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By Ivanyi A. (ed.)

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Xm ) =⇒ (p, a1 a2 . . ak , x1 , x2 . . xm−1 w), if q, (ε, xm /w), p ∈ E. ∗ The reflexive and transitive closure of the relation =⇒ will be denoted by =⇒. Instead of using =⇒, sometimes is considered. How does work such a pushdown automaton? Getting started with the initial configuration (q0 , a1 a2 . . an , z0 ) we will consider all possible next configurations, and after this the next configurations to these next configurations, and so on, until it is possible. 22 Pushdown automaton V accepts (recognizes) word u by final state if there exist a sequence of configurations of V for which the following are true: • the first element of the sequence is (q0 , u, z0 ), • there is a going from each element of the sequence to the next element, excepting the case when the sequence has only one element, • the last element of the sequence is (p, ε, w), where p ∈ F and w ∈ W ∗ .

15, that is we get that q0 ≡ q3 and q1 ≡ q5 . After merging them we get an equivalent minimum state automaton (see Fig. 16). 6. Pumping lemma for regular languages The following theorem, called pumping lemma for historical reasons, may be efficiently used to prove that a language is not regular. It is a sufficient condition for a regular language. 15 (pumping lemma). For any regular language L there exists a natural number n ≥ 1 (depending only on L), such that any word u of L with length at least n may be written as u = xyz such that (1) |xy| ≤ n, (2) |y| ≥ 1, (3) xy i z ∈ L for all i = 0, 1, 2, .

Other equations the number of remaining equations will be diminuted by one. In such a way the system of equation can be solved for each variable. The solution will be given by variables corresponding to final states summing the corresponding regular expressions. In our example from the first equation we get Y = 1. From here Z = 0 + 10 + Z0, or Z = Z0 + (0 + 10), and solving this we get Z = (0 + 10)0∗ . Variable U can be obtained immediately and we obtain U = 11 + (0 + 10)0∗ 1. Using this method in the case of Fig.

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Algorithms of informatics, vol. 1 by Ivanyi A. (ed.)

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