Topological Dynamics of Cellular Automata: Dimension Matters
Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The mai...
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Gespeichert in:
| Autor*in: |
Sablik, Mathieu [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Theory of computing systems - Springer-Verlag, 1997, 48(2010), 3 vom: 10. Apr., Seite 693-714 |
|---|---|
| Übergeordnetes Werk: |
volume:48 ; year:2010 ; number:3 ; day:10 ; month:04 ; pages:693-714 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00224-010-9255-x |
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OLC2061919227 |
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| 520 | |a Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. | ||
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10.1007/s00224-010-9255-x doi (DE-627)OLC2061919227 (DE-He213)s00224-010-9255-x-p DE-627 ger DE-627 rakwb eng 004 510 VZ 510 000 VZ Sablik, Mathieu verfasserin aut Topological Dynamics of Cellular Automata: Dimension Matters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. Multidimensional cellular automata Topological dynamics Complexity of decision problem Theyssier, Guillaume aut Enthalten in Theory of computing systems Springer-Verlag, 1997 48(2010), 3 vom: 10. Apr., Seite 693-714 (DE-627)222610387 (DE-600)1355722-1 (DE-576)056755198 1432-4350 nnns volume:48 year:2010 number:3 day:10 month:04 pages:693-714 https://doi.org/10.1007/s00224-010-9255-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_70 GBV_ILN_105 GBV_ILN_267 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4700 AR 48 2010 3 10 04 693-714 |
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10.1007/s00224-010-9255-x doi (DE-627)OLC2061919227 (DE-He213)s00224-010-9255-x-p DE-627 ger DE-627 rakwb eng 004 510 VZ 510 000 VZ Sablik, Mathieu verfasserin aut Topological Dynamics of Cellular Automata: Dimension Matters 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2010 Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. Multidimensional cellular automata Topological dynamics Complexity of decision problem Theyssier, Guillaume aut Enthalten in Theory of computing systems Springer-Verlag, 1997 48(2010), 3 vom: 10. Apr., Seite 693-714 (DE-627)222610387 (DE-600)1355722-1 (DE-576)056755198 1432-4350 nnns volume:48 year:2010 number:3 day:10 month:04 pages:693-714 https://doi.org/10.1007/s00224-010-9255-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-MAT GBV_ILN_21 GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_70 GBV_ILN_105 GBV_ILN_267 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4700 AR 48 2010 3 10 04 693-714 |
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Topological Dynamics of Cellular Automata: Dimension Matters |
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topological dynamics of cellular automata: dimension matters |
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Topological Dynamics of Cellular Automata: Dimension Matters |
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Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. © Springer Science+Business Media, LLC 2010 |
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Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. © Springer Science+Business Media, LLC 2010 |
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Abstract Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only $\varPi _{2}^{0}$ in dimension 1, but becomes $\varSigma _{3}^{0}$-hard for dimension 3. © Springer Science+Business Media, LLC 2010 |
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