On automatic infinite permutations

An infinite permutation $\alpha$ is a linear ordering of $\mathbb N$. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we try to extend to permutations the notion of automaticity. As we shall show, the standard definitions which are equivalent in the case of words are not equivalent in the context of permutations. We investigate the relationships between these definitions and prove that they constitute a chain of inclusions. We also construct and study an automaton generating the Thue-Morse permutation.

💡 Research Summary

The paper investigates the notion of automaticity for infinite permutations, i.e., linear orderings of the natural numbers, and shows that the several equivalent definitions of automatic sequences for infinite words diverge when transferred to permutations. An infinite permutation is formalised as an equivalence class of sequences of pairwise‑distinct real numbers under the relation that two sequences induce the same pairwise order. The authors first recall the three classic characterisations of k‑automatic words: (i) a deterministic finite automaton (DFA) that reads the base‑k representation of n and outputs the nth symbol, (ii) the image of a fixed point of a k‑uniform morphism under a coding, and (iii) finiteness of the k‑kernel. For permutations they propose three analogous definitions:

1. V‑k‑automatic: a permutation that can be generated from a k‑automatic word, but only when the word defines a valid permutation (i.e., the induced order is a total order on ℕ).

2. A‑k‑automatic: a DFA whose input is a pair of base‑k representations (i, j); the transition function processes the two digit streams in parallel and the output is one of {<, >, =}, which is interpreted as the comparison between the i‑th and j‑th elements of the permutation.

3. K‑k‑automatic: a permutation whose k‑kernel – the set of subsequences of the form α_i, α_{k^n+i}, α_{k^{2n}+i}, … for all n≥0 and 0≤i<k^n – is finite.

The main theorem (Theorem 3) establishes a strict chain of inclusions for every k≥2:
 V_k ⊂ A_k ⊂ K_k.
Thus, unlike the word case where the three definitions are equivalent (Cobham’s theorem, Eilenberg’s theorem), for permutations they define three distinct classes. The strictness is demonstrated by explicit counter‑examples: a monotone permutation belongs to A_k but not V_k (it is generated by a trivial DFA but does not arise from a valid k‑automatic word); a permutation whose comparison between consecutive even‑indexed elements encodes a non‑automatic binary sequence belongs to K_k but not A_k.

The paper then focuses on the Thue‑Morse permutation α_TM, obtained from the classic Thue‑Morse word w_TM (the parity of the number of 1’s in the binary expansion of n). Since w_TM is 2‑automatic, α_TM is V_2‑automatic. The authors construct an explicit 2‑dimensional DFA that, on input (i, j) written in binary, outputs the relation between the infinite binary fractions .w_i w_{i+1}… and .w_j w_{j+1}…. The automaton’s states encode the order of length‑2 factors of the two shifted Thue‑Morse sequences; transitions are defined using the 2‑uniform morphism φ(0)=01, φ(1)=10, and the output function τ selects the first differing factor to decide < or >. They also describe a decidable procedure to verify that any such DFA indeed defines a total order (checking antisymmetry via the square automaton A² and transitivity via the cube automaton A³).

In addition to the theoretical results, the paper provides a constructive method that, given any k‑automatic word generating a valid permutation, builds an A‑k‑automatic DFA. This construction uses the set of length‑2 factors of the morphic fixed point, forms a state space consisting of permutations of these factors together with a special marker, and defines transitions by lifting the morphism’s action to pairs of factors. The output function τ then compares the coded letters of the underlying alphabet to determine the ordering.

Overall, the work establishes a foundational framework for automatic infinite permutations, clarifies how the three natural extensions of automaticity separate into a strict hierarchy, and supplies concrete automata for the most studied example, the Thue‑Morse permutation. It opens the way for further exploration of algorithmic properties (e.g., decidability of membership in each class, closure properties, and connections to combinatorial number theory) for infinite permutations.