By Ivanyi A. (ed.)

ISBN-10: 9638759615

ISBN-13: 9789638759610

Ivanyi A. (ed.) Algorithms of informatics, vol.1.. foundations (2007)(ISBN 9638759615)

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**Extra info for Algorithms of informatics**

**Example text**

Prove that L(G) = L(A) \ {ε}. Let u = a1 a2 . . an ∈ L(A) and u = ε. Thus, since A accepts word u, there is a walk an−1 a1 a2 a3 an q0 −→ q1 −→ q2 −→ · · · −→ qn−1 −→ qn , qn ∈ F. Then there are in P the productions q0 → a1 q1 , q1 → a2 q2 , . . , qn−2 → an−1 qn−1 , qn−1 → an (in the right-hand side of the last production qn does not occur, because qn ∈ F ), so there is the derivation q0 =⇒ a1 q1 =⇒ a1 a2 q2 =⇒ . . =⇒ a1 a2 . . an−1 qn−1 =⇒ a1 a2 . . an . Therefore, u ∈ L(G). 2. 8. 13. Conversely, let u = a1 a2 .

For this operation it is also ∗ true that L(A∗1 ) = L(A1 ) . The denition of these tree operations proves again that regular languages are closed under the regular operations. 5. Minimization of nite automata A DFA A = (Q, Σ, E, {q0 }, F ) is called minimum state automaton if for any equivalent complete DFA A = (Q , Σ, E , {q0 }, F ) it is true that |Q| ≤ |Q |. We give an algorithm which builds for any complete DFA an equivalent minimum state automaton. States p and q of an DFA A = (Q, Σ, E, {q0 }, F ) are equivalent if for arbitrary word u we reach from both either nal or nonnal states, that is u u p −→ r, r ∈ F and q −→ s, s ∈ F or p ≡ q if for any word u ∈ Σ∗ u u p −→ r, r ∈ F and q −→ s, s ∈ F .

12 If L = L(G) is a regular language, then one may construct an NFA that accepts language L. Proof Let G = (N, T, P, S) be the grammar which generates language L. e. Z is a new symbol), • For every production A → aB , dene transition (A, a, B) in E . • For every production A → a, dene transition (A, a, Z) in E . {Z} if production S → ε does not occur in G, •F = {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 =⇒ .

### Algorithms of informatics by Ivanyi A. (ed.)

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