1,146 research outputs found
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
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 . This 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. This extends to deterministic
complexity classes above polynomial time.
This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like , 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, i.e.~by using the framework of analysis
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
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
Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime
The synthesis of classical Computational Complexity Theory with Recursive
Analysis provides a quantitative foundation to reliable numerics. Here the
operators of maximization, integration, and solving ordinary differential
equations are known to map (even high-order differentiable) polynomial-time
computable functions to instances which are `hard' for classical complexity
classes NP, #P, and CH; but, restricted to analytic functions, map
polynomial-time computable ones to polynomial-time computable ones --
non-uniformly!
We investigate the uniform parameterized complexity of the above operators in
the setting of Weihrauch's TTE and its second-order extension due to
Kawamura&Cook (2010). That is, we explore which (both continuous and discrete,
first and second order) information and parameters on some given f is
sufficient to obtain similar data on Max(f) and int(f); and within what running
time, in terms of these parameters and the guaranteed output precision 2^(-n).
It turns out that Gevrey's hierarchy of functions climbing from analytic to
smooth corresponds to the computational complexity of maximization growing from
polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete)
Computation, Hard Analysis, and Information-Based Complexity
Bounded -calculus for cone differential operators
We prove that parameter-elliptic extensions of cone differential operators
have a bounded -calculus. Applications concern the Laplacian and the
porous medium equation on manifolds with warped conical singularities
Decoherence rates for Galilean covariant dynamics
We introduce a measure of decoherence for a class of density operators. For
Gaussian density operators in dimension one it coincides with an index used by
Morikawa (1990). Spatial decoherence rates are derived for three large classes
of the Galilean covariant quantum semigroups introduced by Holevo. We also
characterize the relaxation to a Gaussian state for these dynamics and give a
theorem for the convergence of the Wigner function to the probability
distribution of the classical analog of the process.Comment: 23 page
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