Methods in computational physics 3. Fundamental methods in by Berni Alder, Sidney Fernbach, Manuel Rotenberg

By Berni Alder, Sidney Fernbach, Manuel Rotenberg

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Tests, {We,j,s : e, j, s ∈ N} and stops the enumeration of one if the measure µ(We,j,s ) threatens to exceed 2−(j+1) at any stage s of the simultaneous enumeration. Then we can let Un = ∪e∈N We,n+e+1 . of test, and moreover, a real A passes Then we note that Un is a Martin-L¨ all Martin-L¨ of tests iff A ∈ ∩n∈N Un . We have established the following. 1: (Martin-L¨ of [53]) There exist universal Martin-L¨ of tests: That is there is a Martin-L¨ of test {Un : n ∈ N} such that, for any MartinL¨of test {Vn : n ∈ N}, x ∈ ∩n∈N Vn implies x ∈ ∩n∈N Un .

Hence, x is not Levin-G´acs-Chaitin random. We can use Schnorr’s Theorem to prove that there are many strings that are (weakly) Chaitin random yet are not Kolmogorov random. 2: There are infinitely many n and strings x of length n such that (i) K(x) ≥ n and (ii) C(x) ≤ n − log n. Proof: Let α be Martin-L¨ of random. 4, for all n, K(α n) ≥ n − O(1). 1, for infinitely many n, C(α n) ≤ n − log n. Schnorr’s Theorem also allows us to define some specific kinds of random reals. For instance, the class R = ∪c Rc where Rc = {A : ∀n(K(A n) ≥ n − c}, is the Σ02 class of all Martin-L¨of random reals.

We refer the reader to [39] or Downey and Hirschfeldt [18]. Strictly speaking the proof only uses weak n + 1-randomness. 7. van Lambalgen’s Theorem In this section, we will study independence results for random reals. We begin with a central result concerning randomness, which has turned out to play a very important role in the theory. We begin with a simple Lemma. 14: (van Lambalgen, Kurtz, Kautz) (i) If A ⊕ B is n-random so is A. (ii) If A is n-random so is A[n] , the n-th column of A. (iii) If A ⊕ B is n-random, then A is n − B-random.

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