Mathematical systems theory

, Volume 29, Issue 4, pp 375–386

A note on busy beavers and other creatures

  • A. M. Ben-Amram
  • B. A. Julstrom
  • U. Zwick


Consider Turing machines that read and write the symbols 1 and 0 on a one-dimensional tape that is infinite in both directions, and halt when started on a tape containing all O's. Rado'sbusy beaver function ones(n) is the maximum number of 1's such a machine, withn states, may leave on its tape when it halts. The function ones(n) is noncomputable; in fact, it grows faster than any computable function.

Other functions with a similar nature can also be defined. The function time(n) is the maximum number of moves such a machine may make before halting. The function num(n) is the largest number of 1's such a machine may leave on its tape in the form of a single run; and the function space(n) is the maximum number of tape squares such a machine may scan before it halts.

This paper establishes a variety of bounds on these functions in terms of each other; for example, time(n) ≤ (2n − 1) × ones(3n + 3). In general, we compare the growth rates of such functions, and discuss the problem of characterizing their growth behavior in a more precise way than that given by Rado.


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Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • A. M. Ben-Amram
    • 1
  • B. A. Julstrom
    • 2
  • U. Zwick
    • 1
  1. 1.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Computer ScienceSt. Cloud State UniversitySt. CloudUSA

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