By Ivanyi A. (ed.)

ISBN-10: 9638759607

ISBN-13: 9789638759603

ISBN-10: 9638759615

ISBN-13: 9789638759610

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**Example text**

Finite automata and regular languages 47 First we mark pairs {q2 , q0 }, {q2 , q1 }, {q2 , q3 }, {q2 , q4 } and {q2 , q5 } (because q2 is the single nal state). Then consider all unmarked pairs and examine them as the algorithm requires. Let us begin with pair {q0 , q1 }. Associate with it pairs {elem δ(q0 , 0) , elem δ(q1 , 0) }, {elem δ(q0 , 1) , elem δ(q1 , 1) }, that is {q1 , q4 }, {q4 , q2 }. Because pair {q4 , q2 } is already marked, mark also pair {q0 , q1 }. In the case of pair {q0 , q3 } the new pairs are {q1 , q5 } and {q4 , q4 }.

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 . e. there exists qk ∈ q k ∩F , that is by the denitions of qk ∈ F and q k there is qk−1 such that (qk−1 , ak , qk ) ∈ E . Similarly, there are the states qk−2 , . . , q1 , q0 such that (qk−2 , ak , qk−1 ) ∈ E, . . , (q0 , a1 , q1 ) ∈ E, where q0 ∈ q 0 = I , thus, there is a walk ak−1 ak a1 a2 a3 q0 −→ q1 −→ q2 −→ · · · −→ qk−1 −→ qk , q0 ∈ I, qk ∈ F, so L(A) ⊆ L(A).

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. States p and q of an DFA A = (Q, Σ, E, {q0 }, F ) are equivalent if for arbitrary word u we reach from both either nal or nonnal states, that is u u p −→ r, r ∈ F and q −→ s, s ∈ F or p ≡ q if for any word u ∈ Σ∗ u u p −→ r, r ∈ F and q −→ s, s ∈ F .

### Algorithms of informatics. Foundations by Ivanyi A. (ed.)

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