Computability of probability measures and Martin-Lof randomness over metric spaces

Algorithmic Randomness and Kolmogorov Complexity for Qubits

Nies and Scholz defined quantum Martin-Löf randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, ($QK$) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are…

Universal test for Hippocratic randomness

Hippocratic randomness is defined in a similar way to Martin-Lof randomness, however it does not assume computability of the probability and the existence of universal test is not assured. We introduce the notion of approximation of probability and show the existence of the universal test (Levin-Schnorr theorem) for Hippocratic randomness when the logarithm of the probability is approximated within additive constant.

Every Sequence is Decompressible from a Random One

Kucera and Gacs independently showed that every infinite sequence is Turing reducible to a Martin-Lof random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a Martin-Lof random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this is the optimal ratio of query bits to computed bits achievable with Turing reductions. As an application of this result, a new characterization of constructive dimension is given in terms of Turing reduc…

Quantum algorithmic randomness

Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz. We define a notion of quantum Solovay randomness which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states.

算法随机性与可计算概率测度的弱合并

Algorithmic randomness and the weak merging of computable probability measures

本文通过Kullback-Leibler散度定义算法随机性，探讨可计算概率测度的弱合并特性。

Algorithmic randomness and splitting of supermartingales

Randomness in the sense of Martin-Löf can be defined in terms of lower semicomputable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one) and on the odd bits (the second one) knowing all preceding bits.

Algorithmic tests and randomness with respect to a class of measures

The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof randomness), then consider uniform randomness (test is a function of a sequence and a measure, not necessarily computable) and arbitrary constructive metric spaces. We also consider tests for classes of measures, in particular Bernoulli measures on Cantor space, and show how they are related to uniform tests and original Martin-Lof definition. We show that H…

Ornstein Isomorphism and Algorithmic Randomness

In 1970, Donald Ornstein proved a landmark result in dynamical systems, viz., two Bernoulli systems with the same entropy are isomorphic except for a measure 0 set. Keane and Smorodinsky gave a finitary proof of this result. They also indicated how one can generalize the result to mixing Markov Shifts. We adapt their construction to show that if two computable mixing Markov systems have the same entropy, then there is a layerwise computable isomorphism defined on all Martin-Lof random points in the system. Since the set of Martin-Lof random points forms a measure 1 set, it implies the classica…

Kolmogorov Complexity and Solovay Functions

Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-Löf randomness and K-triviality.

Equivalent characterizations of partial randomness for a recursively enumerable real

A real number αis called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to α. The randomness of a recursively enumerable real αcan be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin's Ωnumber, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rational numbers which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each…