Complexity Theory of Real Functions

Starting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com­ putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some­ times also yield interesting new practical algorithms. A typical exam­ ple is the application of the ellipsoid algorithm to combinatorial op­ timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe­ matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc­ tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per­ mutation groups. In the area of numerical computation, there are also two tradi­ tionally independent approaches: recursive analysis and numerical analysis.

Table of Contents:
Mathematics background.- Notation.- 1 Basics in Discrete Complexity Theory.- 1.1 Models of computation and complexity classes.- 1.2 NP-completeness.- 1.3 Polynomial-time hierarchy.- 1.4 Relativization.- 1.5 Probabilistic complexity classes.- 1.6 Complexity of counting.- 1.7 One-way functions.- 1.8 Polynomial-size circuits and sparse sets.- 2 Computational Complexity of Real Functions.- 2.1 Computable real numbers.- 2.2 Complexity of computable real numbers.- 2.3 Computable real functions.- 2.4 Complexity of computable real functions.- 2.5 Computable multi-dimensional functions.- 2.6 Partial computable real functions and recursively open sets.- 2.7 Computable numerical operators.- 3 Maximization.- 3.1 Computability of the maximum points.- 3.2 Maximization and nondeterminism.- 3.3 Maximum values and NP real numbers.- 3.4 Complexity of NP real numbers.- 3.5 Maximization and NP real functions.- 3.6 Hierarchy of min-max operations.- 3.7 Complexity of NP real functions.- 3.8 Open questions.- 4 Roots and Inverse Functions.- 4.1 Computability of roots.- 4.2 Complexity of roots and inverse modulus of continuity.- 4.3 Complexity of roots and differentiability.- 4.4 Log-space computable real functions.- 4.5 Log-space computability of roots of one-to-one functions.- 4.8 Open questions.- 5 Measure and Integration.- 5.1 Recursive measure theory.- 5.2 Polynomial-time approximation.- 5.3 Polynomial-time approximation and probabilistic computation.- 5.4 Complexity of integration.- 5.5 Open questions.- 6 Differentiation.- 6.1 Computability of derivatives.- 6.2 Derivatives of analytic functions.- 6.3 Functions of bounded variations.- 7 Ordinary Differential Equations.- 7.1 ODEs without the Lipschitz condition.- 7.2 ODEs with the Lipschitz condition: upper bound.- 7.3 ODEs with the Lipschitz condition: lower bound.- 7.4 Open questions.- 8 Approximation by Polynomials.- 8.1 Polynomial Version of the Weierstrass approximation theorem.- 8.2 Best Chebyshev approximation: complexity of the errors.- 8.3 Best Chebyshev approximation: complexity of the approximation functions.- 9 An Optimization Problem in Control Theory.- 9.1 A discrete version.- 9.2 The basic construction.- 9.3 The complexity of LCTEAM.