Complexity theory for operators in analysis

Akitoshi Kawamura, Stephen Cook

Research output: Chapter in Book/Report/Conference proceedingConference contribution

16 Citations (Scopus)

Abstract

We propose a new framework for discussing 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, which we call regular 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 important advantage of using regular functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on regular 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. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem 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 could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomial-space complete.

Original languageEnglish
Title of host publicationSTOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing
Pages495-502
Number of pages8
DOIs
Publication statusPublished - 2010
Event42nd ACM Symposium on Theory of Computing, STOC 2010 - Cambridge, MA, United States
Duration: Jun 5 2010Jun 8 2010

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other42nd ACM Symposium on Theory of Computing, STOC 2010
Country/TerritoryUnited States
CityCambridge, MA
Period6/5/106/8/10

All Science Journal Classification (ASJC) codes

  • Software

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