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
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.MATHCrossRefMathSciNetGoogle 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.MATHMathSciNetGoogle 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.MATHGoogle 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

Personalised recommendations