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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
47 views

Relation Between Different Definitions of Information Distance

I'm reading the fourth edition of An Introduction to Kolmogorov Complexity and Its Applications by Li and Vitanyi. In Section 8.3 of the book, it introduces the concept of "information distance.&...
Ravi Deedwania's user avatar
2 votes
0 answers
69 views

Resource bounded Kolmogorov complexity hardness on average over a non uniform distribution of inputs

$K^{poly}$, as well as other related problems such as $MCSP$, is believed to be hard on average [1, 2] when the input is sampled from a uniform distribution (since otherwise one way functions, pseudo-...
agemO's user avatar
  • 187
6 votes
1 answer
228 views

Fast algorithms for time bounded Kolmogorov complexity

For a universal Turing machine $U$, the time bounded Kolmogorov complexity of a string $x$ is silmilar to the usual Kolmogorov complexity but limited to programs $p$ running in time at most $t(|x|)$: $...
agemO's user avatar
  • 187
1 vote
1 answer
106 views

Clarification sought re Li & Vitanyi's Proof of Godel Incompleteness in Formal System F

(Cross-posted from Computer Science due to lack of response after 1 week) From An Introduction to Kolmogorov Complexity and Its Applications, Li & Vitany, 4th Ed. Example 1.1.1. As you might guess,...
Julian Moore's user avatar
1 vote
0 answers
122 views

How to measure the weirdness of algorithms?

Let $M$ is a polynomial $k$-tape Turing machine and $C^t(x)$ is a time-bounded Kolmogorov complexity. Let $str_M(x)$ be a string of the following form: $$str_M(x)=w_1^1\# w_2^1 \# ... \# w_{m}^1 ■ w_1^...
Ben Tom's user avatar
  • 49
2 votes
0 answers
101 views

Online *detailed* tutorials about Komogorov Complexity

I'm a private math tutor who also tutors some theoretical CS. Last semester I had a student who needed tutoring in Kolmogorov Complexity. I told her that I only know about Kolmogorov Complexity, but ...
Dudley Brooks's user avatar
5 votes
1 answer
234 views

Upper bound on the expected number of correct bits via a "lossy compression"

Consider the following "compression problem" for a pair $(C,D)$ of algorithms: $C$ receives a uniformly random $x \in \{0,1\}^n$ and outputs a smaller bit string $y \in \{0,1\}^s$. Algorithm ...
Marcel Dall'Agnol's user avatar
2 votes
0 answers
53 views

program search with optimization methods for (resource bounded) Kolmogorov complexity

Are there fields of research that look at finding short programs for generating strings (therefore trying to find the (resource bounded) Kolmogorov complexity of the string), but using optimization ...
dorikolmo's user avatar
1 vote
0 answers
55 views

sophistication or logical depth to detect intelligent extra-terrestrial species

From my understanding, Algorithmic information theory (AIT) gives some ways to define the amount of « structure » in a string: for example sophistication or logical depth (see for instance [1]), can ...
dorikolmo's user avatar
3 votes
1 answer
149 views

An invariance theorem for algorithmically random data in statistical learning

Motivation: The following invariance theorem for statistical learning in the setting of algorithmically random data occurred to me yesterday. This theorem uses the fact that the property of ...
Aidan Rocke's user avatar
7 votes
1 answer
200 views

Inferring the Kolmogorov complexity of a string from its substrings' complexity

I know that the Kolmogorov complexity of a substring $v$ of an incompressible string $x$ has $C(v)\geq |v|-O(\log{|x|})$ , but I'm wondering if it is also possible to infer the complexity of a string ...
Ari's user avatar
  • 285
3 votes
1 answer
126 views

how to define "correlation" between languages?

How does one define the concept of correlation between languages? Is there any 'standard' measure of 'correlation' between two (possibly inf) sets of strings / an analogue of the concept in this ...
DeeDee's user avatar
  • 301
4 votes
0 answers
171 views

Alternative exponential definition of Kolmogorov complexity

In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem the Kolmogorov Complexity (KC) of $x$ is defined as $$ K(x) = \mu e (\varphi_e(0) \simeq x) \, , $$ the smallest $e$ such ...
Jori's user avatar
  • 41
1 vote
0 answers
71 views

Reference request on using Kolmogorov complexity to measure the simplicity of models

Have there been any serious attempts to use the notion of Kolmogorov complexity to measure the simplicity of models outside of theoretical CS? I mean models in the english sense - any logical set of ...
ghosts_in_the_code's user avatar
1 vote
0 answers
95 views

Given a program specification, S, what can be said about the size and efficiency of programs that exactly satsify S, with respect to the size of S?

Suppose we are given a program specification, $S$, and we want to reason about programs $P$ that satisfy $S$. One might like to think that if the specification is 'simple', the the program should be '...
user10108's user avatar
5 votes
1 answer
240 views

Kolmogorov Complexity of a Decidable Language

The Kolmogorov Complexity (KC) of a string $y$ is the size of the smallest program $f$ and input $x$ that: $y = f(x)$. Let's define a variation of Kolmogorov's complexity$^1$. Suppose a decidable ...
Raphael Augusto's user avatar
1 vote
0 answers
73 views

existence & characterization of "kolmogorov efficient" programs

$\newcommand{\Prog}{\operatorname{Prog}}\newcommand{\kol}{\operatorname{kol}}$ Disclaimer: I do not have a formal background in algorithmic complexity theory so apologies if I use non-standard ...
Hyperplane's user avatar
1 vote
0 answers
234 views

Fast algorithms for evaluating functions with high Kolmogorov complexity

Motivation: I am motivated by a concrete example that occurs in neuroscience, dendritic computation, which may be approximated by functions computable on binary trees [1]. To be more precise, I ...
Aidan Rocke's user avatar
3 votes
1 answer
225 views

Kolmogorov Complexity of the composition of two computable functions

Let's suppose we encode two computable functions $f$ and $g$ as binary strings so $f,g \in \{0,1\}^*$. What I am curious about is whether we can find good upper and lower bounds for: \begin{equation}...
Aidan Rocke's user avatar
1 vote
1 answer
318 views

Prime factorisation of decidable problems

Disclaimer: I am not a theoretical computer scientist. The set of decidable problems $\mathbb{D}$ is countable so $\lvert \mathbb{D} \rvert = \lvert \mathbb{N} \rvert$ and this led me to the ...
Aidan Rocke's user avatar
1 vote
0 answers
83 views

A question on the Kolmogorov Complexity of Human I/O behaviour

Note: From my Twitter poll I managed to get feedback from AI researchers and neuroscientists so far and I think it would be interesting to get input from theoretical computer scientists on this ...
Aidan Rocke's user avatar
2 votes
0 answers
72 views

How to improve this pseudorandom generator?

Let $f$ be a Boolean function and $\varepsilon > 0$. There exists a pseudorandom generator $G_f: \{0,1 \}^{n^{\varepsilon}} \to \{0,1 \}^n$ with the following property. Let $T$ be a set and $p(n)$...
Alexey Milovanov's user avatar
6 votes
1 answer
374 views

Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?

Disclaimer: I am mostly unfamiliar with theoretical computer science, making it hard for me to navigate literature in the field. I ask the following out of curiosity. Background/Motivation: Coming ...
Julius Kunze's user avatar
3 votes
1 answer
101 views

Commonality of minimal Kolmogorov structure functions

A typical $B$ is difficult to define precisely. Intuitively, it is the sort of bitstring that will be generated most of the time by randomly drawing 0s and 1s. However, the concept can be clarified ...
yters's user avatar
  • 203
6 votes
1 answer
292 views

Algorithmic mutual information between random string and minimal Kolmogorov sufficient statistic

Regarding notation in the following, the function $\ell(B)$ returns the length of bitstring $B$, and the cardinality of set $S$ is denoted by $|S|$. A bitstring $B$ is generated by drawing 0s and 1s ...
yters's user avatar
  • 203
8 votes
0 answers
180 views

Number of circuits computing a given function

Let's say we have some function that can be computed by a minimal circuit of size $m$ (using some metric, say, the number of gates). Other than this minimal circuit, there will be many other circuits ...
interstice's user avatar
4 votes
1 answer
151 views

Relation between different "complexity theories" and complex systems theory

I know of at least 4 fundamentally different uses of the term "complexity theory": the study of how hard a problem is to solve using some sort of computing machine (I am ignoring divisions within ...
lightning's user avatar
  • 433
1 vote
1 answer
97 views

What is the relationship between this notion of symmetry compressibility and Kolmogorov complexity?

If we have some string $x$ and a permutation $g$ which preserves the bits of $x$, we can store the value of $x$ on each of its $g$-orbits as well as $g$ instead of storing $x$ and we can reconstruct $...
Samuel Schlesinger's user avatar
2 votes
1 answer
510 views

Does there exist an ontology for algorithms?

It appears that algorithmic complexity theory has already figured out Kolmogorov complexity, when applied to representations of programs themselves, can already serve as a solid theoretical metric of ...
CinchBlue's user avatar
  • 309
6 votes
1 answer
435 views

Do we know a specific $L_{ZFC}$ such that $K(s) \ge L_{ZFC}$ is unprovable in ZFC for all strings $s$?

Chaitin's incompleteness theorem states for any formal system $F$ (which satisfies various criteria), there is a $L$ such that for any $s$ the statement $$K(s) \ge L_F$$ is unprovable in that formal ...
PyRulez's user avatar
  • 471
2 votes
1 answer
116 views

Reasonable estimate of an asymptotic limit notion of Kolmogorov complexity

Let $$f(c)=\min_{x \ge c} K(x)$$ where $K(x)$ is the kolmogorov complexity of $x$. Since $K(x)$ is always a natural number, there will always be a minimum. My question is, what is the growth rate of $...
PyRulez's user avatar
  • 471
2 votes
0 answers
89 views

What's the distribution of the Kolmogorov complexity of the elements in the set of bitstrings of length n?

take e.g. all possible bitstrings of length n=10,000. there are 2^10,000 of them. imagine that for each of those strings we knew the length of the shortest program that could produce the string. what ...
RoboTeddy's user avatar
1 vote
0 answers
66 views

Parametrically-relaxed Kolmogorov complexity

Consider the following problem: Input: An integer $n$ and a subset $S \subseteq \{0...n-1\}$ in some representation. Output: The encoding of some kind of automaton (say, a Turing machine) which ...
einpoklum's user avatar
  • 173
8 votes
1 answer
347 views

Resource-bounded variant of Kolmogorov complexity

Consider the variant of Kolmogorov complexity, where the program is only allowed to use a bounded amount of resources. This more closely resembles the practical situation, where decompression needs ...
Demi's user avatar
  • 516
4 votes
0 answers
195 views

Worst-Case and Average-Case running-time equal with universal p-distribution with kolmogorov-complexity any applications of this theory?

at the moment I'm reading "Gems of Theoretical Computer Science" from Schöning and Pruim. In Chapter 8 the book defines a "universal probability distribution" in a way that the Average-Case running-...
Senf's user avatar
  • 41
3 votes
0 answers
263 views

A sequence wherein the Kolmogorov complexity of the terms does not increase

I am looking for an algorithm $A$ - which for any non-null input string $s_1$ produces a sequence $s_1, s_2...$ such that : It can be proved in some axiomtic system $S$ that: $\...
ARi's user avatar
  • 405
6 votes
1 answer
450 views

Where does the "intuitive" understanding of Kolmogorov complexity fails

Often, the Kolmogorov complexity of some string $x$ is defined as the length of the shortest program producing $x$, for example on wikipedia. So to give this more formal meaning, define $$ K'(x) := ...
StefanH's user avatar
  • 2,037
3 votes
1 answer
153 views

How to extend Solomonoff Induction to continuous domain

Informally speaking, Solomonoff Induction provides an Bayesian optimal way to predict the next symbol from observed symbol sequences $S_1S_2S_3...S_n$ where each $S_i$ belongs to a finite symbol set $\...
Kh40tiK's user avatar
  • 131
18 votes
1 answer
966 views

Does the uncomputability of Kolmogorov complexity follow from Lawvere's Fixed Point Theorem?

Many theorems and "paradoxes" - Cantor's diagonalization, undecidability of hatling, undeciability of Kolmogorov complexity, Gödel Incompleteness, Chaitin Incompleteness, Russell's paradox, etc. -...
Joshua Grochow's user avatar
3 votes
1 answer
242 views

Compression algorithms for low-complexity strings?

Let $s$ denote a string over a finite alphabet, $n_s = |s|$ be the length of $s$, and $n_s^{*}$ denote the minimum description size of $s$ under a given computational model (TM, CFG, etc.). Are there ...
Jack G.'s user avatar
  • 35
6 votes
1 answer
301 views

Expected Kolmogorov complexity under Kolmogorov complexity distribution

If $K(w)$ is the Kolmogorov complexity of a string $w$, where programs are prefix-encoded so $\sum_{w} 2^{-K(w)} \leq 1$, what is $$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} K(w)}{\sum_{|w|=n} 2^{...
Saram Meti's user avatar
7 votes
1 answer
695 views

Difference between time-bounded and memory-bounded Kolmogorov complexity

Let $x$ be a finite string of length $n$. Denote by $C^t(x)$ the Kolmogorov complexity of $x$ bounded by time $t$ (i.e. the length of a minimal program that outputs $x$ and running at most $t$ steps)....
Alexey Milovanov's user avatar
4 votes
0 answers
107 views

Why primitive rotation is $53.13^\circ$ in the quantum Turing machine used by Vitanyi for Quantum Kolmogrov Complexity?

Right now I am going through Quantum Kolmogorov Complexity Based on Classical Descriptions by Vitanyi. In the introduction, the author assumed the primitive rotation $\theta = 53.13^\circ$ to have ...
Omar Shehab's user avatar
0 votes
0 answers
158 views

Questions about the definition of the Quantum Turing Machine

I am trying to have a better understanding of the definition of the Quantum Turing Machine. My questions: If the output of a quantum program is the eigenvalue of the ground state of a Hamiltonian ...
Omar Shehab's user avatar
5 votes
0 answers
88 views

Computational depth and p-time hard instances

After reading the nice results of the paper: "Worst-Case Running Times for Average-Case Algorithms" by Antunes and Fortnow, I was wondering about the existence of further results linking basic ...
Ramon Bejar Torres's user avatar
5 votes
0 answers
94 views

What is the shortest description of a universal computational structure that includes a meta-circular evaluator?

I am wondering whether there is a minimal (or the shortest known) way of specifying a universal computational structure that includes a specification of a meta-circular evaluator within that structure....
Lenar Hoyt's user avatar
0 votes
3 answers
605 views

Fastest Turing Machine

Recently I have been reading about Kolmogorov Complexity. As such I started thinking about the "fastest turing machine". In particular I am not interested in finding such a machine, I am only ...
user avatar
1 vote
1 answer
110 views

How to quantify the complexity of vectors whose components are real numbers?

I started to study Kolmogorov complexity today, and this question came to mind. Is there any way to use LZW to do this? I'm looking for a guidance to my studies
daemon's user avatar
  • 13
1 vote
1 answer
124 views

"Checking equality for Kolmogorov complexity of two sequences" is computable?

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, ...
Naji's user avatar
  • 111
-3 votes
1 answer
400 views

Are nearly all natural numbers compressible? [closed]

There are numerous proofs that most strings are incompressible. I am interested in strings that represent natural numbers. Assume the Kolmogorov complexity of a natural number, $K(n)$, is the smallest ...
Russell Easterly's user avatar