Open AccessBook
Complexity and Real Computation
TLDR
This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.Abstract:
1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.read more
Citations
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MonographDOI
Planning Algorithms: Introductory Material
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Journal ArticleDOI
On the mathematical foundations of learning
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Computable Analysis : An Introduction
TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
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Mathematical problems for the next century
TL;DR: Arnabels invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay.
References
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Journal ArticleDOI
On the distribution of a scaled condition number
TL;DR: In this paper, the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult is given, and the bounds for the condition number distribution when A has real or complex normally distributed elements are given.
Journal Article
Degree of difficulty of computing a function and a partial ordering of recursive sets
Book ChapterDOI
On the complexity of computations under varying sets of primitives
TL;DR: The goal of the current research is the study of lower bounds on the complexity of a set of searching problems under various restrictions on the nature of the primitive operation used to determine each branch in a search tree.
Proceedings ArticleDOI
Bounds for the computational power and learning complexity of analog neural nets
TL;DR: In this paper, it was shown that neural nets with piecewise-polynomial activation functions and arbitrary real weights can be simulated for Boolean inputs and outputs by neural nets of a somewhat larger size and depth with Heaviside gates and weights from {-1, 0, 1}.
Journal ArticleDOI
An Efficient Algorithm for the Complex Roots Problem
C. Andrew Neff,John H. Reif +1 more
TL;DR: A remarkable property of this algorithm is that it does not require any assumptions about the root separation off, which were either explicitly, or implicitly, required by previous algorithms, and it also has a work-efficient parallel implementation.