计算理论答案ch2

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Undergraduate Course ELEMENTS OF COMPUTATION THEORY College of Computer Science Chapter2ZHEJIANG UNIVERSITY

Fall-Winter,2006

P60

2.1.1Let M be a deterministicfinite 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=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).

2.1.2Describe informally the languages accepted by the following DF A.

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.

2.1.3Construct DF A 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}.

Solution:(c)M=(K,Σ,δ,sF),where

K={q0,q1,q2,q3},Σ={a,b},s=q0,F={q0,q1,q2}

q aδ(q,a)

q0a q1

q0b q2

q1a q3

q1b q2

q2a q1

q2b q3

q3a q3

q3b q3

(e)M=(K,Σ,δ,sF),where

K={q0,q1,q2,q3,q4,q5},Σ={a,b},s=q0,F={q5}

q aδ(q,a)

q0a q1

q0b q2

q1a q1

q1b q3

q2a q4

q2b q2

q3a q5

q3b q3

q4a q4

q4b q5

q5a q5

q5b q5

P74

2.2.2Which regular expression for the languages accepted by the NF A of Problem2.2.1.

Solution:

a)a∗

b)a(ba∪baa)∗(b∪ba)

2.2.6(a)Find a simple NF A accepting(ab∪aab∪aba)∗.

(b)Convert the NF A of part(a)to a DF A by the method in section2.2.

Solution:

(a)M=(K,Σ,∆,sF),where K={q0,q1,q2,q3},Σ={a,b},s=q0,F={q0}

(qσq i)

(q0a q1)

(q1a q2)

(q1b q0)

(q1b q3)

(q2a q0)

(q3b q0)

(b)Determinizing the above machine results in the following DFA:

K={{q0},{q1},{q3},{q0,q1},{q0,q2},{q1,q3},∅},Σ={a,b},s={q0},

F={{q0},{q0,q1},{q0,q2}}

{q}σ{δ(q,σ)}

{q0}a{q1}

{q0}b∅

{q1}a{q3}

{q1}b{q0,q2}

{q0,q2}a{q0,q1}

{q0,q2}b∅

{q0,q1}a{q1,q3}

{q0,q1}b{q0,q2}

{q3}a∅

{q3}b{q0}

{q1,q3}a{q3}

{q1,q3}b{q0,q2}

∅a∅

∅b∅

2.2.10Describe exactly what happens when the construction of this section

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