complexity_of_automatic_sequences

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Complexity of automatic sequences: The subword complexity of a k-automatic sequence is O(N). For non-automatic sequences, the complexity grows faster. Exact complexity functions for specific sequences are open.

Why this node was retired

The posted statement is

import Mathlib

theorem complexity_of_automatic_sequences (a : ℕ → Fin 2) (k : ℕ) (hk : 2 ≤ k)
    (hauto : ∀ n : ℕ, a n = a (k * n) ∨ a n = 1 - a (k * n)) :
    ∀ eps : ℝ, 0 < eps →
    ∃ C : ℝ, 0 < C ∧
    ∀ N : ℕ,
      {w : List (Fin 2) | w.length = N ∧
        ∃ i : ℕ, w = (List.range N).map (fun j => a (i + j))}.ncard ≤ C * N ^ (1 + eps) := by
  sorry

The estimate includes zero word length and the automaticity condition is only a two-symbol tautology, not a finite-automaton definition.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Define k-automaticity through a finite automaton reading base-k digits, and require N≥1 (or bound by C(N+1)). The intended linear subword-complexity theorem can then be stated.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.