The Tribonacci constant and finite automata

TL;DR

This work investigates automata-theoretic properties of Tribonacci representations. It proves that there is no synchronized automaton for mapping $n$ to $a(n)=\lfloor \psi n\rfloor$ and no Tribonacci-automatic for $b(n)$, where $\psi$ is the Tribonacci constant, by deriving a contradiction from a Myhill-Nerode style analysis augmented with Kronecker’s theorem. The results contrast with the Fibonacci case, where such maps are automaton-friendly, and connect to logical expressibility: $\lfloor \psi n\rfloor$ is not first-order definable in the natural Tribonacci-valuation structure, unlike the Fibonacci counterpart. The paper also notes near-miss automata for approximations and discusses implications for broader numeration systems, including cubic Pisot numbers with two complex conjugates.

Abstract

We show that there is no automaton accepting the Tribonacci representations of $n$ and $x$ in parallel, where $ψ= 1.839\cdots$ is the Tribonacci constant, and $x= \lfloor n ψ\rfloor$. Similarly, there is no Tribonacci automaton generating the Sturmian characteristic word with slope $ψ-1$.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof