By Ivanyi A. (ed.)

ISBN-10: 9638759607

ISBN-13: 9789638759603

ISBN-10: 9638759615

ISBN-13: 9789638759610

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**Extra resources for Algorithms of informatics. Foundations**

**Example text**

Qj −→ qm , qm ∈ F . Therefore xy i z ∈ L for all i, and this proves the theorem. 17 We use the pumping lemma to show that L1 = {ak bk | k ≥ 1} is not regular. Assume that L1 is regular, and let n be the corresponding natural number given by the pumping lemma. Because the length of the word u = an bn is 2n, this word can be written as in the lemma. We prove that this leads to a contradiction. Let u = xyz be the decomposition as in the lemma. Then |xy| ≤ n, so x and y can contain no other letters than a, and because we must have |y| ≥ 1, word y contains at least one a.

Wn }, then x = x1 + x2 + . . + xn , where for any i = 1, 2, . . , n expression xi is a regular expression representing language {wi }. This latter can be done by the following rule. If wi = ε, then xi = ε, else if wi = a1 a2 . . am , where m ≥ 1 depends on i, then xi = a1 a2 . . am , where the brackets are omitted. We prove the theorem of Kleene which refers to the relationship between regular languages and regular expression. 20 (Kleene's theorem). Language L ⊆ Σ∗ is regular if and only if there exists a regular expression over Σ representing language L.

2. 24. 23. 23 In the case of Fig. 20 is not necessary to introduce new initial and nal state. The steps of transformations can be seen in Fig. 24. The resulted regular expression can be written also as (0 + 10)0∗ 1 + 11, which is the same as obtained by the previous method. Method 3. The third method for writing regular expressions associated to nite automata uses formal equations. A variable X is associated to each state of the automaton (to dierent states dierent variables). Associate to each state an equation which left side contains X , its right side contains sum of terms of form Y a or ε, where Y is a variable associated to a state, and a is its corresponding input symbol.

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