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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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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How to measure the length of the program on Turing Machine?
In Kolmogorov complexity, there is a notion: length of the shortest program that describes the data. I can use Lempel-Ziv compression to estimate this value. What if I want to estimate this value by ...
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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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Are real computers capable of producing true random series with a program of finite length?
Suppose that one has a Blum–Shub–Smale machine. Is it possible to write down a program of finite length that produces a series of numbers, with each consisting of a finite number of digits, such that ...
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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 ...
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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 \...
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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 $...
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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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When is conditional Kolmogorov complexity zero?
It seems intuitive that conditional Kolmogorov complexity is only zero when the bitstrings are the same, and otherwise is greater than 0. I.e. if $b_1 = b_2$, then $K(b_1|b_2) = 0$, otherwise $K(b_1|...
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Unrolling closures into SAT boolean formula
I need to verify some assertions about the minimalist Turing-complete language Jot.
Many of the assertions I want to investigate are semi-deciable (co-recursively enumerable). So far it's been fairly ...
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If a bitstring is compressible, is the minimal Kolmogorov sufficient statistic most likely large or small?
If a bitstring is incompressible, then its minimal Kolmogorov sufficient statistic (MKSS) is zero, since the bitstring is best represented by a structure function that enumerates bitstrings, and thus ...
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Computing the Kolmogorov complexity of a string
What would be the implications for complexity theory if you could compute the Kolmogorov complexity of a string generated by a psuedorandom generator?
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What do you call a set which has the following enumeration-related machine?
A set $S$ of natural numbers is Recursively Enumerable if there exists a Turing machine which enumerates them, i.e. given no input, outputs the elements of $S$ in increasing order (never halting if $...
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What is The correct terminology for expressing *this* notion of complexity
Say for example, I have an algorithm $Al_i$ that produces output from the set $S = \{s_i\}$ for problems from the set $P = \{p_i\}$, and another algorithm $Al_j$ that also produces output from the set ...
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Is Kolmogorov Complexity Universal?
Is the Kolmogorov complexity of any piece of information with respect to a certain predefined encoding for all pieces of information, or can the encoding vary for each piece of information?
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Prefix complexity of x conditioned on prefix complexity of x
I wonder what is $K(x|K(x))$, $K$ denoting (Kolmogorov-Chaitin) prefix complexity. Naively, it looks like it should just be $K(x)+O(1)$. Is that right? The same question for plain Kolmogorov ...
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Is Kolmogorov-random nonsensical for small numbers?
The Wikipedia definition of Kolmogorov-random defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string.
Aren't nearly1 ...
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Kolmogorov complexity of a random string conditioned on another random string
Given two strings $x$, $y$, both of length $n$, what is the probability that $K(x|y,n)=K(x|n)$ ? (Bounds on this probability would be very interesting too). Here $K$ is Kolmogorov complexity, $x$ and $...
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Conditional Kolmogorov compexity of string concatenation
In what follows $K(x|y)$ is conditional Kolmogorov complexity, $xx$ is $x$ concatenated with itself. It appears to me that $K(xx|yy)=K(xx)$ should be true for infinitely many strings $x$ and $y$. In ...
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Is $K(b|a) \geq 1$ if $a\neq b$?
Since $K(a|a) = 0$, is $K(b|a) \geq 1$ when $a\neq b$, as we need at least one bit to distinguish between $K(a|a)$ and $K(b|a)$? If not true in general, is it true if $a$ and $b$ are elegant programs?...
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Joint Kolmogorov complexity of elegant programs
If $a$ and $b$ are different elegant programs (minimal program for some output), is their joint Kolmogorov complexity the sum of their individual complexities, i.e. $K(a,b) = K(a) + K(b)$?
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For a turing machine which computes $y$ with argument $x$, how much does the descriptional length of a machine producing $y$ without input increases
Let $f$ be some function in some programming language (like C), and we need $n$ bits to store this function. Suppose we have some fixed value $v$ for the argument, then let
g() { f(v) }
be the ...
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Gate/transistor number and program length measures of computational complexity?
Are there scaling concepts regarding the number of components needed and minimum program length needed to solve a problem of some complexity?
For example, problems are characterized by O(.) measures ...
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Number of $1$'s in a Kolmogorov-random number
The problem I'm trying to tackle is to show that for a Kolmogorov-random number of length $n$, the amount of $1$'s in its binary representation is greater than $n/4$.
My only idea so far, is to use a ...
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Kolmogorov complexity of a sequence of prime numbers
Let's say $p_1, p_2, p_3, \ldots $ is the increasing sorted sequence of all prime numbers. Prove that there exists a constant $n_0 \in \mathbb{N}$, so that for all $n \ge n_0$:
$$K(p_n) < \log_2(...
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What's relation between Kolmogorov complexity and pseudorandomness?
In a comment on this question, @Kaveh wondered whether the questioner really wanted to ask "is there a relation between strings with high Kolmogorov complexity and pseudorandomness?" This is not the ...
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An infinite subsequence of random numbers in Kolmogorov sense
Is it possible to construct an infinite increasing sequence of random naturals (in Kolmogorov sense) that is a subsequence of another sequence? Ok, in general I suppose not, e.g. for prime numbers. ...
