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This paper presents a novel and straight formulation, and gives a complete insight towards the understanding of the complexity of the problems of the so called NP-Class. In particular, this paper focuses in the Searching of the Optimal…
In this paper, we show how a construction of an implicit complexity model can be implemented using concepts coming from the core of von Neumann algebras. Namely, our aim is to gain an understanding of classical computation in terms of the…
V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…
We develop theory concerning non-uniform complexity in a setting in which the notion of single-pass instruction sequence considered in program algebra is the central notion. We define counterparts of the complexity classes P/poly and…
This is an expository paper about the Borel complexity of structure and classification theorems. It sorts several classical problems relative to known benchmarks of complexity. As a corollary various problems proposed by people such as von…
In this paper, we propose two second-order methods for solving the \(\ell_1\)-regularized composite optimization problem, which are developed based on two distinct definitions of approximate second-order stationary points. We introduce a…
We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational…
G. Edelman, O. Sporns, and G. Tononi have introduced the neural complexity of a family of random variables, defining it as a specific average of mutual information over subfamilies. We show that their choice of weights satisfies two natural…
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
Recently it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and…
Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While…
In this article, we shall describe some of the most interesting topics in the subject of Complexity Science for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary…
This paper depicts algorithms for solving the decision Boolean Satisfiability Problem. An extreme problem is formulated to analyze the complexity of algorithms and the complexity for solving it. A novel and easy reformulation as a lottery…
Text simplification reduces the language complexity of professional content for accessibility purposes. End-to-end neural network models have been widely adopted to directly generate the simplified version of input text, usually functioning…