$3$ Regular Expressions $($II$)$

a$)$ Let $L$ be the language given by

$L=\{\begin{array}{ccc}\text{w}\text{i}\text{n}& \{a,b,& c{\}}^{**}||w{|}_a=1\end{array}$ or $|w{|}_a=2,$ and every $c$ is preceded and$/$or followed by a $\{$:$b\}$

Give a regular expression for the language $L$ and explain your answer.

b$)$ Here is a very practical application of what we've learned so far. In this exercise you will

have to model the language describing the step of a type of cyber$-$attacks. No knowledge in

cyber$-$security are required for this exercise.

For this exercise we define the $4$ actions that an attacker can do:

use Phishing to get access to a system $(P)$

use a Bug to get access to a system $(B)$

Increase control over the system $(I)$

Extract information from a system $(E)$

In order to start a cyber$-$attack, an attacker needs to start by getting access to a system. To

do so$,$ they need to use Phishing $(P)$ or use a Bug $(B).$ Once the attacker is in the system,

they can Increase their control $(I)$ as much as they want to$.$ However, to get meaningful

information they need to Increase it $(I)$ at least one time. Once they have increased their

control, they can Extract information $(E).$ They can do this as many times as they like, but

if they do this more than twice, their attack will be discovered. To Extract more information

$($without being discovered$),$ they must restart their attack, or attack another system.

Your task is to give a regular expression over the alphabet $\{P,B,$I,$E\}$ describing

all possible attacks that an attacker can do without being discovered and explain

your answer. Don't forget that, after each attack, an attacker can restart their

attack, or switch to attacking another system.