Differential Recursion and Differentially Algebraic Functions

TL;DR

This paper critically examines Moore's class of recursive functions, identifies technical inaccuracies, and demonstrates that the proposed connection to differentially algebraic functions and Shannon's analog computation model does not hold.

Contribution

The paper clarifies issues in Moore's definition of differential recursion and shows the failure of its relation to differentially algebraic functions.

Findings

Moore's recursive functions contain technical inaccuracies.

The relation to differentially algebraic functions fails.

The connection to Shannon's analog computation model is invalid.

Abstract

Moore introduced a class of real-valued "recursive" functions by analogy with Kleene's formulation of the standard recursive functions. While his concise definition inspired a new line of research on analog computation, it contains some technical inaccuracies. Focusing on his "primitive recursive" functions, we pin down what is problematic and discuss possible attempts to remove the ambiguity regarding the behavior of the differential recursion operator on partial functions. It turns out that in any case the purported relation to differentially algebraic functions, and hence to Shannon's model of analog computation, fails.