By Ivanyi A. (ed.)

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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. 10 will have it that to any NFA one may construct an equivalent DFA.

13(a). Initial states are represented by a circle with an arrow, nal states by a double circle. Let A1 = (Q1 , Σ1 , E1 , I1 , F1 ) and A2 = (Q2 , Σ2 , E2 , I2 , F2 ) be NFA. The result of any operation is a FA with ε-moves A = (Q, Σ, E, I, F ). 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).

Nonproductive-States(A) 1 2 3 4 5 6 7 8 9 10 V0 ← F i←0 repeat i←i+1 for all p ∈ Vi−1 do do for all a ∈ Σ do Vi ← Vi−1 ∪ δ −1 (p, a) until Vi = Vi−1 V ← Q \ Vi return V The nonproductive states of the automaton can be eliminated without changing the accepted language. 2. Finite automata and regular languages 31 the running time of the algorithm is also O(n2 m) as in the case of the algorithm Inaccessible-states. The set V given by the algorithm has the property that L(A) = ∅ if and only if V ∩ I = ∅.