By Ivanyi A. (ed.)

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

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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. 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.

Value of IsIn(q, Q) in the algorithm is true if state q is already in Q and is false otherwise. Let a1 , a2 , . . , am be an ordered list of the letters of Σ. Nfa-Dfa(A) 1 2 3 4 5 6 7 q0 ← I Q ← {q 0 } i←0 k←0 repeat ✄ i counts the rows. ✄ k counts the states. for j = 1, 2, . . , m do q ← δ(p, aj ) ✄ j counts the columns. p∈q i 8 if q = ∅ 9 then if IsIn(q, Q) 10 then M [i, j] ← {q} 11 else k ← k + 1 12 qk ← q 13 M [i, j] ← {q k } 14 Q ← Q ∪ {q k } 15 else M [i, j] ← ∅ 16 i←i+1 17 until i = k + 1 18 return transition table M of A Since loop repeat is executed as many times as the number of states of new automaton, in worst case the running time can be exponential, because, if the number of states in NFA is n, then DFA can have even 2n − 1 states.

Sk−1 , ak ) = Sk . Then q1 ∈ S1 , . . , qk ∈ Sk and since qk ∈ F we get Sk ∩ F = ∅, so Sk ∈ F . Thus, there exists a walk a a ak−1 a a 1 2 3 k S0 −→ S1 −→ S2 −→ · · · −→ Sk−1 −→ Sk , S0 ⊆ I, Sk ∈ F . There are sets S0 , . . , Sk for which S0 = I , and for i = 0, 1, . . , k we have Si ⊆ Si , and ak−1 ak a1 a2 a3 S0 −→ S1 −→ S2 −→ · · · −→ Sk−1 −→ Sk is a productive walk. Therefore w ∈ L(A). That is L(A) ⊆ L(A). b) Now we show that L(A) ⊆ L(A). Let w = a1 a2 . . ak ∈ L(A). Then there is a walk ak−1 ak a1 a2 a3 q0 −→ q1 −→ q2 −→ · · · −→ q k−1 −→ q k , q 0 ∈ I, q k ∈ F .