By Jeffrey Shallit

Meant for graduate scholars and complex undergraduates in laptop technological know-how, A moment path in Formal Languages and Automata idea treats themes within the concept of computation no longer often lined in a primary path. After a evaluation of uncomplicated options, the booklet covers combinatorics on phrases, commonplace languages, context-free languages, parsing and popularity, Turing machines, and different language sessions. Many themes frequently absent from different textbooks, comparable to repetitions in phrases, country complexity, the interchange lemma, 2DPDAs, and the incompressibility approach, are coated the following. the writer areas specific emphasis at the assets had to signify yes languages. The publication additionally features a assorted choice of greater than two hundred routines, feedback for time period initiatives, and study difficulties that stay open.

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Show that if x, y ∈ ∗ with xy = yx, then xyxxy is primitive. 38. Show that for all words w ∈ {0, 1}∗ , either w0 or w1 is primitive (or both). 39. Consider the equation in words xXy = z2 . Describe all solutions to this equation. Hint: there are separate cases depending on whether |x|, |y| are even or odd. 10 Notes on Chapter 2 47 40. Let x, y, z be strings. In the Lyndon–Sch¨ utzenberger theorems, we proved a necessary and sufficient condition for xy = yx and xy = yz. Find similar necessary and sufficient conditions for (a) xy = y R x; (b) xy = y R z.

2, w is also a power, a contradiction. Now x < z, since x was lexicographically least among all conjugates of w. Then uvu < uuv, so vu < uv. Then vuu < uvu = x, so we have found a conjugate of w that is lexicographically smaller than x, a contradiction. For the other direction, suppose w is not primitive, and let x be any conjugate of w. 2, x is also a power; that is, x = t k for some nonempty t and integer k ≥ 2. Then x = tt k−2 t, so x is bordered. P1: JsY second CUUS348-Shallit 36 978 0 521 86572 2 2 August 6, 2008 21:1 Combinatorics on words We now apply our results about conjugates and borders to determine the solutions to the equation in words x i y j = zk .

Let w = w1 w2 w3 · · · , and suppose w = wk+1 wk+2 wk+3 · · · for some k > 0. Then w = (w1 w2 · · · wk )ω . 18. Suppose x, y are words with xy = yx. Show that, for all n ≥ 1, at least one of x n y and x n+1 y is primitive. 19. Let x, y, z be words. Show that xyz = zyx if and only if there exist words u, v and integers i, j, k ≥ 0 with x = (uv)i u, y = (vu)j v, and z = (uv)k u. 20. Suppose xyz is a square and xyyz is a square. Show that xy i z is a square for all i ≥ 0. 21. 2 by induction on N .