Morphic Sequences: Complexity and Decidability

TL;DR

This work resolves the decidability of PMClass, the problem of determining the exact complexity class of pure morphic sequences, by integrating Pansiot’s original criteria with Devyatov’s morphic-tools and Harju–Linna’s decidability results. It shows that the complexity of a pure morphic sequence must lie in one of five classes: $Theta(1)$, $Theta(n)$, $Theta(n log log n)$, $Theta(n log n)$, or $Theta(n^2)$, and provides an explicit, recursive algorithm to decide the class. The method hinges on a detailed growth analysis of the morphism, a case split based on whether the morphism is growing, and a constructive reduction via $k$-blocks when not growing, culminating in a strongly 1-periodic morphism with a verifiable condition. Overall, the results connect morphism growth behavior to precise asymptotic complexity and deliver an effective decision procedure in symbolic dynamics.

Abstract

In this work we recall Pansiot's result on the complexity of pure morphic sequences and we use the tools developed by Devyatov for morphic sequences to prove the decidability of the complexity class of pure morphic sequences.

Theorems & Definitions (11)

• Theorem 1.1: Morse-Hedlund, 1938
• Theorem 1.2: J.J. Pansiot, 1984
• Proposition 1.1: J.J. Pansiot, 1985
• Theorem 1.3: R. Devyatov, 2015
• Theorem 2.1
• Theorem 3.1: A. Salomaa, M. Soittola
• Proposition 3.1
• Theorem 4.1
• Lemma 4.1
• Lemma 4.2