Binary Morphisms with Stable Suffix Complexity

Let g, h be marked morphisms of words. A pair (g1, h1) of marked morphisms is called a successor of (g, h) if g ◦ g1(a) = h ◦ h1(a) and g ◦ g1(b) = h ◦ h1(b) and the images of g1 and h1 are shortest possible. Successors play an important role in studying the Post Correspondence Problem. Typically, they are simpler than the original morphisms, measured by the number of suffixes of their images (called the suffix complexity). In some cases, however, the suffix complexity is stable – it does not decrease. In this paper we study the binary case, that is, morphisms defined on a two-letter alphabet. We give a full characterization of binary morphisms with stable suffix complexity. It is a surprising fact that solutions w of the equation g(w) = h(w), where g, h : Σ∗ → ∆∗ are morphisms of a free monoid, are poorly understood even if the cardinality of Σ is two (i.e. in the binary case). The situation is easy only in some special cases, for example when at least one of the morphisms is periodic. The Post Correspondence Problem (PCP), which asks whether the equation has a nonempty solution, is known to be undecidable in general, but decidable in the binary case. The first proof of decidability of the binary case was announced in [1], however, it contains a gap. A complete proof was given in [4], and [5] shows that the decision process is of polynomial time. The structure of the set of all solutions is known for some binary cases. However, even in the binary case, there is still no description of general solutions, in particular, there is no efficient algorithm known that decides whether a given word can be a solution of the above equation. For more details and references see [6, 7, 2]. If a single reason for the complications in the above mentioned research should be pointed out, it could be the existence of so called successor morphisms (called “equality collectors” in [1]). Studying a pair of binary morphisms, sooner or later one may discover that the existence of a solution, as well as its structure, depends essentially on the same question for a different pair of morphisms. Turning the attention to the new pair (the successors) the situation may occur again. The crucial question is whether the resulting chain of reductions can be kept under control. This can be done using so called suffix complexity of morphisms, which in most cases decreases; the concept was first introduced in [1]. However, there are some cases in which the suffix complexity is stable. These cases were once again studied in [1] where a characterization is given, which is sufficient for a proof that the binary PCP is decidable. In this paper we give a complete classification of stable instances for both balanced and unbalanced