The Hausdorff dimension of the halting self-similar sets of T-universal prefix-free machines

Abstract

In our former work [K. Tadaki, Hokkaido Math. J., vol. 31, pp. 219–253, 2002] we studied the notion of the halting self-similar set, which is a self-similar set on [0, 1] defined on the halting set of a universal prefix-free machine. Here the notion of universal prefix-free machine is used to define the notion of program-size complexity. We then showed that the halting self-similar set has a Hausdorff dimension of one. Recently, Calude, et al. [C. S. Calude, N. J. Hay, and F. C. Stephan, Research Report of CDMTCS, 365, May 2009] introduced the notion of T -universal prefix-free machine with a real T ∈ (0, 1] as a generalization of a universal prefix-free machine. In this paper, we first generalize our former results on the Hausdorff dimension of the halting self-similar set of a universal prefix-free machine over general prefix-free sets. We then apply the generalization to evaluate the Hausdorff dimension of the halting self-similar sets of various T -universal prefix-free machines, which include the particular T -universal prefix-free machines V T and V T,k introduced in the work of Calude, et al.