### On Almost Everywhere Complex Recursive FunctionsOn Almost Everywhere Complex Recursive Functions

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 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

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Source: ACM Digital Library