The signature of rational languages

We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the tree. A prefix closed language defines such a tree augmented with labels on arcs, hence is associated with a signature. This way of 'traversing' a language is related to the notion of abstract numeration system, due to Lecomte and Rigo. After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont-Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same. (C) 2016 Elsevier B.V. All rights reserved.