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Unformatted text preview: CS 341: Foundations of Computer Science II Prof. Marvin Nakayama Homework 4 1. Use the procedure described in Lemma 1.55 to convert the regular expression (((00) ∗ (11)) ∪ 01) ∗ into an NFA. Answer: 1 1 00 ε 11 1 ε 1 01 ε 1 (00) ∗ ε ε ε (00) ∗ (11) ε ε ε 1 ε 1 ε ε 1 (00) ∗ (11) ∪ 01 ε ε ε 1 ε 1 ε ε ε 1 ε ε ((00) ∗ (11) ∪ 01) ∗ ε ε ε 1 ε 1 ε ε ε 1 ε ε ε ε ε 2. Use the procedure described in Lemma 1.60 to convert the following DFA M to a regular expression. 1 2 3 a,b a b b a Answer: First convert DFA M into an equivalent GNFA G . 2 s t 1 2 3 ε a ∪ b a b b a ε ε ε Next, we eliminate the states of G (except for s and t ) one at a time. The order in which the states are eliminated does not matter. However, eliminating states in a different order from what is done below may result in a different (but also correct) regular expression. We first eliminate state 3 . To do this, we need to account for the paths 2 → 3 → 1 , which will create an arc from 2 to 1 labelled with ba ; 2 → 3 → 2 , which will create an arc from 2 to 2 labelled with bb ; and 2 → 3 → t , which will create an arc from 2 to t labelled with bε = b . We combine the previous arc from 2 to 2 labelled a with the new one labelled bb to get the new label a ∪ bb . s t 1 2 ε a ∪ b ba a ∪ bb ε ε b We next eliminate state 1 . To do this, we need to account for the following paths: s → 1 → 2 , which will create an arc from s to 2 labelled with ε ( a ∪ b ) = a ∪ b . s → 1 → t , which will create an arc from s to t labelled with εε = ε . 2 → 1 → 2 , which will create an arc from 2 to 2 labelled with ba ( a ∪ b ) . We combine this with the existing 2 to 2 arc to get the new label a ∪ bb ∪ ba ( a ∪ b ) . 2 → 1 → t , which will create an arc from 2 to t labelled with baε = ba . We combine this arc with the existing arc from 2 to t to get the new label b ∪ ba . 3 s t 2 a ∪ b ε a ∪ bb ∪ ba ( a ∪ b ) b ∪ ba Finally, we eliminate state 2 by adding an arc from s to t labelled ( a ∪ b )( a ∪ bb ∪ ba ( a ∪ b )) ∗ ( b ∪ ba ) . We then combine this with the existing s to t arc to get the new label ε ∪ ( a ∪ b )( a ∪ bb ∪ ba ( a ∪ b )) ∗ ( b ∪ ba ) ....
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This note was uploaded on 01/22/2011 for the course CIS 341 taught by Professor Nakayama during the Fall '10 term at NJIT.
 Fall '10
 Nakayama
 Computer Science

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