# Completeness and Reduction in Algebraic Complexity Theory

Book

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

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 105-116
8. Peter Bürgisser
Pages 117-134
9. Peter Bürgisser
Pages 135-147
10. Back Matter
Pages 149-168

### 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

### 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
• 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
• Buy this book on publisher's site
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