عنوان

Algebraic Complexity Theory

(Number (ISBN

9783642082283

(Number (ISBN

9783662033388

Number

b407027

Title Proper

Algebraic Complexity Theory

General Material Designation

[Book]

Other Title Information

With the Collaboration of Thomas Lickteig /

First Statement of Responsibility

by Peter Bürgisser, Michael Clausen, Mohammad Amin Shokrollahi.

Place of Publication, Distribution, etc.

Berlin, Heidelberg :

Name of Publisher, Distributor, etc.

Imprint: Springer,

Date of Publication, Distribution, etc.

1997.

Series Title

Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,

Volume Designation

315

ISSN of Series

0072-7830 ;

Text of Note

1. Introduction -- I. Fundamental Algorithms -- 2. Efficient Polynomial Arithmetic -- 3. Efficient Algorithms with Branching -- II. Elementary Lower Bounds -- 4. Models of Computation -- 5. Preconditioning and Transcendence Degree -- 6. The Substitution Method -- 7. Differential Methods -- III. High Degree -- 8. The Degree Bound -- 9. Specific Polynomials which Are Hard to Compute -- 10. Branching and Degree -- 11. Branching and Connectivity -- 12. Additive Complexity -- IV. Low Degree -- 13. Linear Complexity -- 14. Multiplicative and Bilinear Complexity -- 15. Asymptotic Complexity of Matrix Multiplication -- 16. Problems Related to Matrix Multiplication -- 17. Lower Bounds for the Complexity of Algebras -- 18. Rank over Finite Fields and Codes -- 19. Rank of 2-Slice and 3-Slice Tensors -- 20. Typical Tensorial Rank -- V. Complete Problems -- 21. P Versus NP: A Nonuniform Algebraic Analogue -- List of Notation.

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Text of Note

The algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under standing of the intrinsic computational difficulty of problems.

International Standard Book Number

9783642082283

Title

Springer eBooks

Algorithms.

Combinatorics.

Computer software.

Geometry, Algebraic.

Group theory.

Mathematics.

Matrix theory.

Bürgisser, Peter.

Clausen, Michael.

Shokrollahi, Mohammad Amin.

SpringerLink (Online service)

Date of Transaction

20190301073800.0

Electronic name

[Book]

Y