Completeness and Reduction in Algebraic Complexity Theory.

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.

".... The subject matter of the book is not easy, since it involves prerequisites from several areas, among them complexity theory, combinatorics, analytic number theory, and representations of symmetric and general linear groups. But the author goes to great lengths to motivate his results, to put them into perspective, and to explain the proofs carefully. In summary, this monograph advances its area of algebraic complexity theory, and is a must for people for working on this subject. And it is a pleasure to read."

Joachim von zur Gathen, Mathematical Reviews, Issue 2001g