Characterizing Simpler recognizable sets of integers

peer reviewedFor a given numeration system U, a set X of integers is said to be U-star-free if the language of the normalized U-representations of the elements in X is star-free. Adapting a result of McNaughton and Papert, we give a first-order logical characterization of these sets for various numeration systems including integer base systems and the Fibonacci system. For k-ary systems, the problem of the base dependence of this property is also studied. Finally, the case of k-adic systems is developed

Characterizing simpler recognizable sets of integers

peer reviewedFor the k-ary numeration system, we characterize the sets of integers such that the corresponding representations make up a star-free regular language. This result can be transposed to some linear numeration systems built upon a Pisot number like the Fibonacci system and also to k-adic numeration systems. Moreover we study the problem of the base dependence of this property and obtain results which are related to Cobham's Theorem