Conway's cosmological theorem and automata theory

TL;DR

The paper presents an automata-theoretic, computational proof of Conway's cosmological theorem for the audioactive (look-and-say) sequence. By modeling the derivation with a small set of transducers (Audio, Multi, Mark, Scissors) and constructing split- and atom-focused automata (Splitting, Atom, AtomicF), the authors prove stabilization of the atom set at day $n\ge 24$ via a transducer composition argument and minimization. The approach yields explicit finite-state encodings (e.g., 21-state Splitting, 26-state Atom, and a manageable AtomicF ∘ Audio^n ∘ Src family) and demonstrates that $E_{24}=E_{25}$, hence $E_n=E_{24}$ for all $n\ge 24$, with the full atom list detailed in the appendix. This work illustrates how automata theory and experimental mathematics can provide a transparent, computer-assisted pathway to deep combinatorial results in number theory.

Abstract

John Conway proved that every audioactive sequence (a.k.a. look-and-say) decays into a compound of 94~elements, a statement he termed the cosmological theorem. The underlying audioactive process can be modeled by a finite-state machine, mapping one sequence of integers to another. Leveraging automata theory, we propose a new proof of Conway's theorem based on a few simple machines, using a computer to compose and minimize them.

Figures (8)

• Figure 1: The “multimark”, transducer, denoted ${\newline{\textit{Multi}}}$. The edges are labelled with the convention "input symbol $\mid$ output symbol". The notation $\forall \alpha$ for the input symbol means the corresponding edge should be duplicated for each symbol in the input alphabet. When the output symbol is $\alpha$, it means “copy the input symbol”. The “initial” arrow marks initial state(s). The double stroke marks final state(s).
• Figure 2: The “single mark” transducer, denoted $\newline{\textit{Mark}}$.
• Figure 3: The “scissors” transducer, denoted $\newline{\textit{Scissors}}$.
• Figure 4: The “bounded $a$-counter” transducer, denoted ${\newline{\textit{Cnt}}}_{a}$.
• Figure 5: The "sink" $\newline{\textit{Sink}}$ and "source" $\newline{\textit{Src}}$ transducers.

Theorems & Definitions (9)

• Theorem 1: One-Day Theorem
• proof
• Lemma 2
• proof
• Theorem 3: Splitting Theorem
• proof
• Lemma 4
• Theorem 5: Cosmological Theorem
• proof : Proof of Theorem \ref{['thm:cosmological']}