Program Self-Reference in Constructive Scott Subdomains

Intuitively, a recursion theorem asserts the existence of self-referential programs . Two well-known recursion theorems are Kleene’s Recursion Theorem (krt) and Rogers’ Fixpoint Recursion Theorem (fprt). Does one of these two theorems better capture the notion of program self-reference than the other? In the context of the partial computable functions over the natural numbers (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {PC}$\end{document}), fprt is strictly weaker than krt, in that fprt holds in any effective numbering of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {PC}$\end{document} in which krt holds, but not vice versa. It is shown that, in this context, the existence of self-reproducing programs (a.k.a. quines ) is assured by krt, but not by fprt. Most would surely agree that a self-reproducing program is self-referential. Thus, this result suggests that krt is better than fprt at capturing the notion of program self-reference in\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {PC}$\end{document} .