# ch2 - ELEMENTS OF COMPUTATION THEORY Chapter 2...

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Undergraduate Course ELEMENTS OF COMPUTATION THEORY College of Computer Science Chapter 2 Zhejiang University Fall-Winter, 2007 P 60 2.1.1 Let M be a deterministic ﬁnite automaton. Under exactly what cir- cumstances is e L ( M ) ? Prove your answer. Solution: e L ( M ) if and only if s F . / Suppose e L ( M ) . Then, by deﬁnition of L ( M ) , ( s,e ) * M ( q,e ) , where q F . Because it is not the case that ( s,e ) M ( q,w ) for any conﬁguration ( q,w ) ( w 6 = e ). ( s,e ) * M ( q,e ) must be in the reﬂexive transitive closure of M by virtue of reﬂexivity - that is, ( s,e ) = ( q,e ) . Therefore, s = q and thus s F . / Suppose s F . Because * M is reﬂexive, ( s,e ) * M ( s,e ) . Because s F , we have e L ( M ) by deﬁnition of L ( M ) . 2.1.2 Describe informally the languages accepted by the following DFA. Solution: (c) All strings with the same number of a s and b s and in which no preﬁx has more than two b s than a s, or a s than b s. (d)All strings with the same number of a s and b s and in which no preﬁx has more than one more a than b , or vice-versa. 2.1.3 Construct DFA accepting each of the following languages. (c) { w ∈ { a,b } * : w has neither aa nor bb as a substring } . (e) { w ∈ { a,b } * : w has both ab and ba as a substring } .

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Solution: (c) M = ( K, Σ ,δ,sF ) , where K = { q 0 ,q 1 ,q 2 ,q 3 } , Σ = { a,b } , s
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ch2 - ELEMENTS OF COMPUTATION THEORY Chapter 2...

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