TY - GEN

T1 - Complexity theory for operators in analysis

AU - Kawamura, Akitoshi

AU - Cook, Stephen

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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U2 - 10.1145/1806689.1806758

DO - 10.1145/1806689.1806758

M3 - Conference contribution

AN - SCOPUS:77954752885

SN - 9781605588179

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 495

EP - 502

BT - STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing

T2 - 42nd ACM Symposium on Theory of Computing, STOC 2010

Y2 - 5 June 2010 through 8 June 2010

ER -