# 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.

89 questions
Filter by
Sorted by
Tagged with
77 views

### What are the substitues for Kolmogorov Complexity to analyse Hashing

The paper "Monotone Minimal Perfect Hashing: Searching a Sorted Table with O(1) Accesses" <http://www.itu.dk/people/pagh/papers/sparse.pdf> is the only one that uses Kolmogorov Complexity to obtain ...
52 views

### Martin-Löf randomness characterization

In his 1973 paper On the notion of a random sequence, Levin states (without proving) a characterization of Martin-Löf randomness by writing Theorem 3. A sequence $\alpha$ is random w.r.t. the ...
51 views

### How approximable is time-bounded Kolmogorov Complexity?

Given a Turing Complete Language, the optimization problem would be: Given inputs x and S, where x is a finite binary string and S is a limit on steps, find the shortest program in that TC language ...
125 views

### Universal lower semicomputable semimeasure and Coding Theorem

I'm following Li and Vitanyi's book "An introduction to Kolmogorov complexity and its applications" 3ed. I'll rewrite here the definitions I need for my question. The authors define the reference ...
4k views

### What are very short programs with unknown halting status?

This 579-bit program in the Binary Lambda Calculus has unknown halting status: ...
86 views

### What are the simplest known algorithms to compute PI?

There are many algorithms that compute PI. Some are obviously complex, involving huge formulas and constants. Some formulas are not that complex, but involve operators such as ...
148 views

### Prove that the Kolmogorov complexity function cannot be approached from below

How would one go about proving that Kolmogorov function $K(x)$ cannot be approached from below by any computable function? After some research it seems I must show the function $K(x)$ is not lower ...
85 views

### Kolmogorov complexity vs purposefully inefficient Turing machines

It's a theorem that, although the Kolmogorov complexity of a string is relative to the Turing machine you're working with, it differs by at most a constant (basically the amount of space it takes to ...
83 views

### Does the Kolmogorov complexity of a program $p$ generating a string $x$ equal the complexity of $x$ up to constant?

If $U$ is a universal prefix Turing machine, $U(p)=x$ for some program $p$ and string $x$, is it true that $K(x)=K(p)+O(1)$, with $K$ being the prefix Kolmogorov complexity?
150 views

### Should Kolmogorov complexity include all resources or just program size?

I've been thinking about pi and Kolmogorov complexity (Kc). As the digits of pi are randomly distributed (and infinite) , they can't be compressed with a typical compression program. Through the ...
262 views

### Without fancy math, how does the MC-AIXI algorithm work?

How does the MC-AIXI algorithm work? I tried to read the paper, but I got lost in the math. I understand the concept of AIXI very well, but I don't get two things: 1) Where does MC-AIXI get its ...
307 views

### 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?...
77 views

### Prove $\forall c \in \mathbb{N} \, \exists x,y \in \Sigma^* \, [K(xy) > K(x) + K(y) + c]$

I am trying to prove a theorem (title) given in a starred problem in Sipser's book. I have absolutely no idea how I would go about showing it, and after trying a few different approaches came here ...
148 views

### Kolmogorov complexity of strings in a given language

Consider the language $$L = \{1^i 0^j 1^k \mid i + j = 2k, k ≥ 1\}\,,$$ and let $x_n$ be the canonical $n$'th word in $L$. My problem involves proving that the Kolmogorov complexity of $x_n$ can be ...
424 views

### Are nearly all natural numbers compressible?

A simple counting argument shows most strings can't be compressed to shorter strings. But, compression is usually defined using Kolmogorov complexity. A string is compressible if its Kolmogorov ...
47 views

### (operationalizable) Cost measure for small problems

For what I know of complexity measures in CS, they are aimed at rather large problems. With today's computing power, most people don't care about comparing the complexity of simple problems as they ...
100 views

### Optimization in multivalued logic. Optimal strings with given patterns

This question comes from an application in multivalued logic. Suppose, we are given an alphabet of three letters $A, B, C$ and a set of indices $1,2,3,4,5$. Consider items formed by subscripting the ...
191 views

### How does this proof show that sequences of $O(1)$ polynomially bounded Kolmogorov complexity are NOT the polynomial computable ones?

I'm trying to understand a proof from the paper: Balcázar, José L.; Gavaldà, Ricard; Hermo, Montserrat: Compressiblity of infinite binary sequences, Published in: Complexity, Logic, and Recursion ...
152 views

### Average case algorithm analysis using Kolmogorov Incompressibility Method

The Incompressibility Method is said to simplify the analysis of algorithms for the average case. From what I understand, this is because there is no need to compute all of the possible combinations ...
228 views

### Termination in infinite-time

Does it make sense to speak of algorithms that take an infinite amount of time to terminate? In particular, suppose we have a loop with a bound function that is initially positive and is decreased ...
80 views

### Show that the set of programs whose Kolmorgorov complexity is smaller than their length is recursively enumerable

Define the language $\qquad R = \{x \in \{0,1\}^\ast \mid C(x) \ge |x| \}$ where $C(x)$ is the Kolmorgorov Complexity of $x$ and $|x|$ denotes the length of $x$. Prove that $R$ is co-...
50 views

### Why can't we search lexicographicaly ordered programs to compute Kolmogrov complexity?

Kolmogrov complexity is known to be uncomputable. Why can't we enumerate all programs of size i = 0 in lexicographical order - if any produce string s, that is the Kolmogrov complexity; if not, ...
366 views

### What problem cannot be solved by a short program?

BACKGROUND: Recently I tried to solve a certain difficult problem that gets as input an array of $n$ numbers. For $n=3$, the only solution I could find was to have a different treatment for each of ...
708 views

### Can a transcendental number like $e$ or $\pi$ be compressed as not algorithmically random?

The related and interesting fields of Information Theory, Turing Computability, Kolmogorov Complexity and Algorithmic Information Theory, give definitions of algorithmically random numbers. An ...
197 views

### Is the length of the shortest quine in a programming language computable?

The length of the shortest program in a given (fixed) programming language that produces a given output is that output's Kolmogorov complexity, which is not a computable function on the set of ...
233 views

### What determines the entropy of a program's source code?

a few days ago I asked a question about the limits of compression: Can PRNGs be used to magically compress-stuff? The idea common to all the answers was that if you consider all programs of length &...
340 views

### What is an estimation of the Kolmogorov Complexity for the first N integers?

I'm aware some ints have higher or lower Kolmogorov Complexities. For example, the number 5.41126806512 has a very low complexity as it can be expressed by ...
513 views

### Kolmogorov Complexity: Why would you need more bytes than the string itself?

I was reading Wikipedia's entry on Kolmogorov Complexity (thanks to this question), which states: It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger ...
184 views

### Proving a string is random

I am reading Kolmogorov Complexity by Li and Vitányi: "Let $x$ be a finite binary string. We write '$x$ is random' if the shortest binary description of $x$ with respect to the optimal specification ...
171 views

### Computability of Kolmogorov Complexity

Fix a universal Turing machine $M$. Let $A^*=\{0,1\}^n$ be the set of all binary string of length $n$. Determine the Kolmorogov complexity $K(x)$ of each $x\in A$, w.r.t. $M$. Just for a matter of ...
103 views

### The choice of programming language and the length of a program

I wonder how it's possible that: it can be shown that all reasonable choices of programming languages lead to quantification of the amount of absolute information in individual objects that is ...
82 views

### Interval density of time bounded Kolmogorov complexity

The Kolmogorov complexity of a string $x$ is the size of the smallest Turing machine $M$ that started on empty tape produces $x$. To make it computable, we can add a bound on the time used by $M$ to ...
420 views

### What is an example of complex random string, in the Kolmogorov-Chatin sense?

Any string generated from a PRNG clearly has a very short description. You need the code for the random number generator, the seed, and then the number of times to iterate. So, it seems that all ...
509 views

### Kolmogorov complexity of string concatenation

If $K(s)$ is the Kolmogorov complexity of the string $s \in \{0,1\}^*$, Can we prove (or disprove) the following statement: "Every string $s$ is a prefix of an incompressible string; i.e. for every ...
1k views

### Approximating the Kolmogorov complexity

I've studied something about the Kolmogorov Complexity, read some articles and books from Vitanyi and Li and used the concept of Normalized Compression Distance to verify the stilometry of authors (...