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 resources for Algorithms of informatics
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If in the rst row in any column there occur a pair of states from which one is a nal state and the other not then the algorithm ends, the two automata are not equivalent . If there is no such a pair of states, every new pair is written in the rst column. The algorithm continues with the next un lled row. If no new pair of states occurs in the table and for each pair both of states are nal or both are not, then the algorithm ends and the two DFA are equivalent . If |Q| = n, |Q | = n and |Σ| = m then taking into account that in worst case loop repeat is executed nn times, loop for m times, the running time of the algorithm in worst case will be O(nn m), or if n = n then O(n2 m).
If the DFA accepts also the empty word ε, then in the above grammar we introduce a new start symbol q0 instead of q0 , consider the new production q0 → ε and for each production q0 → α introduce also q0 → α. 13 Let A = ({q0 , q1 , q2 }, {a, b}, E, {q0 }, {q2 }) be a DFA, where E = (q0 , a, q0 ), (q0 , b, q1 ), (q1 , b, q2 ), (q2 , a, q2 ) . The corresponding transition table is δ a b q0 q1 q2 {q0 } ∅ {q2 } {q1 } {q2 } ∅ The transition graph of A is in Fig. 8. 11 we de ne regular grammar G = ({q0 , q1 , q2 }, {a, b}, P, q0 ) with the productions in P q0 → aq0 | bq1 , q1 → bq2 | b, q2 → aq2 | a.
14. (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 de nition of these tree operations proves again that regular languages are closed under the regular operations.
Algorithms of informatics by Ivanyi A. (ed.)
by George
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