Complexity Theory for Operators in Analysis

We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An advantage of using string functions is that we can define their size in a way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.