In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

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. ACM Subject Classification IntroductionThe outcomes of this paper are twofold, and are concerning a priori not closely related topics. Implicit Complexity:Since the introduction of the P and NP complexity classes, much work has been done to build a well-developed complexity theory based on Turing Machines. In particular, classical computational complexity theory is based on limiting resources used by Turing machines, like time and space. Another approach is implicit computational complexity.The term "implicit" in "implicit computational complexity" can sometimes be understood in various ways, but a common point of these characterizations is that they provide (Turing or equivalent) machine-independent alternative definitions of classical complexity. Implicit characterization theory has gained enormous interest in the last decade. This has led to many alternative characterizations of complexity classes using recursive functions, function algebras, rewriting systems, neural networks, lambda calculus and so on.However, most of -if not all -these models or characterizations are essentially discrete: in particular they are based on underlying discrete time models working on objects which are essentially discrete such as words, terms, etc. that can be considered as being defined in a discrete space.Models of computation working on a continuous space have also been considered: they include Blum Shub Smale machines [4], and in some sense Computable A...

In the last decade, there have been several attempts to understand the relations between the many models of analog computation. Unfortunately, most models are not equivalent. Euler's Gamma function, which is computable according to computable analysis, but that cannot be generated by Shannon's General Purpose Analog Computer (GPAC), has often been used to argue that the GPAC is less powerful than digital computation. However, when computability with GPACs is not restricted to real-time generation of functions, it has been shown recently that Gamma becomes computable by a GPAC. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of ଁ Expanded version of the article "The General Purpose Analog Computer and Computable Analysis are two equivalent paradigms of analog computation" presented in the Theory and Applications of Models of Computation Conference (TAMC06).polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models.

Abstract. In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.

Abstract. Let (α, β) ⊆ R denote the maximal interval of existence of solutions for the initial-value problemwhere E is an open subset of R m+1 , f is continuous in E and (t 0 , x 0 ) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t 0 , x 0 ) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the "degree of noncomputability" by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.

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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.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 : R n → R m ); (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 (Shannon, 1941) 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 (Shannon, 1941) over a compact domain to a uniform approximation result over unbounded domains.