Characterizing Computable Analysis with Differential Equations

Abstract

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 [Bournez, O. and E. Hainry, Recursive analysis characterized as a class of real recursive functions, Fundamenta Informaticae 74 (2006), pp. 409–433] 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 [Bournez, O., M. L. Campagnolo, D. S. Graça and E. Hainry, Polynomial differential equations compute all real computable functions on computable compact intervals, Journal of Complexity 23 (2007), pp. 317–335] also characterize the functions of Computable Analysis. We combine the results of [Bournez, O. and E. Hainry, Recursive analysis characterized as a class of real recursive functions, Fundamenta Informaticae 74 (2006), pp. 409–433] and Graça et. al. [Graça, D. S., N. Zhong and J. Buescu, Computability, noncomputability and undecidability of maximal intervals of IVPs, Transactions of the American Mathematical Society (2007), to appear], 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.