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Computing Chaitin's omega. Complexity Explorers Krakow.


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Do even mathematical truths violate the principle of sufficient reason? As bits in Omega number that "are true for no reason. They have the value that they do for no reason simpler than themselves"?
Let’s watch and discuss Chaitin's discoveries made within the Algorithmic Information Theory.

Published in: Science
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Computing Chaitin's omega. Complexity Explorers Krakow.

  1. 1. Computing Chaitin’s Omega Photo by Adam Walanus
  2. 2. src: Photo by Sethwoodworth at English Wikipedia, taken by Bcjordan [CC BY 3.0 (], via Wikimedia Commons Marvin Minsky
  3. 3. 1. Computational Complexity Computational Complexity Theory; time / space complexity 2. Kolmogorov Complexity Measures data compressibility. Part of Algorithmic Information Theory. AIT by Ray Solomonoff, Andrey Kolmogorov, Gregory Chaitin 3. Complex system Part of Systems Theory, rooted in Chaos Theory, emergence phenomenon Complexity term
  4. 4. Math is the worst (in terms of complexity) src: 22:30 - 24:52 - 27:15
  5. 5. “What Science is all about” Computer Universal Turing Machine Experimental dataTheory Finite sequences of zeros and ones Expectations: Theory should be smaller in bits than experimental data
  6. 6. Computer Universal Turing Machine Experimental dataTheory 110100101110 1111000011110000 1010101010101010 Low: complexity, alg. randomness High: complexity, alg. randomness
  7. 7. |Theory| == |Data| ❏ Facts ❏ Borel’s number * ❏ Unexplainable observation ❏ Most (ℵ1 ) real numbers are uncomputable ** Irreducible complexity * Émile Borel, 1927 ** Turing, 1936: “On computable numbers…” There is an app for that |Theory| < |Data| ❏ Compressible data ❏ Scientific theory ❏ Explainable observation ❏ Computable real numbers (to a given precision, ℵ0 ) ½, π, √2 Reducible complexity
  8. 8. How Omega is defined Halting probability Ω = Prob{random program halts} or sum of 2 - lengths of (prefix-free) binary programs p that halts 0 < Ω= .11011100... < 1 Knowing N bits of Ω ⇒ Knowing which ≤ N bit programs halt p halts -|p| 2Ω= 𝛴
  9. 9. Ω = The number of Wisdom Ω- oracle for, stores answers to, Turing’s halting problem Ω is maximally unknowable, almost computable, irreducible, real number. Even though it is precisely defined once you specify the programming language Ω bits of its numerical value look like independent tosses of a fair coin Ω bits are necessary truths but look accidental Ω’s properties suggest that mathematicians should be more willing to postulate new axioms, similar to the way that physicists must evaluate experimental results and assert basic laws that cannot be proved logically. Gregory J Chaitin Thinking About G del and Turing Essays on Complexity, 1970 - 2007
  10. 10. G. Chaitin’s views What is Complexity in Cosmos on Gödel; euclidean geometry has finite complexity but basic arithmetic escapes boundaries of limited set of axioms. Where all complexity comes from? There is no Theory of everything in Mathematics.
  11. 11. 1. Math is build in large on unexplainable truths 2. You cannot tell if your theory is final (elegant), the most concise one 3. Most real number we think of are computable but most real numbers are uncomputable Necessary truths
  12. 12. Can I prove that I have the best theory / the program is elegant? src: 40:35 - 45 P’s output is the output of the first provably elegant program Q larger than P...
  13. 13. 1. Gödel, 1931: No TOE for pure mathematics mathematics is necessarily incomplete 2. Turing, 1936: Unsolvability of the halting problem 3. 1960+ : Infinite complexity, Math & CS as empirical science, no way to tell if a theory is final Limits
  14. 14. src: Is Information Fundamental? on probability is an information. Is matter an epiphenomenon? I don’t believe in continuity. Universe is computation. Understanding is computation. Everything is digital. G. Chaitin’s views 2