set of ...
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Terminology for the Kolmogorov complexity of computable functions, sets, reals?
The Kolmogorov complexity of a binary string can be defined in terms of a prefix-free binary encoding of Turing machines that operate on a binary tape. Then if $x$ is a binary string, $K(x)$ is the ...
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Does Chaitin's constant have infinitely many prime prefixes?
Define $f(n) = \lfloor 2^n \cdot \Omega \rfloor$, that is, $f(n)$ is the first $n$ bits of Chaitin's constant interpreted as a number written in binary.
I am trying to figure out if $f(n)$ can have ...
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The problem $K(x) \le K(y)$ is not decidable for Kolmogorov complexity $K$
Let $X$ be some finite alphabet. Given $(x,y) \in X^{\ast}\times X^{\ast}$, how to show that $K(x) \le K(y)$ is not decidable?
I know that $K(x) \le k$ for some fixed $k$ is not decidable, so I tried ...
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Strong Law of Large Numbers with different n
In this problem, we want to show that $\frac{X_1+...+X_n}{\sqrt{n}(log(n))^{\frac{1}{2}+\epsilon}}$ converges to 0 almost surely as n approaches infinity.
We know mean = 0 and variance = 1 and that ...
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Kolmogorov complexity measure of a formal system
Each formal system can be encoded in a binary string. For instance, you can use the input string that a pre-specified Turing machine needs in order to enumerate all the theorems in a theory in the ...
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Simplest Possible Universal Turing Machine
Consider the Game of Life. Given an infinite two-dimensional plane to run the Game on, and an infinite amount of "matter" so that one can give the automata any initial conditions one wishes, it is ...
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Are there strings with known Kolmogorov complexity?
I just looked into Kolmogorov complexity today and it appears to me that for a binary string of length $1$ (ex. '$0$') the Kolmogorov complexity must be $0$.
It follows that Kolmogorov complexity ...
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Is it meaningful to search for “elegant” representations of mathematical objects?
For centuries we struggled with the concept of spatial rotations. We used to represent them in many different ways: mostly, Euler Angles and matrices. Those all had drawbacks and failed in specific ...
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$K(xy)\leq K(x)+K(y) +c$?
Could anyone show that for any $c$, some strings $x$ and $y$ exist, where $K(xy)>K(x)+K(y)+c$? Here $K(x)$ is the Kolmogorov complexity. I already know that $K(xy) \leq 2K(x) + K(y) +c$ and $K(xy) \...
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What is the shortest LOOP program that outputs 2016? [closed]
Use a minor restriction of the LOOP language described under Wikipedia's "LOOP (Programming Language)". The restriction is to eliminate constants. So, the language contains increment: $x_i++$,
...
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not any computable function f such that x is not in the Halting Problem iff f ( x ) belongs to set of Kolmogorov-random strings
taking clue from this question set of Kolmogorov-random strings is co-re
the paper mentioned in the above link talks about the non existence of a computable function
how can I show that there is ...
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set of Kolmogorov-random strings is co-re
given RC = {x : C(x) ≥ |x|} is a set of Kolmogorov-random strings.
How can I show that RC is co-re
I have been reading this paper What Can be Efficiently Reduced to the
Kolmogorov-Random Strings?...
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Abstract machines that compute primitive recursive functions
What it the simplest (least powerful) abstract machine that can compute primitive recursive sets, i.e. sets whose characteristic or indicator function is primitive recursive?
$$f:\mathbb{N}\...
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Understanding AI through a complexity function
I've been trying to understand in light of a few apparent paradoxes for me. It appears reasonable that we could prove any mathematical problem that has a well defined answer can be solved by a ...
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Kolmogorov complexity when no language is specified
The statement of theorem 3 in "A frequentist understanding of sets of measures" by Fierens, Rêgo, and Fine (pdf available here) requires that the Kolmogorov complexity of a certain function be less ...
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chinese reminder theorem (CRT) time complexity
Let p1,...pk be the k first prime numbers. Denote p1*...*pk by n.
We want to find x mod n, for that asume we found x mod pi for i in {1,...,k} , then use CRT to observe x mod n.
What is the lowest (...
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Computability for equality in Kolmogorov complexity?
It is a known result that Kolmogorov complexity is not computable for every arbitrary sequence. I wonder whether the following problem is computable or not:
"Given $x$ and $y$ as two sequences, ...
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Is the Kolmogorov complexity of a number always its logarithm?
if I have a natural number $a(n,m)$ that depends on some $n$ and $m$, where $m$ is fixed,
isn't then the Kolmogorov complexity of it simply its logarithm?
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Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,
I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here.
Theorem ...
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Why $C(n\mid l(n)) \ge C(n) - C(l(n))$ for Kolmogorov complexity
Denote by $C(n)$ the plain Kolmogorov complexity of $n$ and the length of a binary encoding of $n$ by $l(n)$, why do we have
$$
C(n\mid l(n)) \ge C(n) - C(l(n))?
$$
If I have a shortest program $p$ ...
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Each recursive approximating sequence for Kolmogorov complexity is not uniform
Denote the plain Kolmogorov complexity by $C(x)$.
Let $\phi(t,x)$ be a recursive function and $\lim_{t\to\infty} \phi(t,x) = C(x)$ for all $x$. For each $t$ define $\psi_t(x) := \phi(t,x)$ for all $...
