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**

Q1 , q0 such that (qk−2 , ak , qk−1 ) ∈ E, . . , (q0 , a1 , q1 ) ∈ E, where q0 ∈ q 0 = I , thus, there is a walk ak−1 ak a1 a2 a3 q0 −→ q1 −→ q2 −→ · · · −→ qk−1 −→ qk , q0 ∈ I, qk ∈ F, so L(A) ⊆ L(A). In constructing DFA we can use the corresponding transition function δ : δ(q, a) = δ(q, a) , ∀q ∈ Q, ∀a ∈ Σ. q∈q 1. 5. The equivalent DFA with NFA A in Fig. 3. The empty set was excluded from the states, so we used here ∅ instead of {∅}. 10 to transform NFA A in Fig. 3. Introduce the following notation for the states of the DFA: S0 := {q0 , q1 }, S4 := {q0 , q2 }, S1 := {q0 }, S5 := {q1 , q2 }, S2 := {q1 }, S6 := {q0 , q1 , q2 } , S3 := {q2 }, where S0 is the initial state.

An , thus, u ∈ L(G) and therefore L(A) ⊆ L(G). 14 Let G = ({S, A, B}, {a, b}, {S → aS, S → bA, A → bB, A → b, B → aB, B → a}, S) be a regular grammar. The NFA associated is A = ({S, A, B, Z}, {a, b}, E, S, {Z}), where E = (S, a, S), (S, b, A), (A, b, B), (A, b, Z), (B, a, B), (B, a, Z) . The corresponding transition table is δ a b S A {S} ∅ {A} {B, Z} B E {B, Z} ∅ ∅ ∅ The transition graph is in Fig. 9. This NFA can be simplied, states B and Z can be contracted in one nal state. Using the above theorem we dene an algorithm which associate an NFA A = (Q, T, E, {S}, F ) to a regular grammar G = (N, T, P, S).

Suppose that Q1 ∩Q2 = ∅ always. If not, we can rename the elements of any set of states. Union. A = A1 ∪ A2 , where Q = Q1 ∪ Q2 ∪ {q0 }, Σ = Σ1 ∪ Σ2 , I = {q0 }, F = F1 ∪ F2 , E = E1 ∪ E2 ∪ (q0 , ε, q) . q∈I1 ∪I2 For the result of the union see Fig. 13(b). The result is the same if instead of a single initial state we choose as set of initial states the union I1 ∪ I2 . In this case the result automaton will be without ε-moves. By the denition it is easy to see that L(A1 ∪ A2 ) = L(A1 ) ∪ L(A2 ).

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