2,145 research outputs found
Robust simulations of Turing machines with analytic maps and flows
In this paper, we show that closed-form analytic maps and ows can simulate Turing machines in an error-robust manner. The maps
and ODEs de ning the ows are explicitly obtained and the simulation is performed in real time
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Computability with polynomial differential equations
In this paper, we show that there are Initial Value Problems de ned
with polynomial ordinary di erential equations that can simulate univer-
sal Turing machines in the presence of bounded noise. The polynomial
ODE de ning the IVP is explicitly obtained and the simulation is per-
formed in real time
The Extended Analog Computer and Turing machine
In this paper we compare computational power of two models of analog and classicalcomputers. As a model of analog computer we use the model proposed by Rubel in 1993 called theExtended Analog Computer (EAC) while as a model of classical computer, the Turing machines.Showing that the Extended Analog Computer can robustly generate result of any Turing machinewe use the method of simulation proposed by D.S. Graça, M.L. Campagnolo and J. Buescu [1] in2005
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 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) arbitrarily high computational complexity. 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 maps on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and
Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D
(MDM-2014-0445) via an FPI grant.
Robert Cardona and Eva Miranda are partially supported by the AEI grant PID2019-
103849GB-I00 of MCIN/ AEI /10.13039/501100011033, and AGAUR grant 2017SGR932. Eva
Miranda is supported by the Catalan Institution for Research and Advanced Studies via an
ICREA Academia Prize 2016 and by the Spanish State Research Agency, through the Severo
Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D (project
CEX2020-001084-M).
Daniel Peralta-Salas is supported by the grants CEX2019-000904-S, RED2018-
102650-T, EUR2019-103821 and PID2019-106715GB GB-C21 funded by MCIN/AEI/
10.13039/501100011033.Preprin
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 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 , however
they can be employed to show that an arbitrary tape-bounded Turing machine can
be robustly simulated by a Beltrami flow on (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 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 . Finally, our proof also yields Turing complete
flows and diffeomorphisms on with zero topological entropy, thus
disclosing a certain degree of independence within different hierarchies of
complexity.Comment: overall improvement of the article, proofs revised, 37 pages, 3
figures, final version to appear at J. Math. Pures App
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 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 ChurchTuring 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 complexityPostprint (author's final draft
Computability with polynomial differential equations
Tese dout., Matemática, Inst. Superior Técnico, Univ. Técnica de Lisboa, 2007Nesta dissertação iremos analisar um modelo de computação analógica, baseado
em equações diferenciais polinomiais.
Começa-se por estudar algumas propriedades das equações diferenciais polinomiais, em
particular a sua equivalência a outro modelo baseado em circuitos analógicos (GPAC),
introduzido por C. Shannon em 1941, e que é uma idealização de um dispositivo físico, o
Analisador Diferencial.
Seguidamente, estuda-se o poder computacional do modelo. Mais concretamente,
mostra-se que ele pode simular máquinas de Turing, de uma forma robusta a erros, pelo
que este modelo é capaz de efectuar computações de Tipo-1. Esta simulação é feita em
tempo contínuo. Mais, mostramos que utilizando um enquadramento apropriado, o modelo
é equivalente à Análise Computável, isto é, à computação de Tipo-2.
Finalmente, estudam-se algumas limitações computacionais referentes aos problemas
de valor inicial (PVIs) definidos por equações diferenciais ordinárias. Em particular: (i)
mostra-se que mesmo que o PVI seja definido por uma função analítica e que a mesma,
assim como as condições iniciais, sejam computáveis, o respectivo intervalo maximal de
existência da solução não é necessariamente computável; (ii) estabelecem-se limites para
o grau de não-computabilidade, mostrando-se que o intervalo maximal é, em condições
muito gerais, recursivamente enumerável; (iii) mostra-se que o problema de decidir se o
intervalo maximal é ou não limitado é indecídivel, mesmo que se considerem apenas PVIs
polinomiais
Computation with perturbed dynamical systems
This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability. (C) 2013 Elsevier Inc. All rights reserved.INRIA program "Equipe Associee" ComputR; Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT project [PEst-OE/EEI/LA0008/2011]info:eu-repo/semantics/publishedVersio
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