A second course in formal languages and automata theory by Jeffrey Shallit PDF

By Jeffrey Shallit

ISBN-10: 0511438354

ISBN-13: 9780511438356

ISBN-10: 0521865727

ISBN-13: 9780521865722

Meant for graduate scholars and complex undergraduates in computing device technology, A moment direction in Formal Languages and Automata conception treats subject matters within the idea of computation no longer often coated in a primary direction. After a overview of simple suggestions, the e-book covers combinatorics on phrases, common languages, context-free languages, parsing and popularity, Turing machines, and different language sessions. Many themes frequently absent from different textbooks, corresponding to repetitions in phrases, kingdom complexity, the interchange lemma, 2DPDAs, and the incompressibility process, are lined right here. the writer locations specific emphasis at the assets had to symbolize yes languages. The e-book additionally encompasses a different number of greater than two hundred routines, feedback for time period initiatives, and study difficulties that stay open.

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K) The set of strings over {a, b} containing at least one a and at least one b. 7. Let L = {x ∈ (a + b)∗ : |x|a = |x|b }. Give a regular expression for L2 . 8. Suppose M = (Q, , δ, q0 , F ) is a DFA, and suppose there exists a state q ∈ Q, a string z ∈ ∗ , and integers i, j > 0 such that δ(q, zi ) = q = δ(q, zj ). Prove that δ(q, zgcd(i,j ) ) = q. 9. Let x, y be words. Prove that xy = yx if and only if there exists a word z such that x 2 y 2 = z2 . 9 Exercises 23 10. A regular expression r is said to be in disjunctive normal form if it can be written in the form r = r1 + r2 + · · · + rn for some n ≥ 1, where none of the regular expressions r1 , r2 , .

7071, which you may recognize as 2/2. But how can we prove this? The first step is to recognize the hidden occurrence of the Thue–Morse sequence. 2) , where εn = (−1)tn and tn is the nth symbol√of the Thue–Morsesequence. We now prove that this infinite product equals 2/2. Let P and Q be the infinite products defined by ∞ P = n=0 2n + 1 2n + 2 ∞ εn , 2n 2n + 1 Q= n=1 εn . Then ∞ PQ = n 1 2 n=1 n + 1 ∞ εn = 2n 1 2 n=1 2n + 1 ε2n ∞ 2n + 1 2n + 2 n=0 ε2n+1 . ) Now, since ε2n = εn and ε2n+1 = −εn , we get ∞ PQ = 2n 1 2 n=1 2n + 1 εn ∞ n=0 2n + 1 2n + 2 εn −1 = 1Q .

Mt , we define e(M) = 11111e(q0 )1e(h)1e( )1e( )1e(m1 )1 · · · 1e(mt )11111. 1. There exists a universal TM MU that has the following behavior: on input e(T )e(x), MU simulates T on input x and halts if and only if T halts on x. Proof Idea. The TM MU uses three tapes. The first tape is used to hold the input e(T )e(x). The second tape holds an encoded version of T ’s tape and the third tape holds the encoded state that T is in. A step of MU consists of determining if any moves on tape 1 match the current configuration and then performing the move on tape 2.

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A second course in formal languages and automata theory by Jeffrey Shallit

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