Automatic Sequences (De Gruyter Expositions in Mathematics, by Friedrich Von Haeseler

By Friedrich Von Haeseler

Computerized sequences are sequences that are produced via a finite automaton. even though they don't seem to be random, they might glance as being random. they're advanced, within the experience of now not being eventually periodic. they might additionally glance quite advanced, within the feel that it will possibly no longer be effortless to call the rule of thumb wherein the series is generated; even though, there exists a rule which generates the series. this article bargains with various facets of automated sequences, particularly: a common creation to automated sequences; the elemental (combinatorial) houses of computerized sequences; the algebraic method of automated sequences; and geometric gadgets on the topic of automated sequences.

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F N (γ )). 2), the above defined F is a fixed point of the substitution S. If we define θ : B → A as θ((a1 , . . , aN )) = a1 , we obtain ˆ ) = f = f. θ(F 1 This finishes the proof. 19 is a fundamental tool to establish the finiteness of the kernel of a given sequence f . 2 Substitutions finite set K which contains f and is invariant under all v-decimations, then ker V ,H (f ) being a subset of K is finite. We shall encounter this method of proof several times in the following chapters. 19 shows, we can compute the kernel of a sequence which is a fixed point of a substitution.

T. the norm r ). A (left)-residue set is given by V = {b 1 a 2 c 3 | 1 , 2 ∈ {0, 1} and 3 ∈ {0, 1, 2, 3}}. The residue set V is not a complete digit set. The fixed points of κ are e, a −1 , b−1 , c−1 , b−1 c−1 , a −1 c−1 , b−1 a −1 c−1 . Moreover, these are the only periodic points of κ. The next theorem provides a simple criterion for the existence of complete digit sets. 7. t. the discrete norm with expansion ratio C > 2 and let H ( ) be of index d ∈ N. Then there exists a residue set that is a complete digit set (of H ).

4. If V is any residue set for H , then e is a (V , H )-substitution-invariant subset of and we can speak of the automaticity of elements in ( e , A). The next examples provide some insight how to define a finite automaton which generates a given sequence. Examples. 1. Consider the Thue–Morse sequence t as an element of (N, {0, 1}). Then t is (V , H )-automatic, where H (x j ) = x 2j and V = {x 0 , x 1 }. The state alphabet B is given by the set B = {∅, 1}, the output alphabet is A = {0, 1}, the output function ω : B → A is defined by ω(∅) = 0 and ω(1) = 1, the transition functions αx 0 and αx 1 are given by αx 0 (∅) = ∅, αx 0 (1) = 1, αx 1 (∅) = 1, αx 0 (1) = ∅.

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