From combinatorial games to shape-symmetric morphisms

Siegel suggests in his book on combinatorial games that quite simple games provide us with challenging problems: ``No general formula is known for computing arbitrary Grundy values of Wythoff's game. In general, they appear chaotic, though they exhibit a striking fractal-like pattern.''. This observation is the first motivation behind this chapter. We present some of the existing connections between combinatorial game theory and combinatorics on words. In particular, multidimensional infinite words can be seen as tiling of $N^d$. They naturally arise from subtraction games on $d$ heaps of tokens. We review notions such as $k$-automatic, $k$-regular or shape-symmetric multidimensional words. The underlying general idea is to associate a finite automaton with a morphism

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