Small complexity classes for computable analysis

Akitoshi Kawamura, Hiroyuki Ota

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

11 Citations (Scopus)

Abstract

Type-two Theory of Effectivity (TTE) provides a general framework for Computable Analysis. To refine it to polynomial-time computability while keeping as much generality as possible, Kawamura and Cook recently proposed a modification to TTE using machines that have random access to an oracle and run in time depending on the "size" of the oracle. They defined type-two analogues of P, NP, PSPACE and applied them to real functions and operators. We further refine their model and study computation below P: type-two analogues of the classes L, NC, and P-completeness under log-space reductions. The basic idea is to use second-order polynomials as resource bounds, as Kawamura and Cook did, but we need to make some nontrivial (yet natural, as we will argue) choices when formulating small classes in order to make them well-behaved. Most notably: we use a modification of the constant stack model of Aehlig, Cook and Nguyen for query tapes in order to allow sufficient oracle accesses without interfering with space bounds; representations need to be chosen carefully, as computational equivalence between them is now finer; uniformity of circuits must be defined with varying sizes of oracles taken into account. As prototypical applications, we recast several facts (some in a stronger form than was known) about the complexity of numerical problems into our framework.

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings
PublisherSpringer Verlag
Pages432-444
Number of pages13
EditionPART 2
ISBN (Print)9783662444641
DOIs
Publication statusPublished - Jan 1 2014
Externally publishedYes
Event39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 - Budapest, Hungary
Duration: Aug 25 2014Aug 29 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume8635 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014
CountryHungary
CityBudapest
Period8/25/148/29/14

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Kawamura, A., & Ota, H. (2014). Small complexity classes for computable analysis. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings (PART 2 ed., pp. 432-444). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8635 LNCS, No. PART 2). Springer Verlag. https://doi.org/10.1007/978-3-662-44465-8_37