Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.

Abstract. 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. 1

Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C 2 real elementary computable functions; our result does not require the restriction to C 2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort.

Abstract. We show that, using our more or less established framework of inductive definition of real-valued functions (work started by Cristopher Moore in [9]) together with ideas and concepts of standard computability we can prove theorems of Analysis. Then we will consider our ideas as a bridging tool between the standard Theory of Computability (and Complexity) on one side and Mathematical Analysis on the other, making real recursive functions a possible branch of Descriptive Set Theory. What follows is an Extended Abstract directed to a large audience of

The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to characterize the twice continuously differentiable functions of Computable Analysis, restricted to certain compact domains. In a similar model, Shannon’s General Purpose Analog Computer, Bournez et. al. 2007 [3] also characterize the functions of Computable Analysis. We combine the results of [5] and Graça et. al. [13], to show that a different function algebra also yields Computable Analysis. We believe that our function algebra is an improvement due to its simple definition and because the operations in our algebra are less obviously designed to mimic the operations in the usual definition of the recursive functions using the primitive recursion and minimization operators. 1

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