Here's a cool result from the Pigeon Hole Principle/counting argument that is used in Kolmogorov Theory to show there was at least one string of length n that was not compressible. Here we consider all the binary strings of length n.

The result is that the fraction of strings x with |x| = n such that K(x) < n - k does not exceed 1/2^k.

Ok, but why?

Well, we know that 1 + 2 + ... + 2^(n-k) - 1 < 2^(n-k). We saw this in the previous counting argument. However, there are 2^n strings of length n so the fraction of strings of length n that are compressible is

2^(n-k)/2^n =2^(n-k-n) =2^-k

So no more than 1/2^k of the 2^n strings are compressible. This also means that for strings x where |x| = n, K(x) >= |x| + c (c some constant); that is, if K(x) >= |x| then x is not compressible (also called Kolmogorov random). OTOH, most strings of length n aren't compressible. More specifically, the fraction 1-1/2^k of strings aren't compressible. That is, they have K(x) = n + O(log n). Kinda surprising, really.

#machinelearning #kolmogorov #math #maths

On line sul sito di "Prisma" un mio articolo uscito qualche tempo fa, dove cerco di spiegare il legame fra casualità, comprimibilità e contenuto di informazioni in una sequenza di dati (per esempio il #DNA), trovato da #Kolmogorov negli anni '50.

https://www.prismamagazine.it/2023/05/31/la-complessita-di-kolmogorov/

Just revisiting the subject of why Kolmogorov complexity is incomputable. A 'recent' (2020) paper discusses the subject: https://europepmc.org/article/PMC/PMC7516884
"Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed."
#complexity #kolmogorov #entropy #computation #information_theory #randomness

#DidYouKnow: In #Information theory, the #Kolmogorov #Complexity of a string is defined as the length of the #Shortest #Program that can generate that string as output.

It provides a way to quantify the information content of a string

https://knowledgezone.co.in/kbits/63e8dd1fed03cfa5ce2f4fa4

An interesting and apparently naive question:

Is the Kolmogorov complexity of any string equally low?
https://math.stackexchange.com/questions/4335542/is-the-kolmogorov-complexity-of-any-string-equally-low

No matter what encoding scheme you specify, there will always be, for all n, some string of ≤n bits which requires ≥n bits to encode. This follows inexorably from the pidgeonhole principle.

#complexity #pattern_recognition #kolmogorov #compression #computability

Here we analyze the impact of #facies, #region, #taxonomy, and #collection style over #size #distributions using diameter as a proxy of Late #Devonian #ammonoids in their entirety using non-metric #multidimensional #scaling and #PERMANOVA based on #Kolmogorov distance.

https://doi.org/10.2110/palo.2021.063

@FrohlichMarcel Can we use this to make the #Kolmogorov #complexity of a sequence (with a little noise) computable?

Is the set of #computable #functions equivalent to the set of strings (of #finite or #infinite length) that contain finite #information, i.e. have a finite #Kolmogorov #complexity?

I would like to study Real Analysis.
Any #book suggestions?
I started studying from the Principles Of Mathematical Analysis by #Rudin in order to move to Real And Complex analysis. From baby to papa, one would say. I am also aware that there exists Introductory Real Analysis by #Kolmogorov, who I've heard was a great Mathematician. Has anyone studied his book?Any thoughts?Which book would be more preferable to follow?

#math
#mathematics
#realanalysis