### Unrecognizable Sets of NumbersUnrecognizable Sets of Numbers

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 Author Minsky, Marvin ♦ Papert, Seymour Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1966 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract When is a set $\textit{A}$ of positive integers, represented as binary numbers, “regular” in the sense that it is a set of sequences that can be recognized by a finite-state machine? Let π $\textit{A}(\textit{n})$ be the number of members of $\textit{A}$ less than the integer $\textit{n}.$ It is shown that the asymptotic behavior of π $A(\textit{n})$ is subject to severe restraints if $\textit{A}$ is regular. These constraints are violated by many important natural numerical sets whose distribution functions can be calculated, at least asymptotically. These include the set $\textit{P}$ of prime numbers for which π $\textit{P}(\textit{n})$ @@@@ $\textit{n}/log$ $\textit{n}$ for large $\textit{n},$ the set of integers $\textit{A}(\textit{k})$ of the form $\textit{nk}$ for which π $\textit{A}(\textit{k})\textit{n})$ @@@@ $\textit{nP/k},$ and many others. The technique cannot, however, yield a decision procedure for regularity since for every infinite regular set $\textit{A}$ there is a nonregular set $\textit{A′}$ for which | π $\textit{A}(\textit{n})$ — π $\textit{A′}(\textit{n})$ | ≤ 1, so that the asymptotic behaviors of the two distribution functions are essentially identical. 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 1966-04-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 13 Issue Number 2 Page Count 6 Starting Page 281 Ending Page 286

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Source: ACM Digital Library