Download e-book for iPad: Automatic Sequences by Friedrich Von Haeseler

By Friedrich Von Haeseler

ISBN-10: 3110156296

ISBN-13: 9783110156294

Provides a normal method of learning sequences generated by way of a finite machine. Haeseler (Katholieke Universiteit Leuven, Belgium) first introduces the thoughts of substitution at the area of sequences, increasing staff endomorphisms, and the kernel graph of a series. the most a part of the booklet develops an set of rules for developing a minimum automaton for a given computerized series, investigates the houses of H-automatic sequences and sequences generated by means of substitutions, and considers the answer of Mahler equations

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Let f be generated by a (V , H )-substitution. Then the (V , H )kernel graph of f is a directed, labeled graph defined as follows: The set of vertices is the (V , H )-kernel of f , and the set of labels is the set V . There is a directed edge with label v from f 1 to f 2 if and only if ∂vH (f 1 ) = f2 . The vertex f is called the basepoint of the (V , H )-kernel graph. Remark. If M = {f 1 , . . , f N } is a finite set such that ∂VH (M) ⊂ M for all v ∈ V , then the (V , H )-kernel-graph of the set M is a directed graph with set of vertices v equal to M, label set equal to V , and the set of edges is given by {(f → h) | f , h ∈ M, ∂vH (f ) = h}.

If the matrices Av are induced by a kernel graph of a sequence f ∈ ( , A), then the substitution polynomial P is called the substitution polynomial of f . Examples. 1. , p. 36. 6. If we set f 1 = bf , f 2 = f , and f 3 = h, then the substitution polynomial is given by ⎛ ⎛ ⎞ ⎞ 0 1 0 1 0 0 x0 ⎝ 1 0 0 ⎠ ⊕ x1 ⎝ 0 0 1 ⎠ . 0 0 1 0 0 1 2. The Thue–Morse sequence as a sequence in polynomial for V = {x 0 , x 1 }. x0 1 0 0 1 ⊕ x1 (N, {0, 1}) has the substitution 0 1 1 0 . 3. The paper folding sequence as an element of ( , {0, 1}) and for V = {x 0 , x 1 } has the substitution polynomial ⎛ ⎛ ⎞ ⎞ 0 1 0 0 1 0 0 0 ⎜0 0 0 1⎟ ⎜ ⎟ 1⎜0 0 1 0⎟ ⎟ x0 ⎜ ⎝0 0 1 0⎠ ⊕ x ⎝0 0 1 0⎠.

4. t. E = {x, y}). A residue set V of H is given by V = {x 0 y 0 , x 1 y 0 , x 0 y 1 , x 1 y 1 }, however, V is not a complete residue set. Moreover, there does not exist a complete digit set for H . t, the norm induced by the 2 2 euclidian metric on R ). The set V = {x10 , x11 , x12 , . . , x1b } is a complete digit set, cf. [103]. 5. 2. Consider H :L→L ⎛ ⎞ ⎛ ⎞ 1 x z 1 2x 4z ⎝ 0 1 y ⎠ → ⎝ 0 1 2y ⎠ . 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}}.

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Automatic Sequences by Friedrich Von Haeseler


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