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Unformatted text preview: ECS 120 Lesson 6 – Nondeterministic Finite State Machines, Pt. 2 Oliver Kreylos Wednesday, April 11th, 2001 1 Validity of the Subset Construction Last time, we introduced the subset construction to build a DFA D = ( Q D , Σ ,δ D ,q D ,F D ) that simulates an NFA M = ( Q, Σ ,δ,q ,F ). Today we have to prove that these two automata are in fact equivalent, i. e., that they accept the same language: L ( M ) = L ( D ). We will prove this by showing that the two automata’s extended transition functions are identical, using induction over the length of a word w ∈ Σ * . Claim For any subset of states P ⊂ Q and any word w ∈ Σ * , δ * ( P,w ) = δ * D (ECLOSE( P ) ,w ). Induction Basis Let P ⊂ Q be any subset of states, and let w be the empty word, w = . In this case, | w | = 0. Then δ * ( P,w ) = δ * ( P, ) = ECLOSE( P ). Starting from the other direction, δ * D (ECLOSE( P ) ,w ) = δ * D (ECLOSE( P ) , ) = ECLOSE( P ). Induction Hypothesis Assume the claim is true for all words w ∈ Σ with | w | = n , n ≥ 0. Induction Step Let P ⊂ Q be any subset of states, and let w ∈ Σ * be any word of length | w | = n + 1. Then w can be written as w = ax , where a ∈ Σ and x ∈ Σ * with | x | = n . Then δ * ( P,w ) = δ * ( P,ax ) = δ * ( δ (ECLOSE( P ) ,a ) ,x ) = δ * ( P ,x ) with P := δ (ECLOSE( P ) ,a ). On the other hand, δ * D (ECLOSE( P ) ,w ) = δ * D (ECLOSE( P ) ,ax ) = δ * D ( δ D (ECLOSE( P ) ,a ) ,x ) = δ * D ( ECLOSE ( δ (ECLOSE( P ) ,a ) ) ,x ) = 1 δ * D ( ECLOSE( P ) ,x ) . Since | x | = n , we can now apply the induction hypothesis δ * ( P ,x ) = δ * D ( ECLOSE( P ) ,x ) . To finish the proof of L ( M ) = L ( D ), we now have to consider δ * ( q ,w ) and δ * D ( q D ,w ). Using the result from above and q D = ECLOSE( q ), we have δ * ( q ,w ) = δ * D ( ECLOSE( q ) ,w ) = δ * D ( q D ,w ). Now let w ∈ Σ * be any word. M accepts w , iff δ * ( q ,w ) ∩ F 6 = ∅ . D accepts w , iff δ * D ( q D ,w ) ∈ F D = P ∈ P ( Q ) P ∩ F 6 = ∅ . These two statements are equivalent; therefore, M accepts w exactly if D accepts w . 2 Closure of Regular Languages – Continued Now having the necessary tool in form of nondeterministic automata, we will continue looking at the behaviour of regular languages under the language operations we defined earlier. 2.1 Union We already proved that regular languages are closed under the union op- eration. To show this claim, we had to construct the product automa- ton. Using nondeterministic finite automata, the proof becomes much easier....
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This note was uploaded on 05/20/2010 for the course ECS 120 taught by Professor Filkov during the Spring '07 term at UC Davis.
- Spring '07