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DOI: 10.4230/LIPIcs.STACS.2014.602
URN: urn:nbn:de:0030-drops-44919
URL: https://drops.dagstuhl.de/opus/volltexte/2014/4491/
Nies, André
Differentiability of polynomial time computable functions
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Abstract
We show that a real z is polynomial time random if and only if each nondecreasing polynomial time computable function is differentiable at z. This establishes an analog in feasible analysis of a recent result of Brattka, Miller and Nies, who characterized computable randomness in terms of differentiability of nondecreasing computable functions.Further, we show that a Martin-Loef random real z is a density-one point if and only if each interval-c.e. function is differentiable at z. (To say z is a density-one point means that every effectively closed class containing z has density one at z. The interval-c.e. functions are, essentially, the variation functions of computable functions.)
The proofs are related: they both make use of the analytical concept of porosity in novel ways, and both use a basic geometric fact on shifting dyadic intervals by 1/3.
BibTeX - Entry
@InProceedings{nies:LIPIcs:2014:4491,
author = {Andr{\'e} Nies},
title = {{Differentiability of polynomial time computable functions}},
booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
pages = {602--613},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-65-1},
ISSN = {1868-8969},
year = {2014},
volume = {25},
editor = {Ernst W. Mayr and Natacha Portier},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4491},
URN = {urn:nbn:de:0030-drops-44919},
doi = {10.4230/LIPIcs.STACS.2014.602},
annote = {Keywords: Polynomial time randomness, feasible analysis, differentiability, porosity}
}
| Keywords: | Polynomial time randomness, feasible analysis, differentiability, porosity | |
| Seminar: | 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014) | |
| Issue date: | 2014 | |
| Date of publication: | 05.03.2014 |
