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 $\varphi$ on the reals, and for any positive continuous function $\epsilon(t)$, it has a $\mathcal{C}^\infty$ solution with $| y(t) - \varphi(t) | < \epsilon(t)$ for all $t$. 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 $y^{(k_i)}(a_i)=b_i$. The question whether one can require the solution that approximates $\varphi$ 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 $\varphi$ and $\epsilon(t)$ there exists some initial condition that yields a solution that is $\epsilon$-close to $\varphi$ 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

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 $\operatorname{PTIME}$ 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 $\operatorname{PTIME}$. 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 $\operatorname{PTIME}$, 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

On the functions generated by the general purpose analog computer

PreprintWe 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. The GPAC generates as output univariate functions (i.e. functions f:R→R). In this paper we extend this model by: (i) allowing multivariate functions (i.e. functions f:Rn→Rm); (ii) introducing a notion of amount of resources (space) needed to generate a function, which allows the stratification of GPAC generable functions into proper subclasses. We also prove that a wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain. We prove a few stability properties of this model, mostly stability by arithmetic operations, composition and ODE solving, taking into account the amount of resources needed to perform each operation. We establish that generable functions are always analytic but that they can nonetheless (uniformly) approximate a wide range of nonanalytic functions. Our model and results extend some of the results from [19] to the multidimensional case, allow one to define classes of functions generated by GPACs which take into account bounded resources, and also strengthen the approximation result from [19] over a compact domain to a uniform approximation result over unbounded domains.info:eu-repo/semantics/acceptedVersio

Verification of Timed Automata Using Rewrite Rules and Strategies

ELAN is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. Timed automata is a class of continuous real-time models of reactive systems for which efficient model-checking algorithms have been devised. In this paper, we show that these algorithms can very easily be prototyped in the ELAN system. This paper argues through this example that rewriting based systems relying on rules and strategies are a good framework to prototype, study and test rather efficiently symbolic model-checking algorithms, i.e. algorithms which involve combination of graph exploration rules, deduction rules, constraint solving techniques and decision procedures