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Mathematical systems theory

, Volume 3, Issue 2, pp 186–192 | Cite as

On the base-dependence of sets of numbers recognizable by finite automata

  • Alan Cobham
Article

Abstract

It is known that the set of powers of two is recognizable by a finite automaton if the notational base used for representing numbers is itself a power of two but is unrecognizable in all other bases. On the other hand, the set of multiples of two is recognizable no matter what the notational base. It is shown that the latter situation is the exception, the former the rule: the only sets recognizable independently of base are those which are ultimately periodic; others, if recognizable at all, are recognizable only in bases which are powers of some fixed positive integer.

Keywords

Positive Integer Computational Mathematic Finite Automaton Notational Base 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. R. Büchi, Weak second-order arithmetic and finite automata,Z. Math. Logik Grundlagen Math. 6 (1960), 66–92.Google Scholar
  2. [2]
    J. W. S. Cassels,An Introduction to Diophantine Approximation, Cambridge University Press, 1957.Google Scholar
  3. [3]
    M. Minsky andS. Papert, Unrecognizable sets of numbers,J. Assoc. Comput. Mach. 13 (1966), 281–286.Google Scholar
  4. [4]
    M. O. Rabin andD. Scott, Finite automata and their decision problems,IBM J. Res. Develop. 3 (1959), 114–125.Google Scholar

Copyright information

© Swets & Zeitlinger B.V. 1969

Authors and Affiliations

  • Alan Cobham
    • 1
  1. 1.IBM Watson Research CenterYorktown HeightsUSA

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