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3.4 Figure 3-19 exercises to calculate the failure function KMP algorithm. Illustrated in the case of a known failure function, and how

according to the known keywords b_1b_2 ... b_n construct a DFA n +1 states with the DFA can identify the language. * B_1b_2 ...

b_n. Closer prove construct the DFA's time complexity is O (n).

For illustration, a string abbaabb 3.4.3-3 example, failure function is:

n: 1, 2, 3, 4, 5, 6, 7

f (n): 0, 0, 0, 1, 1, 2, 3

DFA was constructed as follows:

The DFA constructed pseudo-code:

for (i = 0; i <n; i + +) {

move [s [i], c] = {

if (c == b_1b_2 ... b_n [i]) {

goto s [i +1]

} Else {

goto s [f (i)]

}

}

}

Obviously, in a known f (n) of the case, the time complexity of the algorithm is O (n)

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3.6 Exercise

3.6.1!

Answer

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3.3.5 for the exercise of each language to design a DFA or NFA.

NFA shown in Figure 3-29 to identify all labeled aabb path. The NFA accepts aabb it?

(0)-a-> (1)-a-> (2)-b-> (2)-b-> ((3))

(0)-a-> (0)-a-> (0)-b-> (0)-b-> (0)

(0)-a-> (0)-a-> (1)-b-> (1)-b-> (1)

(0)-a-> (1)-a-> (1)-b-> (1)-b-> (1)

(0)-a-> (1)-a-> (2)-b-> (2)-b-> (2)

(0)-a-> (1)-a-> (2)-b-> (2)-ε-> (0)-b-> (0)

(0)-a-> (1)-a-> (2)-ε-> (0)-b-> (0)-b-> (0)

The NFA apparently accepted aabb

For the NFA Figure 3-30, repeat the exercise 3.6.3

In practice is given below of the NFA conversion table:

1. Exercise 3.6.3

2. Exercise 3.6.4

3. Figure 3-26

Table 1

Status a b ε

0 {0,1} {0} ∅

1 {1,2} {1} ∅

2 {2} {2,3} {0}

3 ∅ ∅ ∅

Table 2

Status a b ε

0 {1} ∅ {3}

1 ∅ {2} {0}

2 ∅ {3} {1}

3 {0} ∅ {2}

3.6.2

3.6.3

Answer

3.6.4

3.6.5

Answer

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Table 3

Status a b ε

0 ∅ ∅ {1,2}

1 {2} ∅ ∅

2 {2} ∅ ∅

3 ∅ {4} ∅

4 ∅ {4} ∅

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