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" Completeness and Reduction in Algebraic Complexity Theory "
by Peter Bürgisser.
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BL
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577863
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b407082
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| Main Entry
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Bürgisser, Peter.
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| Title & Author
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Completeness and Reduction in Algebraic Complexity Theory\ by Peter Bürgisser.
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| Publication Statement
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Berlin, Heidelberg :: Springer Berlin Heidelberg :: Imprint: Springer,, 2000.
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| Series Statement
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Algorithms and Computation in Mathematics,; 7
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| ISBN
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9783662041796
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: 9783642086045
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| Contents
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1 Introduction -- 2 Valiant's Algebraic Model of NP-Completeness -- 3 Some Complete Families of Polynomials -- 4 Cook's versus Valiant's Hypothesis -- 5 The Structure of Valiant's Complexity Classes -- 6 Fast Evaluation of Representations of General Linear Groups -- 7 The Complexity of Immanants -- 8 Separation Results and Future Directions -- References -- List of Notation.
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| Abstract
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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.
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Mathematics.
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Information theory.
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Algebra.
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Computer science-- Mathematics.
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SpringerLink (Online service)
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