Algorithmic randomness and splitting of supermartingales
Andrej Muchnik
马丁-洛夫意义下的随机性可以用下半可计算上鞅来定义。我们证明了这样的上鞅不能被一对只赌偶数位（第一个）和奇数位（第二个）的上鞅所取代。
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.
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We show algorithmic randomness versions of the two classical theorems on subsequences of normal numbers. One is Kamae-Weiss theorem (Kamae 1973) on normal numbers, which characterize the selection function that preserves normal numbers. Another one is the Steinhaus (1922) theorem on normal numbers, which characterize the normality from their subsequences. In van Lambalgen (1987), an algorithmic analogy to Kamae-Weiss theorem is conjectured in terms of algorithmic randomness and complexity. In this paper we consider two types of algorithmic random sequence; one is ML-random sequences and the ot…
An instability property of the Birkhoff's ergodic theorem and related asymptotic laws with respect to small violations of algorithmic randomness is studied. The Shannon--McMillan--Breiman theorem and all universal compression schemes are also among them.
We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability, classification of random sets by likelihood ratio tests, decomposition of complexity and independence, Bayesian statistics for individual random sequences. Formerly Lambalgen's theorem for correlated probability is shown under a uniform computability assumption in [H. Takahashi Inform. Comp. 2008]. In this paper we show the theorem without the assumption.
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…
This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and philosophy. Intuitively, randomness is a lack of order or predictability. If randomness is the opposite of determinism, then algorithmic randomness is the opposite of computability. Besides many other things, these concepts have been used to quantify Ockham's razor, solve the induction problem, and define intelligence.
The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy…
Algorithmic complexity and randomness in elastic solids
一个系统，包括一个弹性的固体和它的响应外部随机力序列的表现基于算法的复杂性和随机性理论的原则。固体以与其算法复杂性成比例的方式扭曲输入力序列的随机性。我们证明了这一点，通过数值分析的一维振动弹性固体（系统）上，我们施加一个最大的随机输入力。系统的复杂程度通过外部参数来控制。输出响应是在物体上的几个位置处观察到的位移场。算法的复杂性和随机性的输出位移序列的测量和比较对系统的复杂性。结果表明，系统复杂度越高，输出序列的随机性越差。这与[16]中介绍的理论一致，该理论指出，这样的物理系统表现为算法选择规则，作用于周围环境中的随机行为。
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…
This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are differences between the computability of probability sources, ( which means there is an algorithm that can define the way that random process or probability source generates the numbers ) and the algorithmic randomness of the sequences or strings which are produced by a source. We may have the source without a computable algorithm for that but it can produce compres…
We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of all $\sum_{n=1}^82^n$ bit strings up to 8 bits long, and for some between 9 and 16 bits long. This is done by an exhaustive execution of all deterministic 2-symbol Turing machines with up to 4 states for which the halting times are known thanks to the Busy Beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and t…
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.
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 in Continuous-Time Markov Chains
本文为连续时间马尔可夫链（CTMC）的单个轨迹定义了算法随机性，并通过鞅、算法测度论和Kolmogorov复杂度给出等价刻画，应用于随机化学反应网络证明有界分子数的随机轨迹具有非Zeno性质。
This paper is a survey of applications of the theory of algorithmic randomness to ergodic theory. We establish various degrees of constructivity for asymptotic laws of probability theory. In the framework of the Kolmogorov approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms, we formulate probabilistic laws, i.e. statements which hold almost surely, in a pointwise form, i.e., for Martin-Lof random points. It is shown in this paper that the main statement of ergodic theory - Birkhoff ergodic theorem, is non-constructive in the s…
The Rate-Distortion-Perception Trade-Off with Algorithmic Realism
针对有损压缩中理论上公共随机性对真实性的价值与实践不符的问题，提出基于通用评判器的真实性约束，刻画了对应最优率-失真权衡并证明其无需公共随机性即可渐近实现。
Computability of probability measures and Martin-Lof randomness over metric spaces
在本文中，我们研究算法的随机性更一般的空间比康托空间，即可计算度量空间。要做到这一点，我们首先开发一个统一的框架，允许计算的概率措施。我们表明，任何可计算的度量空间与可计算的概率测度同构的康托空间在可计算和测量理论的意义。我们表明，任何可计算的度量空间承认一个普遍的一致随机性测试（没有进一步的假设）。
A didactical survey of the foundations of Algorithmic Information Theory. These notes are short on motivation, history and background but introduce some of the main techniques and concepts of the field. The "manuscript" has been evolving over the years. Please, look at "Version history" below to see what has changed when.
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…
Von Neumann Entropy and Quantum Algorithmic Randomness
本文研究量子算法随机性与冯·诺依曼熵的关系，证明可计算量子Schnorr随机态的熵密度极限为1，定义量子s-测试并建立熵密度与覆盖s的关系，推导熵密度下界与强量子随机性的关联及量子Schnorr随机性的一致可积等价条件。
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditiona…
An algorithmic complexity interpretation of Lin's third law of information theory
代替静态熵，我们断言静态结构（如固体）的柯尔莫哥洛夫复杂性是无序（或混沌）的适当度量。在周围完全随机的宇宙中，一个静态结构充当了一个干扰实体，它在随机性中引入了局部破坏。这通过选择规则$R$来建模，该选择规则选择命中结构的随机输入序列的子序列。通过随机二元序列随机性和混沌性的不等式，我们认为Lin的稳定性概念对应于所选子序列中1的频率的稳定性。这就解释了为什么更复杂的静态结构更不稳定。林的第三定律被表述为静态结构朝着符合宇宙的完美随机性所经历的必然变化。
The mission of statistics is to provide adequate statistical hypotheses (models) for observed data. But what is an "adequate" model? To answer this question, one needs to use the notions of algorithmic information theory. It turns out that for every data string $x$ one can naturally define "stochasticity profile", a curve that represents a trade-off between complexity of a model and its adequacy. This curve has four different equivalent definitions in terms of (1)~randomness deficiency, (2)~minimal description length, (3)~position in the lists of simple strings and (4)~Kolmogorov complexity wi…
Compression, Regularity, Randomness and Emergent Structure: Rethinking Physical Complexity in the Data-Driven Era
本文提出沿规律性、随机性和复杂性三轴统一组织统计、算法与动力学复杂性度量的框架，并阐明数据驱动方法如何作为经典复杂性理想的可计算近似。
A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and Martin-Lof-random-low sets: three different definitions turns out to be equivalent for very non-trivial reasons. This paper, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of the proof of this equivalence and some related results. We assume that the reader is familiar with basic notions of algorithmic information theory.
The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences $A$ and $B$, $B$ is Martin-Löf random and $A$ is Martin-Löf random relative to $B$ if and only if the interleaved sequence $A \uplus B$ is Martin-Löf random. This implies that $A$ is relative random to $B$ if and only if $B$ is random relative to $A$ \cite{vanLambalgen}, \cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this phenomenon for different notions of time-bounded relative rando…
In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences. In the ideal cases one can obtain the desired un-biasing. This relies critically on the independence of the source, a notion we rigorously define for our model. In real cases, affected by imperfections in measurement and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts sufficiently slowly", the result can be arbitrarily close to un-biasing. For infinite sequences, normalisation can both increase or decrea…
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings $x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having $x$ and the complexity profile of the pair and the other one having $y$ and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, \ldots , x_\ell)$ by $\ell$ parties, each one having one component of the tuple and …
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