Akitoshi Kawamura. Differential recursion. *ACM Transactions on Computational Logic* **10**(3), Article 22, 22 pages, April 2009. DOI = 10.1145/1507244.1507252

The version before proof: Fulltext.pdf (also available here)

We present a redevelopment of
the theory of real-valued “recursive” functions
that was introduced by C. Moore in 1996
by analogy with the standard formulation of the
integer-valued recursive functions.
While his work
opened a new line of research on analog computation,
the original paper contained some technical inaccuracies.
We discuss possible attempts to remove the
ambiguity in the behaviour of the operators on partial functions,
with a focus on his “primitive recursive” functions
generated by the
*differential recursion* operator
that solves initial-value problems.
Under a reasonable reformulation,
the functions in this class are shown to be
analytic and computable in a strong sense in Computable Analysis.
Despite this well-behavedness,
the class turns out to be
too big to have the originally purported relation to
differentially algebraic functions,
and hence to C. E. Shannon's model of analog computation.

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

This article is partly based on the following conference talks. (These are just for record. Please read (or cite) the above journal version.)

- Talk at CiE 2006: Differential recursion and differentially algebraic functions September 2006, revised March 2007
- Talk at Theory Seminar, University of Toronto: Rebuilding Moore's real recursion theory January 2007
- Talk at CCA 2004: Type-2 computability and Moore's recursive functions