### 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 language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 432-444 |

Number of pages | 13 |

Edition | PART 2 |

ISBN (Print) | 9783662444641 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Externally published | Yes |

Event | 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 - Budapest, Hungary Duration: Aug 25 2014 → Aug 29 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 2 |

Volume | 8635 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014 |
---|---|

Country | Hungary |

City | Budapest |

Period | 8/25/14 → 8/29/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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

**Small complexity classes for computable analysis.** / Kawamura, Akitoshi; Ota, Hiroyuki.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings.*PART 2 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 2, vol. 8635 LNCS, Springer Verlag, pp. 432-444, 39th International Symposium on Mathematical Foundations of Computer Science, MFCS 2014, Budapest, Hungary, 8/25/14. https://doi.org/10.1007/978-3-662-44465-8_37

}

TY - GEN

T1 - Small complexity classes for computable analysis

AU - Kawamura, Akitoshi

AU - Ota, Hiroyuki

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84906274009&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906274009&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-44465-8_37

DO - 10.1007/978-3-662-44465-8_37

M3 - Conference contribution

AN - SCOPUS:84906274009

SN - 9783662444641

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 432

EP - 444

BT - Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Proceedings

PB - Springer Verlag

ER -