A Swiss Pocket Knife for Computability

Abstract

This research is about operational- and complexity-oriented aspects of classical foundations of computability theory. The approach is to re-examine some classical theorems and constructions, but with new criteria for success that are natural from a programming language perspective. Three cornerstones of computability theory are the S-m-ntheorem; Turing's "universal machine''; and Kleene's second recursion theorem. In today's programming language parlance these are respectively partial evaluation, self-interpretation, and reflection. In retrospect it is fascinating that Kleene's 1938 proof is constructive; and in essence builds a self-reproducing program. Computability theory originated in the 1930s, long before the invention of computers and programs. Its emphasis was on delimiting the boundaries of computability. Some milestones include 1936 (Turing), 1938 (Kleene), 1967 (isomorphism of programming languages), 1985 (partial evaluation), 1989 (theory implementation), 1993 (efficient self-interpretation) and 2006 (term register machines). The "Swiss pocket knife'' of the title is a programming language that allows efficient computer implementation of all three computability cornerstones, emphasising the third: Kleene's second recursion theorem. We describe experiments with a tree-based computational model aiming for both fast program generation and fast execution of the generated programs

Algebra of Self-Replication

“…This matter was studied by Jones [5] in connection with in-practice computational systems that make use of the Recursion Theorem. It turns out that the two different constructions are correlated with performance differences in a certain algorithm mentioned in [5]. Be that as it may, there is evidently something similar about the two proofs.…”

XMDD as Key Enabling Technology for Integration and Organizational Collaboration: Application to E-Learning Based on NRENs

“…XMDD is compatible with the SCA service-oriented architectures standard [53], and is a service-oriented evoilution of the more traditional component based architectures in software design [54] and of the feature oriented architectures in early telecommunications services design [55]. It would even support higher order processes, allowing for full reconfigurability of the processes at runtime as shown in [56].…”

On the Semantics of Intensionality

“…A series of experiments with the SRT and further discussion are documented in Hansen et al [1989], and Jones [1992Jones [ , 2013.…”

“…The basic impetus behind this thesis was not intensionality itself, but rather its rôle in Kleene's Second Recursion Theorem, which we exhaustively studied in §2. These questions were brought to the author's attention by Abramsky [2012Abramsky [ , 2014, whoalong with Jones [2013]-had identified the mysterious nature of this theorem in the 1980s. Abramsky [2012] also remarks that the theorem is very powerful, and-even though very simple-its proof is opaque, and provides no intuition.…”