Differential Recursion and Differentially Algebraic Functions
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.
💡 Research Summary
The paper revisits the class of real‑valued “recursive” functions introduced by Moore in the mid‑1990s, a formulation that was meant to parallel Kleene’s classic theory of computable functions but in an analog setting. Moore’s construction rests on two pillars: a collection of elementary real functions (constants, arithmetic, comparisons, etc.) and a single higher‑order operator called differential recursion (DR). DR takes a differential equation of the form y′ = f(x, y) together with an initial condition y(x₀) = y₀ and declares the unique solution (when it exists) to be a new function. By iterating the elementary operations, composition, restriction of domain, and DR, Moore defined a hierarchy he called “primitive recursive” (PR) real functions.
The authors of the present paper identify a fundamental flaw in Moore’s definition: the behavior of DR on partial functions is left unspecified. In the discrete setting, Kleene’s primitive recursion is always total because the recursion is over natural numbers. In the continuous case, however, a differential equation may fail to have a solution on the whole restricted domain, may have multiple solutions, or may blow up in finite time. Moore’s text merely says that “undefined cases are ignored,” which is mathematically insufficient for a rigorous theory of function composition.
To make the discussion concrete, the authors reconstruct Moore’s PR class in formal notation. They show that the restriction operator can produce a function whose domain is a bounded interval I, and then DR is applied to a differential equation whose right‑hand side f may be undefined or non‑Lipschitz on I. For instance, f(x, y) = √(1 – y²) together with the initial condition y(0) = 0 leads to a solution that exists only while |y| ≤ 1; any extension beyond that point is undefined. The original definition provides no rule for what the resulting PR function should be on points where the solution ceases to exist.
The paper proposes two possible remedies. The first is a “global existence” restriction: DR may be used only when the underlying differential equation admits a unique solution on the entire intended domain. This turns DR into a partial operator whose applicability must be checked beforehand, effectively pruning many otherwise interesting functions. The second remedy embraces nondeterminism: DR returns the set of all possible solutions (when multiple exist) and a separate “choice” operator selects one. While this restores completeness, it departs from the deterministic nature of classical recursion theory and complicates the notion of a function as a single-valued mapping.
Having clarified the technical gap, the authors turn to the claimed equivalence between Moore’s PR functions and differentially algebraic (DA) functions. A DA function satisfies a polynomial differential equation P(x, f, f′, …, f⁽ⁿ⁾) = 0 for some finite n; this class coincides with the functions computable by Shannon’s General Purpose Analog Computer (GPAC). Moore suggested that PR = DA, thereby linking his model to the GPAC. The paper disproves this claim by constructing explicit counterexamples. The function
g(x) = ∫₀ˣ e^{t²} dt
is obtained by a single application of DR to the differential equation y′ = e^{x²} with y(0) = 0. g is known to be transcendental and does not satisfy any algebraic differential equation, so it is not DA. Moreover, DR can generate solutions of nonlinear differential equations that are not closed under the algebraic operations that define the DA class. Consequently, the PR class strictly contains non‑DA functions, and the inclusion in the opposite direction also fails because many DA functions (e.g., solutions of linear differential equations with rational coefficients) cannot be expressed using only Moore’s elementary operations and a single DR step without additional mechanisms.
The authors conclude that Moore’s original “primitive recursive = differentially algebraic” identification is untenable. The ambiguity in the definition of DR on partial functions undermines the claimed correspondence with the GPAC, and the existence of non‑DA functions within PR demonstrates a genuine mismatch. They recommend that future work on analog recursion should (i) make the existence and uniqueness conditions of DR explicit, (ii) treat restriction and composition of partial functions with a rigorous domain‑tracking discipline, and (iii) possibly introduce a richer set of operators (e.g., integration, inversion) that align more closely with the closure properties of the DA class. Only with such refinements can an analog recursion theory be placed on a solid mathematical footing and meaningfully compared to existing models of analog computation.