After 100 Years, Can We Finally Crack Post's Problem of Tag? A Story of Computational Irreducibility, and More

TL;DR

This paper investigates Post's problem of tag through empirical, theoretical, and historical analysis, revealing strong computational irreducibility and complex behaviors in tag systems, with implications for understanding computational universality.

Contribution

It provides new evidence of computational irreducibility in tag systems and explores their complex behaviors, connecting historical insights with modern computational theory.

Findings

Strong evidence of computational irreducibility in tag systems

All initial conditions eventually halt, some after over 6*10^11 steps

Identification of minimal complex behaviors including a Collatz-like problem

Abstract

Empirical, theoretical and historical aspects of Post's "problem of tag" from 1921 are explored. Evidence of strong computational irreducibility is found. Despite their deterministic origin, the lengths of successive sequences generated seem to closely approximate random walks. All 10^25 smallest initial conditions are found to eventually halt, although sometimes in > 6*10^11 steps. Implications of the Principle of Computational Equivalence are discussed, along with examples of identifiable computational capabilities of tag systems. Various minimal examples of complex behavior are found, including a less-biased analog of the 3n+1 Collatz problem. There is also discussion of the history of Emil Post and of tag systems in the context of ideas about the foundations of mathematics and computation.