An algorithm for enumerating all infinite repetitions in a D0L-system

Abstract

Abstract. We describe a simple algorithm which, for a given D0L-system, returns all words v such that v k is a factor of the language of the system for all k. This algorithm can be used to decide whether a D0L-system is repetitive.

Characterization of circular D0L-systems

“…The following claim follows from Proposition 4.3 in [KO00] and Theorem 2 in [KtS13]. In fact, if the condition in the previous claim is satisfied for some injective simplification, then it is satisfied for all injective simplifications.…”

“…(⇐): Proposition 10 implies that there is a positive integer ℓ and a letter a such that (ϕ ℓ ) ∞ (a) = w ω for some w ∈ A + . In [KtS13] it is proved that the word w can be taken so that it contains the letter a only once at its beginning. It follows that ϕ ℓ (w) = w k for some k > 1.…”

Synchronizing delay for binary uniform morphisms

“…Because of the simple structure of D0L-systems with a k-uniform morphism, we can find an explicit list of morphisms that are not circular. We state a proof here as it is quite simple using the results from [7]. However, a complete characterisation of repetitive (and so non-circular) D0L-systems with binary morphisms was done in [8].…”

“…Clearly, a k-uniform morphism ϕ is not injective if and only if ϕ(a) = ϕ(b) (condition (i)). It follows from [7] that G is repetitive if and only if there is a primitive word u (i.e., a word for which u = z implies = 1) in S (L(G)) such that ϕ j (u) = u for some integers j ≥ 1 and ≥ 2. Moreover, the factor u cannot contain any unbounded letter twice and j is bounded by the number of letters in the alphabet: hence we must have u ∈ {a, b, ba, ab} and j ∈ {1, 2}.…”

“…By the results from [2,3], non-circular systems are repetitive (and by [6] also strongly repetitive). In fact, a D0L-system injective on S (L(G)) is not circular if and only if it is unboundedly repetitive [3], i.e., there exists v containing an unbounded letter such that v k ∈ S (L(G)) for all k ∈ N. Since this property can be checked by a simple algorithm [7], we can easily verify whether a given D0L-system injective on S (L(G)) is circular or not.…”