Complexity and real computation 计算复杂性与实计算

作者简介

The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

书籍目录

ForewordPrefaceⅠBasic Development　1 Introduction　2  Definitions and First Properties of Computation　3  Computation over a Ring　4  Decision Problems and Complexity over a Ring　5  The Class NP and NP-Complete Problems　6  Integer Machines　7  Algebraic Settings for the Problem "P≠ NP?"　Appendix AⅡ  Some Geometry of Numerical Algorithms　8  Newton's Method　9  Fundamental Theorem of Algebra: Complexity Aspects　10 Bezout's Theorem　11 Condition Numbers and the Loss of Precision of Linear Equations　12 The Condition Number for Nonlinear Problems　13 The Condition Number in P（H（d））　14 Complexity and the Condition Number　15 Linear Programming　Appendix BⅢ  Complexity Classes over the Reals　16 Deterministic Lower Bounds　17 Probabilistic Machines　18 Parallel Computations　19 Some Separations of Complexity Classes　20 Weak Machines　21 Additive Machines　22 Nonuniform Complexity Classes　23 Descriptive ComplexityReferencesIndex

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