First-Order Logic and Numeration Systems

  • Émilie Charlier
Part of the Trends in Mathematics book series (TM)


The Büchi-Bruyère theorem states that a subset of \(\mathbb {N}^d\) 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 subsets of \(\mathbb {N}^d\) and β-recognizable subsets of \(\mathbb {R}^d\). In the second case, real numbers are represented by infinite words, and hence, the notion of β-recognizability is defined by means of Büchi automata. Again, logic-based characterization of U-recognizable (resp. β-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.



I thank Julien Leroy and Narad Rampersad for a careful reading of this chapter and for many helpful comments.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LiègeLiègeBelgium

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