# Improved Bounds for Functions Related to Busy Beavers

## Authors

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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

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## 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.