Related papers: A Complexity Theory for Hard Enumeration Problems
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case…
This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it…
This article surveys quantum computational complexity, with a focus on three fundamental notions: polynomial-time quantum computations, the efficient verification of quantum proofs, and quantum interactive proof systems. Properties of…
A method is suggested for treating those complicated physical problems for which exact solutions are not known but a few approximation terms of a calculational algorithm can be derived. The method permits one to answer the following rather…
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.
The best way to model, understand, and quantify the information contained in complex systems is an open question in physics, mathematics, and computer science. The uncertain relationship between entropy and complexity further complicates…
The term {\em complexity} is used informally both as a quality and as a quantity. As a quality, complexity has something to do with our ability to understand a system or object -- we understand simple systems, but not complex ones. On…
We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem…
We consider training probabilistic classifiers in the case of a large number of classes. The number of classes is assumed too large to perform exact normalisation over all classes. To account for this we consider a simple approach that…
This survey aims at demonstrating that the structure of precedence constraints plays a tremendous role on the complexity of scheduling problems. Indeed many problems can be NP-hard when considering general precedence constraints, while they…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that…
The question What is Complexity? has occupied a great deal of time and paper over the last 20 or so years. There are a myriad different perspectives and definitions but still no consensus. In this paper I take a phenomenological approach,…
For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…
Nowadays there are several classes of constrained codes intended for different applications. The following two large classes can be distinguished. The first class contains codes with local constraints; for example, the source data must be…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
The authors discuss information-based complexity theory, which is a model of finite-precision computations with real numbers, and its applications to numerical analysis.
A set of general physical principles is proposed as the structural basis for the theory of complex systems. First the concept of harmony is analyzed and its different aspects are uncovered. Then the concept of reflection is defined and…
In classification problems, especially those that categorize data into a large number of classes, the classes often naturally follow a hierarchical structure. That is, some classes are likely to share similar structures and features. Those…
Sentence simplification is the task of rewriting texts so they are easier to understand. Recent research has applied sequence-to-sequence (Seq2Seq) models to this task, focusing largely on training-time improvements via reinforcement…