Analog Computers and the Iteration Functional

Abstract

In this paper we extend the class of differentially algebraic functions computed by Shannon's General Purpose Analog Computer (GPAC). We relax Pour-El's definition of GPAC to obtain new operators and we use recursion theory on the reals to define a new class of analog computable functions. We show that a function F(t,x) which simulates t time-steps of a Turing machine on input x, and more generally a functional that allows us to define the t'th iterate of a definable function, are definable in this system. Therefore, functions like Gamma which are not generable by GPAC become computable in this extension