Fractal Dimension versus Process Complexity
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 τ and any particular input x, we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations...
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/5030593 |
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| author | Joost J. Joosten Fernando Soler-Toscano Hector Zenil |
| author_facet | Joost J. Joosten Fernando Soler-Toscano Hector Zenil |
| author_sort | Joost J. Joosten |
| collection | DOAJ |
| description | 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 τ and any particular input x, we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation τ(x). 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 O(xn), we have empirically verified that the corresponding dimension is (n+1)/n, 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. |
| format | Article |
| id | doaj-art-ffde989bd56d4d6f9d80bcbe25fbca8d |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-ffde989bd56d4d6f9d80bcbe25fbca8d2025-02-03T01:21:27ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/50305935030593Fractal Dimension versus Process ComplexityJoost J. Joosten0Fernando Soler-Toscano1Hector Zenil2Departamento de Lògica, Història i Filosofia de la Ciéncia, Universitat de Barcelona, 08001 Barcelona, SpainAlgorithmic Nature Group, LABORES, 75006 Paris, FranceAlgorithmic Nature Group, LABORES, 75006 Paris, FranceWe 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 τ and any particular input x, we consider what we call the space-time diagram which is basically the collection of consecutive tape configurations of the computation τ(x). 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 O(xn), we have empirically verified that the corresponding dimension is (n+1)/n, 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.http://dx.doi.org/10.1155/2016/5030593 |
| spellingShingle | Joost J. Joosten Fernando Soler-Toscano Hector Zenil Fractal Dimension versus Process Complexity Advances in Mathematical Physics |
| title | Fractal Dimension versus Process Complexity |
| title_full | Fractal Dimension versus Process Complexity |
| title_fullStr | Fractal Dimension versus Process Complexity |
| title_full_unstemmed | Fractal Dimension versus Process Complexity |
| title_short | Fractal Dimension versus Process Complexity |
| title_sort | fractal dimension versus process complexity |
| url | http://dx.doi.org/10.1155/2016/5030593 |
| work_keys_str_mv | AT joostjjoosten fractaldimensionversusprocesscomplexity AT fernandosolertoscano fractaldimensionversusprocesscomplexity AT hectorzenil fractaldimensionversusprocesscomplexity |