The TMs are fed with a series of inputs where the

input consists of

consecutive black cells on an otherwise white tape. By convention, the TM computation starts with its head in the rightmost cell in state 1. The TMs are numbered using Wolfram's enumeration scheme [1]. Small

and big

notation indicates the asymptotic behavior of space or time usage. For example,

denotes linear limit behavior and

denotes super-polynomial limit behavior (in the limit, growing faster than any polynomial).

The Busy Beaver is the TM in this space of three states and two colors that has the largest computation time at each input. It can be seen as TM number 599603, which lives in

-space and

-time. Since the runtimes grow so astronomically fast, only the computations corresponding to the first five inputs are shown.

The fractal dimension is computed by using an approximation of the box-counting dimension. Since the convergence rates of these approximations are in general very slow, the sequences are analyzed and small values are extrapolated to the larger setting [4].

[2] J. J. Joosten, F. Soler-Toscano, and H. Zenil, "Program-size versus Time Complexity Slowdown and Speed-up Phenomena in the Micro-cosmos of Small Turing Machines,"

*International Journal of Unconventional Computing*,

**7**(5), 2011 pp. 353-387.

[4] J. J. Joosten, F. Soler-Toscano, and H. Zenil. "Fractal Dimension versus Computational Complexity."

arxiv.org/abs/1309.1779.

[5] H. Zenil, F. Soler-Toscano, and J. J. Joosten. "Empirical Encounters with Computational Irreducibility and Unpredictability,"

*Minds and Machines*,

**22**(3), 2012 pp. 149-165.

doi:dx.doi.org/10.1007/s11023-011-9262-y.