Related papers: Algorithmic Complexity for Short Binary Strings Ap…
We introduce a complexity measure for symbolic sequences. Starting from a segmentation procedure of the sequence, we define its complexity as the entropy of the distribution of lengths of the domains of relatively uniform composition in…
The use of random sampling in decision-making and control has become popular with the ease of access to graphic processing units that can generate and calculate multiple random trajectories for real-time robotic applications. In contrast to…
We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…
The concept of information has emerged as a language in its own right, bridging several disciplines that analyze natural phenomena and man-made systems. Integrated information has been introduced as a metric to quantify the amount of…
There are several ways to measure the compressibility of a random measure; they include general approaches such as using the rate-distortion curve, as well as more specific notions, such as the Renyi information dimension (RID). The RID…
In this paper we define a notion of automatic randomness tests (ART) which capture measure theoretic typicalness of infinite binary sequences within the framework of automata theory. An individual ART is found to be equivalent to a…
Randomness is fundamental in quantum theory, with many philosophical and practical implications. In this paper we discuss the concept of algorithmic randomness, which provides a quantitative method to assess the Borel normality of a given…
The problem addressed concerns the determination of the average number of successive attempts of guessing a word of a certain length consisting of letters with given probabilities of occurrence. Both first- and second-order approximations…
As is the case of many signals produced by complex systems, language presents a statistical structure that is balanced between order and disorder. Here we review and extend recent results from quantitative characterisations of the degree of…
Infinite words, also known as streams, hold significant interest in computer science and mathematics, raising the natural question of how their complexity should be measured. We introduce cellular automaton reducibility as a measure of…
A practical measure for the complexity of sequences of symbols (``strings'') is introduced that is rooted in automata theory but avoids the problems of Kolmogorov-Chaitin complexity. This physical complexity can be estimated for ensembles…
In statistics and machine learning, logistic regression is a widely-used supervised learning technique primarily employed for binary classification tasks. When the number of observations greatly exceeds the number of predictor variables, we…
We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that $P=BPP$, it also…
How to achieve an arbitrary real-valued probability amplitude in the general single-partite or multipartite quantum system without measuring any other quantum state's probability amplitude? How to achieve an arbitrary real-valued…
In this paper, we study randomized methods for feedback design of uncertain systems. The first contribution is to derive the sample complexity of various constrained control problems. In particular, we show the key role played by the…
Algorithmic modeling relies on limited information in data to extrapolate outcomes for unseen scenarios, often embedding an element of arbitrariness in its decisions. A perspective on this arbitrariness that has recently gained interest is…
Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the…
A simple, intuitive approach to the assessment of probabilistic inferences is introduced. The Shannon information metrics are translated to the probability domain. The translation shows that the negative logarithmic score and the geometric…
Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information…