8. More examples

DFA $A_3$

digraph {
	rankdir=LR
	init[shape=point]
	node[shape=doublecircle]; r;
	node[shape=circle];
	init->s
	s->s [label=b]
	s->q [label=a]
	q->s [label=b]
	q->r [label=a]
	r->r [label=a]
	r->r [label=b]
}

$$\begin{array}{r}L(A_3)=\text{all the words that contain two consecutive }a\text{ symbols}\end{array}$$

DFA $A_4$

digraph {
	rankdir=LR
	init[shape=point]
	node[shape=doublecircle]; s, q;
	node[shape=circle];
	init->s
	s->s [label=b]
	s->q [label=a]
	q->s [label=b]
	q->r [label=a]
	r->r [label=a]
	r->r [label=b]
}

$$\begin{array}{rl}L(A_4)& =\text{all the words that do not contain two consecutive }a\text{ symbols}\\ \text{it is also the case that }& =\{w|w\text{ is any word of }a\text{ and }b\}-L(A_3)\end{array}$$

DFA $A_5$

digraph {
	rankdir=LR
	init[shape=point]
	node[shape=doublecircle]; t;
	node[shape=circle];
	init->s
	s->q [label=a]
	s->t [label=b]
	q->r [label=b]
	t->r [label=a]
	
	q->s [label=a]
	t->s [label=b]
	r->t [label=a]
	r->q [label=b]
}

Some experiments:

• $abbabbabaa$ ends up in $r⇝$ rejected
• $abbaabbabaa$ ends up in $r⇝$ accepted
• $abbababbabaa$ ends up in $r⇝$ rejected
• $babbababbabaa$ ends up in $r⇝$ accepted
• $babbababbabba$ ends up in $r⇝$ rejected

$$\begin{array}{rl}L(A_5)& =\text{all the words that contain an even number of }a\text{ and odd number of }b\end{array}$$