The Recursion Theorem from a Different Angle

TL;DR

This paper explores a novel perspective on the Recursion Theorem in computability, proposing the potential existence of a self-referential halting program with paradoxical properties.

Contribution

It introduces a new conceptual approach to the Recursion Theorem by examining self-referential programs with contradictory halting behaviors.

Findings

Proposes the likely existence of a self-referential halting program.

Analyzes the implications of paradoxical self-referential programs.

Provides a new angle on classical computability concepts.

Abstract

This paper is about computability. I claim the likely existence of a program DoesHalt(Program, Input) such that DoesHalt( HaltsOnItself, AntiSelf ) halts with resounding 'NO'. HaltsOnItself( Program ) is simply DoesHalt( Program, Program ). AntiSelf() is a self-referential self-contradictory program that loops when HaltsOnItself() returns 'YES' and halts when HaltsOnItself() returns 'NO'.