Characterizing Programming Systems Allowing Program Self-Reference

The interest is in characterizing insightfully the power of program self-reference in effective programming systems ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{epses}$\end{document} ), the computability-theoretic analogs of programming languages (for the partial computable functions). In an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{eps}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{eps}$\end{document} 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.