The Tribonacci constant and finite automata

2025-10-12

AI Summary

This paper addresses the existence of two classes of automata in the Tribonacci numeration system: (1) whether a finite automaton can simultaneously accept the Tribonacci representations of an integer $n$ and $lfloor npsi floor$, where $psi approx 1.839$ is the Tribonacci constant; and (2) whether a Tribonacci automaton can generate the characteristic Sturmian word of slope $psi-1$. Leveraging tools from formal language theory, finite automata models, combinatorial analysis of Tribonacci representations, number-theoretic arguments, and sequence complexity theory, we establish—rigorously for the first time—the nonexistence of both automata. This result exposes a fundamental limitation on the computational power of automata over transcendental (non-rational) numeration systems and establishes new theoretical boundaries for automata-based number representation and the computability of Sturmian words.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

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

Problem

• No automaton accepts Tribonacci representations of n and x simultaneously
• No Tribonacci automaton generates Sturmian word with slope ψ-1
• Studies limitations of automata processing Tribonacci constant representations

Innovation

• No automaton for parallel Tribonacci representations
• Tribonacci constant automaton limitations demonstrated
• No automaton generates Sturmian word with slope