Characterizing Computable Analysis with Differential Equations

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.