Titlebook: Sequences, Groups, and Number Theory; Valérie Berthé,Michel Rigo Book 2018 Springer International Publishing AG, part of Springer Nature 2

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First-Order Logic and Numeration Systems,gic-based characterization of .-recognizable (resp. .-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.

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Normal Numbers and Symbolic Dynamics,urn our attention to non-normal numbers. Since they are not completely random, we have to find a different measurement for analyzing their structure. The Hausdorff dimension will provide us with an interesting parameter in this context.

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About the Domino Problem for Subshifts on Groups,f the art concerning the domino problem in this extended framework. We also discuss different notions of effectiveness in subshifts defined over groups, that is, the ways in which these dynamical objects can be described through Turing machines.

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General Framework,d results from combinatorics on groups and words, formal language theory, morphic words, .-automatic and .-regular sequences, and dynamical systems. Our aim is not to be exhaustive. The reader can consult this chapter when studying other parts of this book.

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First-Order Logic and Numeration Systems,t many properties of .-automatic sequences are decidable. Going a step further, first-order logic can be used to show that many enumeration problems of .-automatic sequences can be described by .-regular sequences. The latter sequences can be viewed as a generalization of .-automatic sequences to in

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Some Applications of Algebra to Automatic Sequences,enauer of automatic and regular sequences in terms of two properties, which we call the shuffle property and the power property. These properties show that if one views an automatic sequence . as a map on a free monoid on .-letters to a finite subset of a ring, then the values of . are closely relat