Stochasticity Randomness and Kolmogorov complexity Our result Kolmogorov-Loveland stochasticity and Kolmogorov complexity Laurent Bienvenu Laboratoire d’Informatique Fondamentale Marseille, France Symposium on Theoretical Aspects of Computer Science, 2007 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Outline 1 Stochasticity 2 Randomness and Kolmogorov complexity 3 Our result Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Algorithmic randomness The goal of algorithmic randomness is to define what it means for an individual object to be random. Here we consider infinite binary sequences. What makes a sequence random ? Look at the following two sequences. Which one is random? 00100101111010011010011110011010111000110001101111001010101100110... 00000000000000000100000001000000001000000000000000100000000001000... Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Algorithmic randomness The goal of algorithmic randomness is to define what it means for an individual object to be random. Here we consider infinite binary sequences. What makes a sequence random ? Look at the following two sequences. Which one is random? 00100101111010011010011110011010111000110001101111001010101100110... 00000000000000000100000001000000001000000000000000100000000001000... The second one does not satisfy the Law of Large Numbers! Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result A naive definition It is tempting to say that a sequence A ∈ 2 ω is random if it satisfies the Law of Large Numbers. But this is too naive: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result A naive definition It is tempting to say that a sequence A ∈ 2 ω is random if it satisfies the Law of Large Numbers. But this is too naive: 01010101010101010101010101010101010101010101010101010101010101010... This sequence does not look random at all! Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result The first attempt to define randomness was made by R. von Mises. It is based on the Law of Large Numbers and avoids the problem we had with our naive definition: Definition (von Mises, 1919) A sequence A ∈ 2 ω is random ( kollektiv ) if every sequence B extracted from A via a reasonable process satisfies the Law of Large Numbers. Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result ? ? ? ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result read ? ? ? ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 ? ? ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result read 0 ? ? ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 ? 1 ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result select 0 ? 1 ? ? ? ? ? selected subsequence: Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 ? 1 1 ? ? ? ? selected subsequence: 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result read 0 ? 1 1 ? ? ? ? selected subsequence: 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 1 1 1 ? ? ? ? selected subsequence: 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result select 0 1 1 1 ? ? ? ? selected subsequence: 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 1 1 1 0 ? ? ? selected subsequence: 1 0 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result select 0 1 1 1 0 ? ? ? selected subsequence: 1 0 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 1 1 1 0 ? ? 1 selected subsequence: 1 0 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result select 0 1 1 1 0 ? ? 1 selected subsequence: 1 0 1 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 1 1 1 0 0 ? 1 selected subsequence: 1 0 1 0 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result read 0 1 1 1 0 0 ? 1 selected subsequence: 1 0 1 0 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result 0 1 1 1 0 0 1 1 selected subsequence: 1 0 1 0 Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Definition (von Mises, 1919) A sequence A ∈ 2 ω is random ( kollektiv ) if every sequence B extracted from A via a reasonable process satisfies the Law of Large Numbers. Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Definition (von Mises, 1919) A sequence A ∈ 2 ω is random ( kollektiv ) if every sequence B extracted from A via a reasonable process satisfies the Law of Large Numbers. Definition (Church, 1940) A sequence A ∈ 2 ω is random (we now say Church stochastic) if every sequence B extracted from A via a computable process satisfies the Law of Large Numbers. Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result J. Ville (1939) claimed that stochasticity is not a good notion of randomness: Theorem (Ville 1939) There exists a sequence A ∈ 2 ω such that: A is Church stochastic Every prefix of A has more 0 ’s than 1 ’s Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Ville’s argument is essentially a diagonalization. Kolmogorov and Loveland proposed the following improvement of the notion of stochasticity: Definition (Kolmogorov-Loveland 19) A sequence is Kolmogorov-Loveland stochastic if for every sequence B extracted from A via a computable non-monotonic process, B satisfies the Law of Large Numbers. Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Martin-L¨ of randomness Definition (Martin-L¨ of 1966) A sequence A ∈ 2 ω is Martin-L¨ of random if it does not belong to any effective G δ set (i.e. Π 0 2 ) which is effectively of measure 0. Theorem (Levin-Schnorr ∼ 1970) A is Martin-L¨ of random iff ∃ c ∀ n K ( A ↾ n ) ≥ n − c Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result KL stochasticity and ML randomness It is not difficult to prove that ML randomness implies KL stochasticity. The converse has been a longstanding open question, and was finally answered negatively: Theorem (van Lambalgen-Shen 1989) There exists a sequence A ∈ 2 ω such that: A is Kolmogorov-Loveland stochastic Every prefix of A has more 0 ’s than 1 ’s Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Kolmogorov complexity and stochasticity How complex are stochastic sequences? Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Kolmogorov complexity and stochasticity How complex are stochastic sequences? Theorem (An. A. Muchnik 1998, Merkle 2003) There exists a Church stochastic sequence A such that K ( A ↾ n ) ≤ log log n Laurent Bienvenu KL stochasticity and Kolmogorov complexity
Stochasticity Randomness and Kolmogorov complexity Our result Kolmogorov complexity and stochasticity How complex are stochastic sequences? Theorem (An. A. Muchnik 1998, Merkle 2003) There exists a Church stochastic sequence A such that K ( A ↾ n ) ≤ log log n Theorem (Merkle, Miller, Nies, Stephan, Reimann 2005) For every Kolmogorov-Loveland stochastic sequence A: K ( A ↾ n ) dim 1 ( A ) = lim inf = 1 n n → + ∞ Laurent Bienvenu KL stochasticity and Kolmogorov complexity
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