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Field name	Details
Dewey Class	518
Title	Completeness and Reduction in Algebraic Complexity Theory ([EBook] /) / by Peter Bürgisser.
Author	Bürgisser, Peter
Other name(s)	SpringerLink (Online service)
Publication	Berlin, Heidelberg : : Springer Berlin Heidelberg : : Imprint: Springer, , 2000.
Physical Details	XII, 168 p. : online resource.
Series	Algorithms and computation in mathematics 1431-1550 ; ; 7
ISBN	9783662041796
Summary Note	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.:
Contents note	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.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-3-662-04179-6
Links to Related Works	Subject References: Algebra . Computational Mathematics and Numerical Analysis . Computer mathematics . Computers . Mathematics . Theory of Computation . Authors: Bürgisser, Peter . Corporate Authors: SpringerLink (Online service) . Series: Algorithms and computation in mathematics . Classification: 518 .

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Completeness and Reduction in Algebraic Complexity Theory

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