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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 5030593, 21 pages
Research Article

Fractal Dimension versus Process Complexity

1Departamento de Lògica, Història i Filosofia de la Ciéncia, Universitat de Barcelona, 08001 Barcelona, Spain
2Algorithmic Nature Group, LABORES, 75006 Paris, France
3Departamento de Filosofía, Lógica y Filosofía de la Ciencia, Universidad de Sevilla, 41018 Seville, Spain
4Unit of Computational Medicine, SciLifeLab, Department of Medicine Solna, Center for Molecular Medicine, Karolinska Institute, 171 76 Stockholm, Sweden
5Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK

Received 1 May 2016; Accepted 29 June 2016

Academic Editor: Joao Florindo

Copyright © 2016 Joost J. Joosten et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machine and any particular input , we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation . In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in time , we have empirically verified that the corresponding dimension is , a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.