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Complexity and Real Computation

TL;DR: 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.
Citations
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MonographDOI
01 Jan 2006
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

6,340 citations


Cites background from "Complexity and Real Computation"

  • ...Other models of computation also exist, such as integer RAM and real RAM (see [120]); there are debates as to which model is most appropriate, especially when performing geometric computations with real numbers....

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  • ...Computation models over the reals are covered in [120]....

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Book
26 Jun 2003
TL;DR: Preface, Notations 1.Introduction to Time-Delay Systems I.Robust Stability Analysis II.Input-output stability A.LMI and Quadratic Integral Inequalities Bibliography Index
Abstract: Preface, Notations 1.Introduction to Time-Delay Systems I.Frequency-Domain Approach 2.Systems with Commensurate Delays 3.Systems withIncommensurate Delays 4.Robust Stability Analysis II.Time Domain Approach 5.Systems with Single Delay 6.Robust Stability Analysis 7.Systems with Multiple and Distributed Delays III.Input-Output Approach 8.Input-output stability A.Matrix Facts B.LMI and Quadratic Integral Inequalities Bibliography Index

4,200 citations


Cites background from "Complexity and Real Computation"

  • ...[17] focuses on the development of the theory in numerical analysis....

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Journal ArticleDOI
TL;DR: A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference) and relations of the theory of learning to the mainstream of mathematics are emphasized.
Abstract: (1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares, and for tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is anticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alphabet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cognitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought processes (how the brain works). Mathematics has already played a big role towards the goal of giving a universal foundation of studies in these disciplines. We mention as examples the theory of Neural Networks going back to McCulloch and Pitts [25] and Minsky and Papert [27], the PAC learning of Valiant [40], Statistical Learning Theory as developed by Vapnik [42], and the use of reproducing kernels as in [17] among many other mathematical developments. We are heavily indebted to these developments. Recent discussions with a number of mathematicians have also been helpful. In

1,651 citations


Cites background from "Complexity and Real Computation"

  • ...Lemma 7 of Chapter 14 of [8]; this reference gives more background to this discussion)....

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Book
12 Mar 2014
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.
Abstract: Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

1,330 citations

Journal ArticleDOI
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.
Abstract: V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's 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. I have listed 18 problems, chosen with these criteria:

1,310 citations

References
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Book
01 Jan 1978
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Abstract: A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.

8,196 citations


"Complexity and Real Computation" refers background or methods in this paper

  • ...Karmarkar [1984] gave a proof employing interior point methods. Gonzaga [1989] proves the theorem using the barrier method. Our analysis of the barrier method using the robust ex theorem is from [Renegar and Shub 1992]. For the reduction of the feasibility problem to the barrier method and the bit analysis we have largely followed Vavasis [1991]. Renegar [1995a, 1995b] proves that LPF~ is in P when the condition of the problem is taken into account....

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  • ...See [Griffiths and Harris 1978] for background on projective space and complex algebraic geometry....

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  • ...Karmarkar [1984] gave a proof employing interior point methods. Gonzaga [1989] proves the theorem using the barrier method....

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  • ...Karmarkar [1984] gave a proof employing interior point methods....

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Book
01 Jan 1965
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Abstract: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.

7,422 citations

Book
01 Dec 1986
TL;DR: Introduction and Preliminaries.
Abstract: Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

7,005 citations

Proceedings ArticleDOI
03 May 1971
TL;DR: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology.
Abstract: It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology. Here “reduced” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed.

6,675 citations


"Complexity and Real Computation" refers background or result in this paper

  • ...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it....

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  • ...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]....

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  • ...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula....

    [...]

  • ...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]. In this latter article, a recently discovered letter of G6del to von Neumann dated 1956 is reproduced in which G6del stated the P versus NP question in the form of the time required by a Turing machine to test whether a formula of the predicate calculus has a proof of a given length. The rise of complexity issues in the numerical tradition is less attached to the advent of the digital computer. Early in 1937, in a short note of Scholz [1937], complexity questions arose under the form of the number of additions needed to produce a given integer starting from 1....

    [...]

  • ...At the beginning of the 1970s Cook [1971] defined a reduction for a class of decision problems corresponding to these search problems and proved the existence of complete problems. Concretely, he proved the completeness of the Satisfiability Problem of propositional logic. Independently Levin [1973] obtained similar results for a class of search problems and proved the existence of six complete problems, including the one of finding a satisfying truth assignment for a given propositional formula. This is the search version of Cook's completeness result for decision problems. Shortly afterwards, Karp [1972] considered the class of decision problems dealt with by Cook and coined the name NP for it. Moreover he showed that a series of familiar decision problems from different areas of discrete mathematics were also complete, coining the name NP-complete. This gave strong impetus to the subject that was reflected in work exhibiting hundreds of NP-complete problems and, on the other hand, in attempts to prove the inequality P =f=. NP leading to results on the structure of the class NP. A lively exposition on the P versus NP question (containing a large list of NP-complete problems) can be found in the already classic book by Garey and Johnson [1979]. A survey of the state of the art of this question is given in [Sipser 1992]. In this latter article, a recently discovered letter of G6del to von Neumann dated 1956 is reproduced in which G6del stated the P versus NP question in the form of the time required by a Turing machine to test whether a formula of the predicate calculus has a proof of a given length. The rise of complexity issues in the numerical tradition is less attached to the advent of the digital computer. Early in 1937, in a short note of Scholz [1937], complexity questions arose under the form of the number of additions needed to produce a given integer starting from 1. Seventeen years later Ostrowski [1954] conjectured the optimality of Homer's rule for evaluating univariate polynomials....

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