Tribonacci Automaton and Its Combinatorial Applications

• Tribonacci Automaton is a finite automaton based on Tribonacci numeral representations that efficiently analyzes abelian complexity and pattern occurrences in morphic sequences.
• It employs deterministic finite automata with output mapping to compute complexity functions in O(log n) time via recursive state transitions.
• It underpins automated decision procedures and mechanical proofs for combinatorial properties such as squares, palindromes, and repeated factors in infinite words.

A Tribonacci Automaton is a finite automaton whose state transitions, operations, or output functions are directly linked to the Tribonacci sequence or associated morphic words. Such automata have become central to the paper of combinatorics on words, numeration systems, and automatic complexity, particularly in connection with the Tribonacci word, its factor structure, and related complexity functions.

1. Foundational Principles: The Tribonacci Sequence and Morphic Words

The Tribonacci sequence is defined by the recurrence $T_0 = 0$, $T_1 = 1$, $T_2 = 1$, and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \geq 0$. The Tribonacci word—a substitution sequence—is the fixed point of the morphism $\varphi: 0 \mapsto 01, 1 \mapsto 02, 2 \mapsto 0$.

A Tribonacci Automaton, as formalized in (Turek, 2013), leverages the structure of the Tribonacci numeration system, where every $n$ has a unique representation as a sum of non-consecutive Tribonacci numbers, encoded as a binary word $\langle n \rangle_T$ over $\{0,1\}$. The automaton operates naturally over these representations, rendering certain combinatorial and factor complexity functions "automatic" in the generalized sense.

2. Automaton Construction for Abelian Complexity Evaluation

A prominent application is the evaluation of the abelian complexity function $\rho^{\mathrm{ab}}(n)$ of the Tribonacci word. This function counts the number of distinct Parikh vectors (frequency distributions of letters) for the word factors of length $n$.

The automaton is built as a deterministic finite automaton with output (DFAO):

• States correspond to equivalence classes of abelian co-decompositions of factors.
• For each $n$, the set $(n)$ is computed using the morphism (co-decomposition) method, analyzing $\varphi^{K+R}(0)$ where $K$ and $R$ are determined by $n$ and the substitution's growth.
• Transition functions $\delta(q, d)$ are established for each state $q$ and incoming digit $d$ (from the Tribonacci representation $\langle n \rangle_T$), encoding how the set $(n)$ evolves as new digits are appended.
• An output map $\tau(q)$ returns the complexity value as the cardinality of the set of relative Parikh vectors in $(n)$.

The central evaluation formula is:

$$\rho^{\mathrm{ab}}(n) = \tau\left(\delta(0, \langle n \rangle_T)\right)$$

This process requires $O(\log n)$ steps, yielding efficient computation of the abelian complexity even for very large $n$ (Turek, 2013).

3. Decision Procedures and Mechanical Proofs

Generalization of mechanical decision procedures has been explored for the Tribonacci word in (Mousavi et al., 2014). By encoding combinatorial properties as first-order predicates in extended Presburger arithmetic (over the Tribonacci numeration system), one converts logical assertions into automata:

• Properties such as avoidance of higher powers (e.g., fourth powers), location of squares and cubes, distribution of palindromic factors, and quasiperiodicity become decidable via automaton construction.
• Tribonacci-automatic sequences can thus encode regularities (such as factor complexity, abelian complexity, or specific word patterns), and automata can recognize or enumerate these efficiently.
• Mechanical proofs, including enumeration of pattern occurrences and calculation of critical exponents (e.g., the supremum for $|w|/\text{period}(w)$ over all factors), are realized by algorithmic translation to automata (Mousavi et al., 2014).

4. Enumeration Algorithms and Recursive Structures

Automata constructions have been explicitly used to enumerate repeated patterns and factor occurrences in the Tribonacci sequence (Yuke et al., 2016, Huang et al., 2016, Huang et al., 2016, Huang et al., 2016):

• Algorithms for counting repeated palindromes, squares, and cubes in prefixes $T[1, n]$ rely on recursive decompositions—kernel words, gap sequences, and "square trees" (a recursive tree structure covering positions of repeated squares).
• In automata-theoretic terms, states often capture these recursive decompositions; transitions correspond to the application of substitution rules or navigation through tree structures.
• These methods result in closed-form expressions for the quantities of repeated patterns and enable efficient computation via automata or transducers reflecting the inherent recurrence (Yuke et al., 2016, Huang et al., 2016, Huang et al., 2016, Huang et al., 2016).

5. Automata Theory: Complexity, Synchronization, and Limitations

Tribonacci automata reflect both the strengths and boundaries of automata theory in handling sequences defined by linear recurrences of order higher than two:

• Intermediate complexity: The nondeterministic automatic complexity rate for the Tribonacci word is strictly between 0 (ultimately periodic) and $1/2$ (random), as shown in (Kjos-Hanssen, 2020). The recursive, self-similar nature of the Tribonacci word allows its recognition by automata of moderate state complexity.
• Representation synchronizability: Unlike the Fibonacci case (Zeckendorf numeration and golden ratio), there does not exist an automaton accepting pairs $(n, x)$ with $x = \lfloor n \psi \rfloor$ for $\psi$ the Tribonacci constant, nor an automaton generating the Sturmian characteristic word with slope $\psi-1$ from Tribonacci representations (Shallit, 12 Oct 2025). The analytic complexity arising from the cubic minimal polynomial and complex roots obstructs synchronization and automaticity for certain number-theoretic functions.
• Automata construction is strongly dependent on the algebraic properties of the underlying recurrence; cubic recurrences exhibit non-periodic oscillations in key fractional parts, leading to an infinite Myhill-Nerode complexity—no finite automaton is sufficient.

6. Extensions, Generalizations, and Applications

The automaton-based approach, originally designed for the Tribonacci word, generalizes directly to other $k$-bonacci words (e.g., the $4$-bonacci word), as the substitution and numeration system structures admit analogous representations (Turek, 2013). For the $4$-bonacci word, the output values for abelian complexity differ fundamentally from the Tribonacci case, and only finitely many values may occur for specific parameter choices.

In practical terms, Tribonacci automata have contributed to:

• Efficient mechanical evaluation of complexity functions (e.g., $\rho^{\mathrm{ab}}(n)$) for infinite sequences with $O(\log n)$ complexity per query.
• Complete algorithmic solutions to characterization problems for word complexity—explicit description of $n$ yielding prescribed complexity values.
• Structural understanding of factor decompositions, pattern occurrences, and avoidance properties in combinatorics on words.
• Direct translation of recursion-based combinatorial properties into automata-theoretic algorithms suitable for symbolic computation, combinatorial enumeration, and even applications in coding theory and symbolic dynamics.

7. Summary Table: Algorithmic Workflow for Tribonacci Automaton Evaluation

Step	Operation	Output
1. Tribonacci representation	Compute $\langle n \rangle_T$ (greedy)	Binary word encoding of $n$
2. Initialization	Start at initial automaton state $q_0$	Current state $q$
3. Transition processing	For each digit $d$, apply $\delta(q, d)$	Update state $q$
4. Output extraction	Compute $\tau(q)$	Returns $\rho^{\mathrm{ab}}(n)$

The above automaton process encapsulates mechanical evaluation for the abelian complexity function, but similar algorithms exist for repeated pattern enumeration and factor analysis.

---

In summary, Tribonacci Automata embody the intersection of combinatorial recursion, numeration systems, and automata theory, providing mechanical, efficient, and extensible frameworks for evaluating and characterizing complex properties of morphic sequences generated by higher-order substitutions. The analytic and algorithmic machinery developed for these automata has illuminated both the scope and the limitations of automaticity in symbolic sequence analysis and extended the reach of automata-theoretic approaches far beyond classical (Fibonacci-based) systems.