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Extra resources for Algorithms of informatics, vol. 1

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14. {Z} if production S → ε does not occur in G, {Z, S} if production S → ε occurs in G. Prove that L(G) = L(A). Let u = a1 a2 . . an ∈ L(G), u = ε. Then there is in G a derivation of word u: S =⇒ a1 A1 =⇒ a1 a2 A2 =⇒ . . =⇒ a1 a2 . . an−1 An−1 =⇒ a1 a2 . . an . This derivation is based on productions S → a1 A1 , A1 → a2 A2 , . . , An−2 → an−1 An−1 , An−1 → an . Then, by the definition of the transitions of NFA A there exists a walk •F = a a a an−1 a 1 2 3 n A2 −→ · · · −→ An−1 −→ Z, Z ∈ F .

For example the computation tree associated to the initial configuration (q0 , 1001, z0 ) can be seen in Fig. 34. From this computation tree we can observe that, because (q1 , ε, ε) is a leaf of the tree, pushdown automaton V2 accepts word 1001 by empty stack. The computation tree in Fig. 35 shows that pushdown automaton V2 does not accept word 101, because the configurations in leaves can not be continued and none of them has the form (q, ε, ε). 3. 27). 27 does not accept word 101. 24 A language L is accepted by a nondeterministic pushdown automaton V1 by empty stack if and only if it can be accepted by a nondeterministic pushdown automaton V2 by final state.

64 1. 27 Because the initial state is also a final state, the empty word is accepted by final state, but not by empty stack. To show that word abab is not accepted, we need to study all possibilities. It is easy to see that in our case there is only a single possibility: (q0 , abab, z0 ) =⇒ (q1 , bab, z0 z1 ) =⇒ (q2 , ab, z0 ) =⇒ (q0 , ab, ε), but there is no further going, so word abab is not accepted. 27 The transition table ({q0 , q1 }, {0, 1}, {z0 , z1 , z2 }, E, q0 , z0 , ∅) is: Σ ∪ {ε} of the 0 pushdown automaton 1 V2 ε W z0 z1 z2 z0 z1 z2 z0 q0 (z0 z1 , q0 ) (z1 z1 , q0 ) (ε, q1 ) (z2 z1 , q0 ) (z0 z2 , q0 ) (z1 z2 , q0 ) (z2 z2 , q0 ) (ε, q1 ) (ε, q1 ) (ε, q1 ) (ε, q1 ) q1 (ε, q1 ) = The corresponding transition graph can be seen in Fig.

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