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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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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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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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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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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Complexity of Roman numeral evaluation

I came up with a result the other day that arbitrary length Roman numeral evaluation can be modeled as a monoid: https://gist.github.com/4542999 1) Is this a known result? 2) If not, any ...
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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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85 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 ...
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155 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-...
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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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1answer
219 views

Is “normalized distance” (as per Li & Vitanyi, Kolmogorov Complexity) a reasonable thing?

In "The Similarity Metric" (Li, Vitanyi, et. al) they define a normalized distance (or similarity distance) as a function $\Omega \times \Omega \to [0,1]$ which is both symmetric and satisfies the ...
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254 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: $\...
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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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120 views

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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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 ...
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55 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 $p(n)$...
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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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71 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 $\...
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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....
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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 ...
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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 '...
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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 ...
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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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53 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 ...
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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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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 ...