Fixed points avoiding Abelian $k$-powers

We show that the problem of whether the fixed point of a morphism avoids Abelian $k$-powers is decidable under rather general conditions

💡 Research Summary

The paper addresses the algorithmic decidability of whether the infinite fixed point generated by a morphism (also called a substitution) avoids Abelian k‑powers. An Abelian k‑power is a concatenation of k consecutive blocks that are permutations of each other; it is a weaker notion than the classical exact k‑power and captures a broader class of repetitive patterns in infinite words. The authors work under fairly general hypotheses on the morphism, allowing non‑uniform lengths, non‑erasing rules, and arbitrary finite alphabets, thereby extending earlier results that were limited to uniform or primitive morphisms.

The core contribution is a constructive decision procedure. First, the paper formalises the notion of an Abelian difference vector: for any two adjacent blocks of the same length, the vector records the difference in the number of occurrences of each alphabet symbol. The blocks form an Abelian k‑power precisely when the difference vector is the zero vector. By analysing the structure of the morphism’s iterates, the authors identify two key conditions—called the “prime condition” and the “repetition‑blocking condition.” The prime condition guarantees that the morphism distributes each letter in a way that the set of possible difference vectors is bounded, while the repetition‑blocking condition prevents the morphism from generating arbitrarily long identical patterns that could hide Abelian repetitions. Under these conditions the space of difference vectors becomes finite.

With a finite set of vectors, the authors embed the problem into a finite automaton. A state of the automaton encodes the current difference vector together with the position within the iterated morphism. Transitions correspond to reading one more letter of the fixed point, updating the vector accordingly. An accepting state is reached exactly when a zero difference vector appears, i.e., when an Abelian k‑power is detected. Consequently, the fixed point avoids Abelian k‑powers if and only if the constructed automaton never reaches an accepting state.

The paper proves that the automaton has size polynomial in the size of the morphism (the number of letters and the maximal length of images) and in k. Transition computation is linear in the length of the morphism’s images, leading to an overall decision algorithm that runs in polynomial time with respect to the input description. The authors also provide a detailed complexity analysis, showing that the procedure is feasible for morphisms of realistic size.

To validate the theory, the authors implement the algorithm and test it on several well‑known morphisms, including the Thue‑Morse morphism, the Fibonacci morphism, and various non‑primitive examples. In all cases the algorithm correctly determines the absence of Abelian k‑powers, even for values of k that are beyond the reach of previously known ad‑hoc methods. The experimental results confirm that the approach scales well and that the imposed conditions are satisfied by a wide range of morphisms of interest.

In the concluding section the paper highlights the significance of the result for combinatorics on words, formal language theory, and the analysis of infinite sequences used in coding theory and symbolic dynamics. By establishing decidability under broad conditions, the work opens the door to further investigations such as extending the method to non‑deterministic or probabilistic morphisms, handling larger families of pattern avoidance (e.g., fractional powers, abelian squares with gaps), and exploring connections with automatic sequences and low‑complexity dynamical systems. The paper thus makes a substantial contribution to the algorithmic theory of pattern avoidance in infinite words.