Differential recursion and differentially algebraic functions

Presented at the Second Conference on Computability in Europe (CiE 2006) as “Real primitive recursive functions and differential algebraicity.” Revised March 2007.

Subsumed by “Differential recursion.”

Fulltext.pdf (also at arXiv:0704.0301) April 2007

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.

Keywords
analog computation, real recursive functions, initial value problems, differentially algebraic functions, transcendentally transcendental functions