Amaury Pouly

On the complexity of solving initial value problems

In this paper we prove that computing the solution of an initial-value problem y = <i>p</i>(<i>y</i>) with initial condition <i>y</i>(<i>t</i>₀) = <i>y</i>₀ ∈ R^<i>d</i> at time <i>t</i>₀ + <i>T</i> with precision 2^−μ where <i>p</i> is a vector of polynomials can be done in time polynomial in the value of <i>T</i>, μ and <i>Y</i> = [equation]. Contrary to existing results, our algorithm works over any bounded or unbounded domain. Furthermore, we do not assume any Lipschitz condition on the initial-value problem.

Solving analytic differential equations in polynomial time over unbounded domains

In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of Rn and Cn, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane.

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

The outcomes of this article 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 elegant and simple characterization of P. We believe it is the first time complexity 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 models 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 in terms of computability and complexity, 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 in terms of computability and complexity.

On the Functions Generated by the General Purpose Analog Computer

We consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that time. We extend the model properly to a model of computation not restricted to univariate functions (i.e. functions

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.

f: \mathbb{R} \to \mathbb{R}

On The Complexity of Bounded Time Reachability for Piecewise Affine Systems

) but also to the multivariate case of (i.e. functions

Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem

f: \mathbb{R}^n \to \mathbb{R}^m

Model Checking Flat Freeze LTL on One-Counter Automata

), and establish some basic properties. In particular, we prove that a very wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain. Technically: we generalize some known results about the GPAC to the multidimensional case: we extend naturally the notion of \emph{generable} function, from the unidimensional to the multidimensional case. We prove a few stability properties of this class, mostly stability by arithmetic operations, composition and ODE solving. We establish that generable functions are always analytic. We prove that generable functions include some basic (useful) generable functions, and that we can (uniformly) approximate a wide range of functions this way. This extends some of the results from \cite{Sha41} to the multidimensional case, and this also strengths the approximation result from \cite{Sha41} over a compact domain to a uniform approximation result over unbounded domains. We also discuss the issue of constants, and we prove that involved constants can basically assumed to always be polynomial time computable numbers.

Computing with polynomial ordinary differential equations

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class

Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems

\operatorname{PTIME}