Multidimensional Generalized Automatic Sequences and Shape-symmetric Morphic Words

An infinite word is S-automatic if, for all n>=0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=2, we state that a multidimensional infinite word x : N^d \to \Sigma over a finite alphabet \Sigma is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.

💡 Research Summary

The paper extends the classical notion of automatic sequences from one dimension to arbitrary dimensions d ≥ 2 and establishes a precise equivalence with the class of shape‑symmetric infinite words introduced by Arnaud Maes. An S‑automatic multidimensional word is defined with respect to an abstract numeration system S built on a regular language L that contains the empty word ε. Each point n = (n₁,…,n_d) ∈ ℕ^d is uniquely encoded as a word w ∈ L; a deterministic finite automaton (DFA) reads w and, via its transition function and output labeling, produces the symbol x(n). Thus the infinite array x : ℕ^d → Σ is S‑automatic if a DFA together with L yields the correct symbol for every multidimensional index.

The second central concept is shape‑symmetry. A shape‑symmetric infinite word arises from a morphic construction: there exists a d‑dimensional morphism μ : Σ → Σ^{*^d} and a coding τ : Σ → Γ such that, starting from a seed letter a, the iterates μ^k(a) converge (as k → ∞) to an infinite word y whose pattern is invariant under permutation of the d coordinate axes. In other words, the growth of the word is governed by a substitution rule that treats each dimension in a symmetric way, producing a globally self‑similar structure.

The main theorem states that for any d ≥ 2, a multidimensional infinite word x is S‑automatic for some abstract numeration system S (with L regular and ε ∈ L) if and only if x is the image under a coding of a shape‑symmetric infinite word. The proof proceeds in two directions.

1. From S‑automatic to shape‑symmetric.
   Given a DFA M and regular language L, the authors construct a morphism μ whose alphabet consists of the states of M. For each state q and each symbol a of the input alphabet (which encodes a step in one coordinate), the transition δ(q,a) determines the substitution of q in the corresponding dimension. Because L contains ε, the origin (0,…,0) is represented and the initial state is well defined. The output labeling of M becomes the coding τ. By iterating μ, one obtains a multidimensional word whose projection under τ coincides with the original S‑automatic array. The construction guarantees that the substitution treats all dimensions uniformly, yielding shape‑symmetry.

2. From shape‑symmetric to S‑automatic.
   Starting from a shape‑symmetric word y = τ(μ^∞(a)), the authors encode each multidimensional index using the regular language L that mirrors the substitution structure of μ. They then design a DFA M that, on reading the L‑representation of an index, simulates the application of μ to the corresponding state and finally outputs the symbol given by τ. Because μ is shape‑symmetric, the encoding is consistent across all dimensions, and the DFA’s transition function is well defined. Consequently, the resulting (S,L,M) triple reproduces y, proving that y is S‑automatic.

Key technical tools include the closure properties of regular languages, the deterministic nature of DFA transitions, and a lexicographic ordering on ℕ^d that ensures a unique encoding of each index. The paper also clarifies the distinction between the classical k‑automatic sequences (based on a fixed base‑b numeration) and the present S‑automatic framework, which allows arbitrary regular numeration systems and thus accommodates non‑standard, even non‑positional, representations.

The authors illustrate the theory with examples such as two‑dimensional cellular automaton patterns, fractal lattices, and multidimensional Sturmian‑type sequences, all of which can be expressed as shape‑symmetric morphic words and consequently as S‑automatic arrays. These examples demonstrate the practical relevance of the equivalence: it provides a unified language for describing self‑similar multidimensional structures in terms of finite automata.

In conclusion, the paper delivers a robust bridge between multidimensional automatic sequences and shape‑symmetric morphic words. By proving that the two notions are exactly the same up to coding, it unifies two previously separate research streams—abstract numeration systems and multidimensional morphic dynamics—and opens new avenues for exploring self‑similarity, decidability, and complexity in higher‑dimensional combinatorial objects.