Theory of Computing Systems

, Volume 35, Issue 1, pp 1–11

Improved Bounds for Functions Related to Busy Beavers


  • A. M. Ben-Amram
    • The Academic College of Tel-Aviv-Yaffo, 4 Antokolski St.,
  • H. Petersen
    • The University of Stuttgart, Institute of Computer Science,

DOI: 10.1007/s00224-001-1052-0

Cite this article as:
Ben-Amram, A. & Petersen, H. Theory Comput. Systems (2002) 35: 1. doi:10.1007/s00224-001-1052-0


Consider Turing machines that use a tape infinite in both directions, with the tape alphabet {0,1} . Rado's busy beaver function, ones(n), is the maximum number of 1's such a machine, with n states, started on a blank (all-zero) tape, may leave on its tape when it halts. The function ones(n) is non-computable; in fact, it grows faster than any computable function. Other functions with a similar nature can be defined also. All involve machines of n states, started on a blank tape. 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 new bounds on these functions in terms of each other. Specifically, we bound time(n) by num(n+o(n)), improving on the previously known bound num(3n+6) . This result is obtained using a kind of ``self-interpreting'' Turing machine. We also improve on the trivial relation space(n) ≤ time(n) , using a technique of counting crossing sequences.

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© Springer-Verlag New York 2002