Thumbnail
Access Restriction
Subscribed

Author Gill, John ♦ Blum, Manuel
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1974
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract Let $\textit{h}$ be a recursive function. A partial recursive function $\textit{&psgr;}$ is i.o. (infinitely often) $\textit{h}-complex$ if every program for $\textit{&psgr;}$ requires more than $\textit{h}(\textit{&khgr;})$ steps to compute $\textit{&psgr;}(\textit{&khgr;})$ for infinitely many inputs $\textit{&khgr;}.$ A more stringent notion is that of $\textit{&psgr;}$ being a.e. (almost everywhere) $\textit{h}-complex:$ $\textit{&psgr;}$ is a.e. $\textit{h}-complex$ if every program for $\textit{&psgr;}$ requires more than $\textit{h}(\textit{&khgr;})$ steps to compute $\textit{&psgr;}(\textit{&khgr;})$ for all but finitely many inputs $\textit{&khgr;}.These$ two definitions of $\textit{h}-complex$ functions do not yield the same theorems. Although it is possible to prove of every i.o. $\textit{h}-complex$ recursive function that it is i.o. $\textit{h}-complex,$ it is not possible to prove of every a.e. $\textit{h}-complex$ recursive function that it is a.e. $\textit{h}-complex.$ Similarly, recursive functions not i.o. $\textit{h}-complex$ can be proven to be such, but recursive functions not a.e. $\textit{h}-complex$ cannot be so proven.The construction of almost everywhere complex recursive functions appears much more difficult than the construction of infinitely often complex recursive functions. There have been found no “natural” examples of recursive functions requiring more than polynomial time for all but finitely many inputs. It is shown that from a single example of a moderately a.e. complex recursive function, one can obtain a.e. very complex recursive functions.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1974-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 21
Issue Number 3
Page Count 11
Starting Page 425
Ending Page 435


Open content in new tab

   Open content in new tab
Source: ACM Digital Library