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Unformatted text preview: Sep. 17, 2010 (Friday) Chapter 1: Regular Languages 7.6 Equivalent ﬁnite automaton M For any regular expression R, there is an equivalent ﬁnite automaton M recognizing language L(R). We prove it by constructing an equivalent ﬁnite automaton for each of the six cases in the deﬁnition of a regular expression. •a
a •ε •∅ • (R1 ∪ R2 )
R1 ε R2 ε 23 • (R1 ◦ R2 ) R1 ε ε R2 ∗ • (R1 ) R1 ε ε ε 0 The new start state and its transition accept the empty string (i.e. R1 .), and the other + transitions accept the R1 . What if we do not add a new start state, just change the start state from a non-ﬁnal state (if it is) to a ﬁnal state? Considering the following ﬁnite automaton which accepts all strings ending with b, what is the star of it? The star contains all strings ending with b and . start a 24 b done a b 7.7 Examples of Equivalent ﬁnite automata Example: a∗ b∗ a*
ε ε a b*
ε ε b a* b*
ε ε a ε ε ε b ε Example: (ab ∪ a)∗
ε a ε ε ε a ε b ε 25 ...
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- Spring '10