53 research outputs found
Boundedness of the domain of definition is undecidable for polynomial odes
Consider the initial-value problem with computable parameters
dx
dt = p(t, x)
x(t0) = x0,
where p : Rn+1 ! Rn is a vector of polynomials and (t0, x0) 2 Rn+1.
We show that the problem of determining whether the maximal interval
of definition of this initial-value problem is bounded or not is in general
undecidable
Computational bounds on polynomial differential equations
In this paper we study from a computational perspective some prop-erties of the solutions of polynomial ordinary di erential equations.
We consider elementary (in the sense of Analysis) discrete-time dynam-ical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time
dynamical systems which can be expanded into fully polynomial ordinary diferential equations with coe cients in Q[ ]. This sets a computational lower bound on polynomial ODEs since the former class is large enough
to include the dynamics of arbitrary Turing machines.
We also apply the previous methods to show that the problem of de-termining whether the maximal interval of defnition of an initial-value problem defned with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most
56.
Combined with earlier results on the computability of solutions of poly-nomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines
Abstraction of Elementary Hybrid Systems by Variable Transformation
Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing
elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in
practice, especially in safety-critical domains. Due to the non-polynomial
expressions which lead to undecidable arithmetic, verification of EHSs is very
hard. Existing approaches based on partition of state space or
over-approximation of reachable sets suffer from state explosion or inflation
of numerical errors. In this paper, we propose a symbolic abstraction approach
that reduces EHSs to polynomial hybrid systems (PHSs), by replacing all
non-polynomial terms with newly introduced variables. Thus the verification of
EHSs is reduced to the one of PHSs, enabling us to apply all the
well-established verification techniques and tools for PHSs to EHSs. In this
way, it is possible to avoid the limitations of many existing methods. We
illustrate the abstraction approach and its application in safety verification
of EHSs by several real world examples
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
Computability of ordinary differential equations
In this paper we provide a brief review of several results about the
computability of initial-value problems (IVPs) defined with ordinary differential
equations (ODEs). We will consider a variety of settings and analyze
how the computability of the IVP will be affected. Computational
complexity results will also be presented, as well as computable versions
of some classical theorems about the asymptotic behavior of ODEs.info:eu-repo/semantics/publishedVersio
Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations
The outcomes of this paper are twofold.
Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple characterization of P. We believe it is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis.
Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations.
Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog model of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both at the computability and complexity level, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both at a computability and at a computational complexity level
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