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Mathematical systems theory

, Volume 29, Issue 4, pp 375–386 | Cite as

A note on busy beavers and other creatures

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

Abstract

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.

Keywords

Turing Machine Computable Function Bell System Technical Journal Marked Zone Blank Tape 
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]
    Brady, Allen H. (1983). The determination of the value of Rado's noncomputable function ∑(k) for four-state Turing machines.Mathematics of Computation, Vol. 40, No. 162, pp. 647–665.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Julstrom, Bryant A. (1992). A bound on the shift function in terms of the Busy Beaver function.SIGACT News, Vol. 23, No. 3 (Summer), pp. 100–106.CrossRefGoogle Scholar
  3. [3]
    Julstrom, Bryant A. (1993). Noncomputability and the Busy Beaver problem (UMAP Unit 728).The UMAP Journal, Vol. 14, No. 1, pp. 39–74.Google Scholar
  4. [4]
    Lin, Shen, and Tibor Rado (1965). Computer studies of Turing machine problems.Journal of the Association for Computing Machinery, Vol. 12, pp. 196–212.zbMATHMathSciNetGoogle Scholar
  5. [5]
    Marxen, Heiner, and Jürgen Buntrock (1990). Attacking the Busy Beaver Problem 5.Bulletin of the European Association for Theoretical Computer Science, Vol. 40, pp. 247–251.zbMATHGoogle Scholar
  6. [6]
    Rado, Tibor (1962). On non-computable functions.The Bell System Technical Journal, Vol. 41, pp. 877–884.MathSciNetGoogle Scholar

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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