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Completeness and Reduction in Algebraic Complexity Theory

  • Peter Bürgisser

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

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

Bibliographic information

  • 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
  • Print ISBN 978-3-642-08604-5
  • Online ISBN 978-3-662-04179-6
  • Series Print ISSN 1431-1550
  • Buy this book on publisher's site
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