First-order Logic and Numeration Systems

The Büchi-Bruyère theorem states that a multidimensional set of non-negative integers is b-recognizable if and only if it is b-definable. This result is a powerful tool for showing that many properties of b-automatic sequences are decidable. Going a step further, first-order logic can be used to show that many enumeration problems of b-automatic sequences can be described by b-regular sequences. The latter sequences can be viewed as a generalization of b-automatic sequences to integer-valued sequences. These techniques were extended to two wider frameworks: U-recognizable multidimensional sets of non-negative integers and multidimensional beta-recognizable sets of reals. In the second case, real numbers are represented by infinite words, and hence, the notion of beta-recognizability is defined by means of Büchi automata. Again, logic-based characterizations of $U$-recognizable (resp. beta-recognizable) sets allows us to obtain various decidability results. The aim of this chapter is to present a survey of this very active research domain.