# 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.

28 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
157 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 ...
162 views

### 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 ...
144 views

### 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 ...
35 views

### 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 ...
187 views

### 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 ...
195 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 ...
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 ...
160 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 ...
172 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-...
100 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 ...
67 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....
229 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 ...
256 views