# Some applications of logic to feasibility in higher types

Ignjatovic, Aleksandar & Sharma, Arun
(2004)
Some applications of logic to feasibility in higher types.
*ACM Transactions on Computational Logic*, *5*(2), pp. 332-350.

## Abstract

While it is commonly accepted that computability on a Turing machine in polynomial time represents a correct formalization of the notion of a feasibly computable function, there is no similar agreement on how to extend this notion on functionals, that is, what functionals should be considered feasible. One possible paradigm was introduced by Mehlhorn, who extended Cobham's definition of feasible functions to type 2 functionals. Subsequently, this class of functionals (with inessential changes of the definition) was studied by Townsend who calls this class POLY, and by Kapron and Cook who call the same class basic feasible functionals. Kapron and Cook gave an oracle Turing machine model characterisation of this class. In this article, we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalise the corresponding properties of the class of feasible functions, thus giving further evidence that the notion of feasibility of functionals mentioned above is correctly chosen. We also improve the Kapron and Cook result on machine representation.Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from NN which suitably characterises basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible programs from mathematical proofs that use nonfeasible functions.

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ID Code: | 37247 |
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Item Type: | Journal Article |

Refereed: | Yes |

Additional Information: | Published version available via DOI link. |

Keywords: | bounded arithmetic, functionals, higher-order complexity, second-order theories |

DOI: | 10.1145/976706.976713 |

ISSN: | 1529-3785 |

Subjects: | Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200) |

Divisions: | Current > QUT Faculties and Divisions > Division of Research and Commercialisation |

Deposited On: | 27 Sep 2010 00:08 |

Last Modified: | 29 Feb 2012 14:10 |

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