Homework for chapter $12$

$1.$ Let $\backslash (\backslash$Sigma$=\backslash \{0,1\backslash \}\backslash ),$ and let $\backslash ($ L $\backslash )$ be the language over $\backslash (\backslash$Sigma $\backslash )$ consisting of all strings of $0\text{'}$s and $1\text{'}$s of length $4$ with an equal number of $0\text{'}$s and $1\text{'}$s$.$ List the elements of $\backslash ($ L $\backslash )$

$2.$ Let $\backslash ($ L $\backslash )$ be the language defined by the regular expression $\backslash (0(0\backslash$mid $1)^\{*\}1(0\backslash$mid $1)^\{*\}\backslash ).$

$($a$)$ Write $3$ strings that belong to $\backslash ($ L $\backslash )$

$($b$)$ Use words to describe $\backslash ($ L $\backslash )$

$3.$

Consider the finite$-$state automaton given by the following transition diagram:

$($a$)$ What is $\backslash ($ N$\backslash$left$($s$\_\{2\},$ a$\backslash$right$)\backslash )?$

$($b$)$ To what state does the automaton go if the string babag is input to it$?$

$($c$)$ Describe the language accepted by this automaton

$($d$)$ Find a regular expression that defines the same language.

$4.$

Consider the finite$-$state automaton given by the following next$-$state table: :$-.....$

sta

$($a$)$ Draw the transition diagram for this automaton.

$($b$)$ Describe the language accepted by the automaton