International audienceVery large databases are a ma jor opp ortunity for science and data analytics is a remarkable new field of investigation in computer science. The effectiveness of these toolsis used to support a “philosophy” against the scientific method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to givescientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: “with enough data, the numbers speak for themselves”. The “end of science” is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this “philosophy” is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in “randomly” generated, large enough databases, which - as we will prove - implies that most correlations are spurious. Too much information tends to behave like very little information. The scientific method can be enriched by computer mining in immense databases, but not replaced by it

The Kochen-Specker theorem shows the impossibility for a hidden variable
theory to consistently assign values to certain (finite) sets of observables in
a way that is non-contextual and consistent with quantum mechanics. If we
require non-contextuality, the consequence is that many observables must not
have pre-existing definite values. However, the Kochen-Specker theorem does not
allow one to determine which observables must be value indefinite. In this
paper we present an improvement on the Kochen-Specker theorem which allows one
to actually locate observables which are provably value indefinite. Various
technical and subtle aspects relating to this formal proof and its connection
to quantum mechanics are discussed. This result is then utilized for the
proposal and certification of a dichotomic quantum random number generator
operating in a three-dimensional Hilbert space.Comment: 31 pages, 5 figures, final versio

Abstract. A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay 23 and Chaitin 10 we s a y that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's 23 -like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability o f a universal self-delimiting Turing machine Chaitin's number, 9 is also a random r.e. real. Solovay showed that any Chaitin numberis -like. In this paper we show that the converse implication is true as well: any -like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

The aim of this paper is to provide a probabilistic, but non-quantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N -bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k > 0, we can effectively compute a time bound T such that the probability that an N -bit program will eventually halt given that it has not halted by T is smaller than 2 −k .We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability.Finally, we show that "long" runtimes are effectively rare. More formally, the set of times at which an N -bit program can stop after the time 2 N + constant has effectively zero density.

The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave noncontextually, one can nonetheless only conclude that some observables (in this set) are value indefinite.In this paper we prove a variant of the Kochen-Specker theorem showing that, under the same assumption of noncontextuality, if a single one-dimensional projection observable is assigned the definite value 1, then no one-dimensional projection observable that is incompatible (i.e., non-commuting) with this one can be assigned consistently a definite value. Unlike standard proofs of the Kochen-Specker theorem, in order to localise and show the extent of value indefiniteness this result requires a constructive method of reduction between KochenSpecker sets.If a system is prepared in a pure state |ψ , then it is reasonable to assume that any value assignment (i.e., hidden variable model) for this system assigns the value 1 to the observable projecting onto the one-dimensional linear subspace spanned by |ψ , and the value 0 to those projecting onto linear subspaces orthogonal to it. Our result can be interpreted, under this assumption, as showing that the outcome of a measurement of any other incompatible one-dimensional projection observable cannot be determined in advance, thus formalising a notion of quantum randomness.

In contrast with software-generated randomness (called pseudo-randomness), quantum randomness can be proven incomputable; that is, it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability-an asymptotic property-of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.

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