Periodicity, repetitions, and orbits of an automatic sequence

Abstract

a b s t r a c tWe revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope α, have automatic continued fraction expansions if α does.

The Critical Exponent is Computable for Automatic Sequences

“…Minor i leads to an accepting condition. This construction was already given in [4], and the details can be found there.…”

“…Given a k-automatic sequence a = (a i ) i≥0 , we can, using the techniques of [4,8], (effectively) create a two-dimensional DFA M accepting…”

Syntactic Complexity of Ultimately Periodic Sets of Integers

“…This problem was settled positively for integer base systems by Honkala in [9]. See also [2] and in particular [5] for a first order logic approach. Recently this decision problem was settled positively in [3] for a large class of numeration systems based on linear recurrence sequences.…”

A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems

“…For the restricted case of the D0L periodicity problem, where only the morphism f is considered, decision procedures are well-known [24,33]. Finally, questions connected to the ones addressed here have independently and recently gained interest [3]. In particular, a simple proof of Honkala's original result based on the construction of some automata is given in that paper.…”

“…In particular, a simple proof of Honkala's original result based on the construction of some automata is given in that paper. As for the logical approach considered by Muchnik and Leroux, the arguments given in [3] rely on the recognizability of addition by automata (which can be done for the classical k-ary system but not necessarily for an arbitrary linear numeration system).…”