Infinitely growing configurations in Emil Post's tag system problem

TL;DR

This paper identifies an infinite family of non-halting configurations in Emil Post's tag system, demonstrating their behavior through a finite, bounded verification process, advancing understanding of the system's computational properties.

Contribution

It introduces a new family of configurations in Post's tag system that grow infinitely without halting, with a finite verification proof of their behavior.

Findings

Constructed a family of configurations of the form A^n B C^m that grow indefinitely.

Proved these configurations do not halt using a finite, bounded verification process.

Extended understanding of the computational complexity of Post's tag system.

Abstract

Emil Post's tag system problem posed the question of whether or not a tag system $\{N=3, P(0) = 00, P(1) = 1101\}$ has a configuration, simulation of which will never halt or end up in a loop. Over the subsequent decades, there were several attempts to find an answer to this question, including a recent study, during which the first $2^{84}$ initial configurations were checked. This paper presents a family of configurations of this type in the form of strings $A^{n} B C^{m}$ that evolve to $A^{n+1} B C^{m+1}$ after a finite number of steps. The proof of this behavior for all non-negative $n$ and $m$ is described later in this paper as a finite verification procedure, which is computationally bounded by 20 000 iterations of tag.