Characterizing Programming Systems Allowing Program Self-reference

Abstract

The interest is in characterizing insightfully the power of program self-reference in effective programming systems (epses), the computability-theoretic analogs of programming languages. In an 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 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 epses in which KRT holds. One main result herein, proven by a finite injury priority argument, is that the 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 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.

Know thyself.

– Greek proverb

This paper received support from NSF Grant CCR-0208616.

An Erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-540-73001-9_87