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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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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 ...
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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 $...
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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 ...
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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 ...
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230 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 ...
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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 ...
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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 ...
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66 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 $...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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-...
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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: $\...
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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) := ...
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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 $\...
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1answer
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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. -...
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213 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 ...
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243 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^{...
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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)....
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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1answer
104 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
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“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, ...
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182 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 ...
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What is the problem of finding a largest subset of smallest Kolmogorov complexity?

What do you call the problem of finding a largest possible subset of strings with smallest possible information content? I'm studying a particular instantiation of this problem in a different setting ...
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Is joint Kolmogorov Complexity order invariant?

Due to the symmetry of information, it follows up to an additive constant that K(X,Y) = K(Y,X) Does this hold for more than two data objects as well?
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What is the Kolmogorov complexity of arithmetic?

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(n) > L$ where $K(n)$ is the Kolmogorov complexity of the number $n$ and $L$ is a sufficiently large ...
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Can we not output the Kolmogorov complexity?

Let us fix a prefix-free encoding of Turing-machines and a universal Turing-machine $U$ that on input $(T,x)$ (encoded as the prefix-free code of $T$ followed by $x$) outputs whatever $T$ outputs on ...
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Proof for Kolmogorov complexity is uncomputable using reductions

I am looking for a proof that Kolmogorov complexity is uncomputable using a reduction from another uncomputable problem. The common proof is a formalization of Berry's paradox rather than a reduction, ...
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Comparing the Kolmogorov complexity of theories - Part 2

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(x) > L$ where $K(x)$ is the Kolmogorov complexity of natural number $x$ and $L$ is a sufficiently ...
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430 views

Comparing the Kolmogorov complexity of theories

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(s) > L$ where $K(s)$ is the Kolmogorov complexity of string $s$ and $L$ is a sufficiently large ...
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178 views

Is Kolmogorov complexity quasi-surjective?

For Kolmogorov complexities $\hspace{.02 in}K$ induced by essentially-optimal description languages, does there exist an integer $c$ such that for all positive integers $n$, there exists a string $x$...
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1answer
234 views

Kolmogorov Complexity vs Running Time (Edited)

Let $U$ be a universal Turing Machine. Suppose I have a Kolmogorov incompressible string $s$ of length $n$. Let $A:\{1,...,n\} \to \{0,1\}$ be an algorithm such that $A(i) = s_i$. I believe that the ...
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95 views

The concept of binary description by Ray Solomonoff

In this article, http://world.std.com/~rjs/rayfeb60.pdf in page 7 of the PDF, Solomonoff gives an a-priori probability for a string which is $pr(S)=2^{-|S|}$. My question - why are shorter strings ...
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Is Kolmogorov complexity a surjective function?

Let us fix an encoding of Turing-machines and a universal Turing-machine, U, that on input (T,x) outputs whatever T outputs on input x (possibly both running forever). Define the Kolmogorov complexity ...
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Different definitions of optimal decompressors

Let $B^{<\omega}$ be the set of finite binary strings. I will only consider functions from $B^{<\omega}$ to $B^{<\omega}$. I recall the definition of the algorithmic complexity of a string ...
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3answers
959 views

Using Kolmogorov complexity as input “size”

Say we have a computational problem, e.g. 3-SAT, that has a set of problem instances (possible inputs) $S$. Normally in the analysis of algorithms or computational complexity theory, we have some ...
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Entropy criterion of efficiency for (comparison using hashing)

I understand that hash is effective iff the "domain" size is smaller than the size of the "general set" - set of all possible objects. E.g., "domain" is the set of valid english phrases with length ...
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Is there an $\mathcal{L}$-theory and a formula $\phi$ for which Kolmogorov(proof($\phi$)) $<$ Kolmogorov($\phi$)?

Are there a complete decidable $\mathcal{L}$-theory, a formula $\phi$ and a proof of $\phi$ for which the Kolmogorov complexity of the proof of $\phi$ is less than the Kolmogorov complexity of $\phi$? ...
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246 views

Kolmogorov Complexity of an arbitrary string

For an arbitrary string sampled from $\{0,1\}^n$, what would be the probability that the string has a Kolmogorov Complexity $\geq$ length of the string, $n$. In other words, how many strings are ...
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876 views

Kolmogorov Complexity applications in Number Theory

What are the applications of Kolmogorov Complexity in Number Theory and on proofs related fields? (The monograph by Li & Vitanyi doesn't have many applications related to Number Theory.) One of ...
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1answer
116 views

What information can a program give about the Kolmogorov Complexity of its accepting Language

Suppose we have a program(algorithm) that solves some decision problem $(\Sigma,L)$. Given the program, can we always say something about the Kolmogorov complexity of the words in $L$ ?
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How to measure programming language succinctness?

I want to explore the notion of quantifying the amount of succinctness a programming language provides. That is, the amount a high-level language reduces the complex. This idea of "simplification" ...
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Boltzmann sampling software

I'm looking for an implementation of Boltzmann sampling for combinatorial structures. Recent paper in the area for context: http://hal.inria.fr/docs/00/74/77/09/PDF/NonRedundantGeneration-TCS-2010....