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Additional info for Algorithms of informatics, vol. 1

Example text

Finite automata and regular languages 35 In the first column of the first row we put (q0 , q0 ) and complete the first row using the above method. If in the first row in any column there occur a pair of states from which one is a final 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 first column. The algorithm continues with the next unfilled row. If no new pair of states occurs in the table and for each pair both of states are final or both are not, then the algorithm ends and the two DFA are equivalent.

42 1. 11 Finite automata -moves. The FA with ε-moves in Fig. 11 accepts words of form uvw, where u ∈ {1}∗ , v ∈ {0}∗ and w ∈ {1}∗ . 14 To any FA with ε-moves one may construct an equivalent NFA (without ε-moves). Let A = (Q, Σ, E, I, F ) be an FA with ε-moves and we construct an equivalent NFA A = (Q, Σ, E, I, F ). The following algorithm determines sets F and E. For a state q denote by Λ(q) the set of states (including even q) in which one may go from q using ε-moves only. This may be extended also to sets Λ(S) = Λ(q), q∈S ∀S ⊆ Q .

Xm , then the pushdown automaton can change its configuration in two ways: • (q, a1 a2 . . ak , x1 x2 . . xm−1 xm ) =⇒ (p, a2 a3 . . ak , x1 , x2 . . xm−1 w), if q, (a1 , xm /w), p ∈ E • (q, a1 a2 . . ak , x1 x2 . . xm ) =⇒ (p, a1 a2 . . ak , x1 , x2 . . xm−1 w), if q, (ε, xm /w), p ∈ E. ∗ The reflexive and transitive closure of the relation =⇒ will be denoted by =⇒. Instead of using =⇒, sometimes is considered. How does work such a pushdown automaton? Getting started with the initial configuration (q0 , a1 a2 .

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