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Capacity of quasigroups for generating information

Capacity of quasigroups for generating information. Danilo Gligoroski Institute of Informatics Faculty of Natural Sciences Skopje. Transformation of strings. Transformation of strings with 4x4 Qs 576 quasigroups

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Capacity of quasigroups for generating information

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  1. Capacity of quasigroups for generating information Danilo Gligoroski Institute of Informatics Faculty of Natural Sciences Skopje

  2. Transformation of strings • Transformation of strings with 4x4 Qs • 576 quasigroups • For every s{0,1,2,3}nn=1..6, there is at least one Q and k such that Qk(s)=00…0 • For n=7 there are 45 strings (0.27%) that CAN NOT be transformed in 00…0 • For n=8 there are 2,517 strings (3.84%) that CAN NOT be transformed in 00…0 • For n=9 there are 34,455 strings (13.14%) that CAN NOT be transformed in 00…0

  3. For n=10 there are 255,732 strings (24.39%) that CAN NOT be transformed in 00…0 • For n=11 there are 2,042,895 strings (48.71%) that CAN NOT be transformed in 00…0 • For n=12 there are 10,122,285 strings (60.33 %) that CAN NOT be transformed in 00…0 • Transformation of strings with 5x5 Qs • 161280 quasigroups • I have checked for every s{0,1,2,3,4}n, n=1..8,9,10,11, and 12, andALWAYSthere is at least one Q and ksuch that Qk(s)=00…0 • What is the capacity of the quasigroups of order 5, i.e. what is the smallest length of a string s{0,1,2,3,4}n that can not be transformed in 00..0 ?

  4. Transformation of strings with 256x256 Qs • 1058000 quasigroups Hypothesis For every s{0,1,..255}nn1000000 there is at least one Q (256x256) and ksuch that Qk(s)=00…0

  5. Fractals & Symmetry Chaos What is happening when you process a string 00…0 of length more then 200, with a quasigroup?

  6. Q174= Fractals & Symmetry

  7. Q175= Chaos

  8. Related work • “A new kind of science” – Stephen Wolfram • “Algorithmic information” - Chaitin • “Process Physics: Modeling Reality as Self-Organising Information” – R.T.Cahill, M.Klinger, K.Kitto

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