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© 2000

Completeness and Reduction in Algebraic Complexity Theory

Book

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 7)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Peter Bürgisser
    Pages 1-9
  3. Peter Bürgisser
    Pages 11-36
  4. Peter Bürgisser
    Pages 37-60
  5. Peter Bürgisser
    Pages 61-79
  6. Peter Bürgisser
    Pages 81-103
  7. Peter Bürgisser
    Pages 117-134
  8. Peter Bürgisser
    Pages 135-147
  9. Back Matter
    Pages 149-168

About this book

Introduction

One of the most important and successful theories in computational complex­ ity is that of NP-completeness. This discrete theory is based on the Turing machine model and achieves a classification of discrete computational prob­ lems according to their algorithmic difficulty. Turing machines formalize al­ gorithms which operate on finite strings of symbols over a finite alphabet. By contrast, in algebraic models of computation, the basic computational step is an arithmetic operation (or comparison) of elements of a fixed field, for in­ stance of real numbers. Hereby one assumes exact arithmetic. In 1989, Blum, Shub, and Smale [12] combined existing algebraic models of computation with the concept of uniformity and developed a theory of NP-completeness over the reals (BSS-model). Their paper created a renewed interest in the field of algebraic complexity and initiated new research directions. The ultimate goal of the BSS-model (and its future extensions) is to unite classical dis­ crete complexity theory with numerical analysis and thus to provide a deeper foundation of scientific computation (cf. [11, 101]). Already ten years before the BSS-paper, Valiant [107, 110] had proposed an analogue of the theory of NP-completeness in an entirely algebraic frame­ work, in connection with his famous hardness result for the permanent [108]. While the part of his theory based on the Turing approach (#P-completeness) is now standard and well-known among the theoretical computer science com­ munity, his algebraic completeness result for the permanents received much less attention.

Keywords

NP-completeness Notation algebra complexity complexity theory

Authors and affiliations

  1. 1.Fachbereich 17 • Mathematik-InformatikUniversität-Gesamthochschule PaderbornPaderbornGermany

Bibliographic information

  • Book Title Completeness and Reduction in Algebraic Complexity Theory
  • Authors Peter Bürgisser
  • Series Title Algorithms and Computation in Mathematics
  • DOI https://doi.org/10.1007/978-3-662-04179-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2000
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-66752-0
  • Softcover ISBN 978-3-642-08604-5
  • eBook ISBN 978-3-662-04179-6
  • Series ISSN 1431-1550
  • Edition Number 1
  • Number of Pages XII, 168
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Computational Mathematics and Numerical Analysis
    Theory of Computation
    Algebra
  • Buy this book on publisher's site
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Reviews

".... The subject matter of the book is not easy, since it involves prerequisites from several areas, among them complexity theory, combinatorics, analytic number theory, and representations of symmetric and general linear groups. But the author goes to great lengths to motivate his results, to put them into perspective, and to explain the proofs carefully. In summary, this monograph advances its area of algebraic complexity theory, and is a must for people for working on this subject. And it is a pleasure to read."

Joachim von zur Gathen, Mathematical Reviews, Issue 2001g