Peter Linz Edition 4 Exercise 2.1 Question 6 (Page No. 47)

Question

With $Σ =  \{a,b\},$ give a DFA for $L = \{w_1aw_2: |w_1|\geq 3, |w_2|\leq 5\}.$

Answer 1

$L=\left \{ w1\, a\, w2 \right \},\, \sum =\left \{ a,b \right \},\left | w1 \right |\geq 3,\left | w2 \right |\leq 5$

Assuming $w1,w2\epsilon \left \{ a,b \right \}$

Given DFA is correct .Following points may he;lp you 

$\Rightarrow$Regular expression is $\left ( a+b \right )\left ( a+b \right )\left ( a+b \right )b^{*}a\left ( a+b \right )^{n},n\leq 5$

$\Rightarrow$ $a$ is your boundary,$\left | w2 \right |\leq 5$  which comprises $\left ( a+b \right )^{n},n \leq 5$

$\Rightarrow$ $a$ is your boundary,$\left | w1 \right |\geq 3$ which comprises $\left ( a+b \right )\left ( a+b \right )\left ( a+b \right )b^{*}$

Take any example-:

Let us take a least one $\left | w1 \right |=3,\left | w2 \right |=1,$

abbab $\Rightarrow$ accepted

abbbbbbb,here $\left | w1 \right |=7,\left | w2 \right |=1,$$\Rightarrow$ not accepted

Here only point to be noted this machine needs 'a' to be crossed.

Comments

at state D you make it NFA

pls correct it

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how u have differentiate that till state "I" a's and b's not from W1 ??

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Reg expression given by u generates the string bbbabaabb
w1=bbb {|w1|>=3}   w2=baabbb {|w2|<=5}  but it is not accepting bythe given machine 
plzz clarify me 
thnx.......

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bbbaaaaaaaa not accept

Answer 2

States      a         b

->A               B         B

B                C         C

C               D          D

D               E          D

*E                E          E