Computability and Beltrami fields in Euclidean space
In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solution...
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| Published in: | Journal de mathématiques pures et appliquées Vol. 169; pp. 50 - 81 |
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Elsevier Masson SAS
01.01.2023
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| ISSN: | 0021-7824 |
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| Abstract | In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and diffeomorphisms on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.
Dans cet article, on poursuit la recherche des liens entre la théorie de la computation et l'hydrodynamique. On démontre l'existence de solutions stationnaires des équations d'Euler dans l'espace euclidien, de type Beltrami, qui peuvent simuler une machine de Turing universelle. En particulier, ces solutions possèdent des trajectoires indécidables. Jusqu'à présent, les constructions Turing complètes connues des écoulements d'Euler stationnaires en dimension 3 ou supérieure n'étaient pas associées à une métrique prescrite. Nos solutions ne sont pas d'énergie finie, et leur construction utilise de manière essentielle la non-compacité de R3, même si elles peuvent être utilisées pour montrer qu'une machine de Turing arbitrairement bornée peut être simulée de manière robuste par un flot de Beltrami sur T3 (avec la métrique plate standard). Cela montre l'existence de solutions stationnaires aux équations d'Euler sur le tore plat présentant des phénomènes dynamiques à complexité computationnelle (robuste) arbitraire. On quantifie le coût énergétique d'un champ de Beltrami sur T3 pour simuler une machine de Turing avec une bande bornée, fournissant ainsi un support supplémentaire pour la thèse de Church-Turing à espace borné. Une autre conséquence de notre construction est qu'un champ de Beltrami gaussien aléatoire sur l'espace euclidien présente une complexité computationnelle arbitrairement élevée à probabilité 1. Enfin, notre preuve produit également des flots et des difféomorphismes complets de Turing sur S2 à entropie topologique nulle, révélant ainsi un certain degré d'indépendance entre les différentes hiérarchies de complexité. |
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| AbstractList | In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and diffeomorphisms on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.
Dans cet article, on poursuit la recherche des liens entre la théorie de la computation et l'hydrodynamique. On démontre l'existence de solutions stationnaires des équations d'Euler dans l'espace euclidien, de type Beltrami, qui peuvent simuler une machine de Turing universelle. En particulier, ces solutions possèdent des trajectoires indécidables. Jusqu'à présent, les constructions Turing complètes connues des écoulements d'Euler stationnaires en dimension 3 ou supérieure n'étaient pas associées à une métrique prescrite. Nos solutions ne sont pas d'énergie finie, et leur construction utilise de manière essentielle la non-compacité de R3, même si elles peuvent être utilisées pour montrer qu'une machine de Turing arbitrairement bornée peut être simulée de manière robuste par un flot de Beltrami sur T3 (avec la métrique plate standard). Cela montre l'existence de solutions stationnaires aux équations d'Euler sur le tore plat présentant des phénomènes dynamiques à complexité computationnelle (robuste) arbitraire. On quantifie le coût énergétique d'un champ de Beltrami sur T3 pour simuler une machine de Turing avec une bande bornée, fournissant ainsi un support supplémentaire pour la thèse de Church-Turing à espace borné. Une autre conséquence de notre construction est qu'un champ de Beltrami gaussien aléatoire sur l'espace euclidien présente une complexité computationnelle arbitrairement élevée à probabilité 1. Enfin, notre preuve produit également des flots et des difféomorphismes complets de Turing sur S2 à entropie topologique nulle, révélant ainsi un certain degré d'indépendance entre les différentes hiérarchies de complexité. |
| Author | Cardona, Robert Miranda, Eva Peralta-Salas, Daniel |
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| Title | Computability and Beltrami fields in Euclidean space |
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