20 research outputs found
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
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
Turing machines can be efficiently simulated by the General Purpose Analog Computer
The Church-Turing thesis states that any sufficiently powerful computational
model which captures the notion of algorithm is computationally equivalent to
the Turing machine. This equivalence usually holds both at a computability
level and at a computational complexity level modulo polynomial reductions.
However, the situation is less clear in what concerns models of computation
using real numbers, and no analog of the Church-Turing thesis exists for this
case. Recently it was shown that some models of computation with real numbers
were equivalent from a computability perspective. In particular it was shown
that Shannon's General Purpose Analog Computer (GPAC) is equivalent to
Computable Analysis. However, little is known about what happens at a
computational complexity level. In this paper we shed some light on the
connections between this two models, from a computational complexity level, by
showing that, modulo polynomial reductions, computations of Turing machines can
be simulated by GPACs, without the need of using more (space) resources than
those used in the original Turing computation, as long as we are talking about
bounded computations. In other words, computations done by the GPAC are as
space-efficient as computations done in the context of Computable Analysis
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
A Universal Ordinary Differential Equation
An astonishing fact was established by Lee A. Rubel (1981): there exists a
fixed non-trivial fourth-order polynomial differential algebraic equation (DAE)
such that for any positive continuous function on the reals, and for
any positive continuous function , it has a
solution with for all . Lee A. Rubel
provided an explicit example of such a polynomial DAE. Other examples of
universal DAE have later been proposed by other authors. However, Rubel's DAE
\emph{never} has a unique solution, even with a finite number of conditions of
the form .
The question whether one can require the solution that approximates
to be the unique solution for a given initial data is a well known open problem
[Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we
solve it and show that Rubel's statement holds for polynomial ordinary
differential equations (ODEs), and since polynomial ODEs have a unique solution
given an initial data, this positively answers Rubel's open problem. More
precisely, we show that there exists a \textbf{fixed} polynomial ODE such that
for any and there exists some initial condition that
yields a solution that is -close to at all times.
In particular, the solution to the ODE is necessarily analytic, and we show
that the initial condition is computable from the target function and error
function
The Structure of Differential Invariants and Differential Cut Elimination
The biggest challenge in hybrid systems verification is the handling of
differential equations. Because computable closed-form solutions only exist for
very simple differential equations, proof certificates have been proposed for
more scalable verification. Search procedures for these proof certificates are
still rather ad-hoc, though, because the problem structure is only understood
poorly. We investigate differential invariants, which define an induction
principle for differential equations and which can be checked for invariance
along a differential equation just by using their differential structure,
without having to solve them. We study the structural properties of
differential invariants. To analyze trade-offs for proof search complexity, we
identify more than a dozen relations between several classes of differential
invariants and compare their deductive power. As our main results, we analyze
the deductive power of differential cuts and the deductive power of
differential invariants with auxiliary differential variables. We refute the
differential cut elimination hypothesis and show that, unlike standard cuts,
differential cuts are fundamental proof principles that strictly increase the
deductive power. We also prove that the deductive power increases further when
adding auxiliary differential variables to the dynamics