Algebraic characterizations of the computational aspects of functions defined over the real numbers provide very effective tool to understand what computability and complexity over the reals, and generally over continuous spaces, mean. This is relevant for both communities of computer scientists and mathematical analysts, particularly the latter who do not understand (and or like) the language of machines and string encodings. Recursive analysis can be considered the most standard framework of computation over continuous spaces, it is however defined in a very machine specific way which does not leave much to intuitiveness. Recently several characterizations, in the form of function algebras, of recursively computable functions and some sub-recursive classes were introduced. These characterizations shed light on the hidden behavior of recursive analysis as they convert complex computational operations on sequences of real objects to "simple" intuitive mathematical operations such as integration or taking limits. The authors previously presented a framework for obtaining algebraic characterizations at the complexity level over compact domains. The current paper presents a comprehensive extension to that framework. Though we focus our attention in this paper on functions defined over the whole real line, the framework, and accordingly the obtained results, can be easily extended to functions defined over arbitrary domains.
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. 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 polynomial differential equations then we show that all real computable functions over compact intervals can be defined by such models.