Computational Complexity of Real Functions

Abstract

The aim of this chapter is to present a natural definition of computational complexity of real functions and to study the relationship between complexity and analytical properties of real functions. The field of computational complexity has been largely concerned with discrete problems. The chapter defines the notion of computable real function and establishes basic continuity results. The chapter also defines computational complexity of recursive real numbers and functions and relates this to continuity, proving, for example, that a polynomial time computable function has a polynomially bounded modulus of continuity. A real number is considered a sequence of dyadic rational numbers that converges to it. Computable real functions are to be defined not only on the computable real numbers so a natural approach is via oracle Turing machines (OTM). A root of a recursive real function must be recursive, but a root of a polynomial time computable function need not be polynomial time computable. A polynomial time computable function need not be differentiable; a standard example of an “everywhere continuous nowhere differentiable” function has been shown to be polynomial time computable. Because all polynomial time computable functions are continuous, they are also Riemann integrable. The maximum value of a recursive real function is known (e.g., Lacombe to be a recursive real number). Step functions are a useful tool in analysis and if they have recursive jump points and the jump points are ignored, they are partial recursive.