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

By Ivanyi A. (ed.)

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

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A) Product of two FA. (b) Iteration of an FA. Product. A = A1 · A2 , where Q = Q1 ∪ Q2 , Σ = Σ1 ∪ Σ2 , F = F2 , I = I1 , E = E1 ∪ E2 ∪ p ∈ F1 q ∈ I2 (p, ε, q) For the result automaton see Fig. 14(a). Here also L(A1 · A2 ) = L(A1 )L(A2 ). Iteration. 2. 15. Minimization of DFA. E = E1 ∪ (q0 , ε, p) ∪ p∈I1 (q, ε, p) . q ∈ F1 p ∈ I1 The iteration of an FA can be seen in Fig. 14(b). 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.

States S1 , S4 , S5 , S6 are inaccessible and can be removed from the DFA. The transition table of the resulted DFA is δ 0 1 S0 S2 S3 {S2 } ∅ {S3 } {S3 } {S3 } {S3 } The corresponding transition graph is in Fig. 5. 10 can be simplied. It is not necessary to consider all subset of the set of states of NFA. The states of DFA A can be obtained successively. Begin with the state q 0 = I and determine the states δ(q 0 , a) for all a ∈ Σ. For the newly obtained states we determine the states accessible from them.

Conversely, any DFA is also an NFA by denition. So, the nondeterministic nite automata accepts the same class of languages as the deterministic nite automata. 1. 2. Equivalence of deterministic nite automata In this subsection we will use complete deterministic nite automata only. In this case δ(q, a) has a single element. In formulae, sometimes, instead of set δ(q, a) we will use its single element. We introduce for a set A = {a} the function elem(A) which give us the single element of set A, so elem(A) = a.

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