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Theory of Computing Systems

, Volume 35, Issue 1, pp 1–11 | Cite as

Improved Bounds for Functions Related to Busy Beavers

  • A. M. Ben-Amram
  • H. Petersen
Article

Abstract.

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.

Keywords

Turing Machine Computable Function Main Program Transition Table Improve Bound 
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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Copyright information

© Springer-Verlag New York 2002

Authors and Affiliations

  • A. M. Ben-Amram
    • 1
  • H. Petersen
    • 2
  1. 1.The Academic College of Tel-Aviv-Yaffo, 4 Antokolski St., amirben@mta.ac.ilIL
  2. 2.The University of Stuttgart, Institute of Computer Science, petersen@informatik.uni-stuttgart.deDE

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