# A note on busy beavers and other creatures

- Received:
- Accepted:

DOI: 10.1007/BF01192693

- Cite this article as:
- Ben-Amram, A.M., Julstrom, B.A. & Zwick, U. Math. Systems Theory (1996) 29: 375. doi:10.1007/BF01192693

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## 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's*busy beaver* function ones(*n*) is the maximum number of 1's such a machine, with*n* 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*) ≤ (2*n* − 1) × ones(3*n* + 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.