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  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 . 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.
What people are saying - Write a review
We haven't found any reviews in the usual places.
Other editions - View all
algebraic algorithm arithmetic operations assume Biirgisser bijection Boolean circuits bounded BP(VNPfc BSS-model Chap claim class VP Cn(h coefficients Comp complete graph complexity class compute constants contained Corollary corresponding countable cr-limit sets Cut9 cycle covers cycle format decision problem defined diagonal digraph disjoint union Dn(h edge weighted graph evaluate finite field fn(x formula Gelfand-Tsetlin basis graph G graph property Hamilton cycle Hence homogeneous immanants implies indeterminates induction integer polynomial irreducible k[Xi Lemma Let f linear matrix Moreover multiplication natural numbers nonscalar nonuniform number of variables obtain p-bounded function p-computable families p-definable families p-degrees p-family p-projection P/poly partition permanent planar planar graphs polynomial f polynomial hierarchy poset projection Prop prove representation result s-t-paths satisfying Sect sequence straight-line program string functions subgraph subsets Theorem theory Turing machine Valiant's hypothesis vector VNP-complete VNPh weighted graph