Title
Completeness and Reduction in Algebraic Complexity Theory (Algorithms and Computation in Mathematics, 7),Used
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One of the most important and successful theories in computational complex ity is that of NPcompleteness. 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 NPcompleteness over the reals (BSSmodel). Their paper created a renewed interest in the field of algebraic complexity and initiated new research directions. The ultimate goal of the BSSmodel (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 BSSpaper, Valiant [107, 110] had proposed an analogue of the theory of NPcompleteness 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 (#Pcompleteness) is now standard and wellknown among the theoretical computer science com munity, his algebraic completeness result for the permanents received much less attention.
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