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Paper Abstract and Keywords
Presentation 2011-10-21 14:25
Complexity Theory for Operators in Analysis
Akitoshi Kawamura (Univ. of Tokyo), Stephen Cook (Univ. of Toronto) COMP2011-32
Abstract (in Japanese) (See Japanese page) 
(in English) We propose a new 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 important advantage of using string 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 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. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of functions on 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 and show that the operator itself is polynomial-space complete.
Keyword (in Japanese) (See Japanese page) 
(in English) computable analysis / computational complexity / representations / second-order polynomials / / / /  
Reference Info. IEICE Tech. Rep., vol. 111, no. 256, COMP2011-32, pp. 25-32, Oct. 2011.
Paper # COMP2011-32 
Date of Issue 2011-10-14 (COMP) 
ISSN Print edition: ISSN 0913-5685    Online edition: ISSN 2432-6380
Copyright
and
reproduction		 All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
Download PDF COMP2011-32

Conference Information
Committee COMP  
Conference Date 2011-10-21 - 2011-10-21 
Place (in Japanese) (See Japanese page) 
Place (in English) Tohoku Univ. 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
Paper Information
Registration To COMP 
Conference Code 2011-10-COMP 
Language Japanese 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Complexity Theory for Operators in Analysis 
Sub Title (in English)  
Keyword(1) computable analysis  
Keyword(2) computational complexity  
Keyword(3) representations  
Keyword(4) second-order polynomials  
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1st Author's Name Akitoshi Kawamura  
1st Author's Affiliation University of Tokyo (Univ. of Tokyo)
2nd Author's Name Stephen Cook  
2nd Author's Affiliation University of Toronto (Univ. of Toronto)
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Speaker Author-1 
Date Time 2011-10-21 14:25:00 
Presentation Time 35 minutes 
Registration for COMP 
Paper # COMP2011-32 
Volume (vol) vol.111 
Number (no) no.256 
Page pp.25-32 
#Pages 8 
Date of Issue 2011-10-14 (COMP) 

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