By Jean-Paul Allouche

Combining innovations of arithmetic and desktop technology, this publication is ready the sequences of symbols that may be generated via basic versions of computation referred to as ''finite automata''. compatible for graduate scholars or complex undergraduates, it starts off from effortless ideas and develops the elemental conception. The examine then progresses to teach how those rules will be utilized to unravel difficulties in quantity concept and physics.

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**Additional info for Automatic Sequences: Theory, Applications, Generalizations**

**Sample text**

20. Find an English word containing two distinct squares, each of length ≥ 4. 21. Is the decimal√expansion of π squarefree? Cubefree? How about the decimal expansion of 2, or e? 22. Let u(n) denote the number of overlap-free binary words of length n. Show that u(n) is bounded by a polynomial in n. 23. Let w = c0 c1 c2 · · · = 0010011010010110011 · · · be the inﬁnite word deﬁned by cn = the number of 0’s (mod 2) in the binary expansion of n. Show that w is overlap-free. 24. Show how to extend the Thue–Morse inﬁnite word t on the left to a two-sided inﬁnite word that is still overlap-free.

Morse [1921] introduced the sequence t independently, as did Mahler [1927] and the Dutch chess master Max Euwe [1929]; see Exercise 18. ) Arshon [1937] introduced the Thue–Morse sequence in a slightly disguised form (as a ﬁxed point of the map 1 → 12, 2 → 21) and observed that it was cubefree. Gardner [1961a, 1961b] popularized the problem of ﬁnding an inﬁnite squarefree string in his Mathematical Games column; the columns were reprinted in Gardner [1967a, pp. 32–33, 90–95]. Noland [1962] proposed the problem of ﬁnding an inﬁnite squarefree word over three symbols, and a solution was given by Braunholtz [1963].

Show how to generate a random primitive word in expected linear time. 47. Let k ≥ 2 be an integer. Show that the set of all subwords of the primes expressed in base k is {0, 1, . . , k − 1}∗ . 48. Let µ : 0 → 01, 1 → 10 be the Thue–Morse morphism. (a) Show that the lexicographically least inﬁnite overlap-free word over 2 = {0, 1} starting with 1 is µω (1). (b) Show that the lexicographically least inﬁnite overlap-free word starting with 0 is 001001ϕ ω (1). (c) What is the lexicographically least inﬁnite overlap-free word over 2 that can be extended to a two-sided inﬁnite overlap-free word?