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Complexity Theory for Operators in Analysis

Complexity Theory for Operators in Analysis AKITOSHI KAWAMURA, University of Tokyo STEPHEN COOK, University of Toronto 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 in nite 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 de ne 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 de ne 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computation Theory (TOCT) Association for Computing Machinery

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