# Rebuilding Moore's real recursion theory

Informal seminar of Theory group, University of Toronto, January 2007.

Journal version: “Differential recursion”

## Abstract

Analog computers, in which both state and time are continuous, were
once considered another plausible computing mechanism until digital
dominance became clear in the fifties. Recently, theoretical models
of such computers enjoy resurgence of interest with the aim to
understand computation carried out by non-artificial systems. We
study one such model, Moore's real (primitive) recursive functions.
These classes of real functions are defined as the closure under
certain operators, including what are supposed to be real-number
versions of primitive recursion and minimization. We give a sound
foundation to his theory by removing technical inaccuracies in the
original work, and study the analytic properties of the class. It
turns out, contrary to common belief, that there are real primitive
recursive functions which are not differentially algebraic. We also
discuss the relationship between Moore's classes and the standard
(digital) Computability Theory.