Characterizing Programming Systems Allowing Program Self-Reference

“Know thyself.”

Greek proverb

Abstract

The interest is in characterizing insightfully the power of program self-reference in effective programming systems ( $\mathsf{epses}$ ), the computability-theoretic analogs of programming languages (for the partial computable functions). In an $\mathsf{eps}$ in which the constructive form of Kleene’s Recursion Theorem (KRT) holds, it is possible to construct, algorithmically, from an arbitrary algorithmic task, a self-referential program that, in a sense, creates a self-copy and then performs that task on the self-copy. In an $\mathsf{eps}$ in which the not-necessarily-constructive form of Kleene’s Recursion Theorem (krt) holds, such self-referential programs exist, but cannot, in general, be found algorithmically.

In an earlier effort, Royer proved that there is no collection of recursive denotational control structures whose implementability characterizes the $\mathsf{epses}$ in which KRT holds. One main result herein, proven by a finite injury priority argument, is that the $\mathsf{epses}$ in which krt holds are, similarly, not characterized by the implementability of some collection of recursive denotational control structures.

On the positive side, however, a characterization of such $\mathsf{epses}$ of a rather different sort is shown herein. Though, perhaps not the insightful characterization sought after, this surprising result reveals that a hidden and inherent constructivity is always present in krt.