Algebra of Self-Replication

TL;DR

This paper explores the algebraic structure of self-replicating programs using decidable equational logics, classifies quine programs within these logics, and links combinatory logic fragments to computability theory.

Contribution

It introduces small decidable equational logics to analyze computability and classifies quine programs within these frameworks, connecting combinatory logic to self-replication.

Findings

Classified quine programs in two decidable fragments.

Linked normal forms in term rewriting to self-replication.

Provided insights into the structure of programs that output themselves.

Abstract

Typical arguments for results like Kleene's Second Recursion Theorem and the existence of self-writing computer programs bear the fingerprints of equational reasoning and combinatory logic. In fact, the connection of combinatory logic and computability theory is very old, and this paper extends this connection in new ways. In one direction, we counter the main trend in both computability theory and combinatory logic of heading straight to undecidability. Instead, this paper proposes using several very small equational logics to examine results in computability theory itself. These logics are decidable via term rewriting. We argue that they have something interesting to say about computability theory. They are closely related to fragments of combinatory logic which are decidable, and so this paper contributes to the study of such fragments. The paper has a few surprising results such as a classification of quine programs (programs which output themselves) in two decidable fragments. The classification goes via examination of normal forms in term rewriting systems, hence the title of the paper. The classification is an explanation of why all quine programs (in any language) are "pretty much the same, except for inessential details." In addition, we study the relational structure whose objects are the programs with the relation "p expresses q" meaning that if the program p is run on nothing, then it eventually outputs the program q.