$1.$ Construct a PDA for the following languages

$($a$)|,j,k1$ and $\{$:$i+j=k\}$

$($b$)L_2=\{xin\{a,b,c,{\}}^{*}||x{|}_a+|x{|}_b=|x{|}_c\}$

$($c$)L_3=\{a^n{b}^{2n}|n1\}$

$($d$)L_4=\{xin\{a,b{\}}^{**}||x{|}_a=|x{|}_b\}$

$($e$)$ Given an example of a finite language that cannot be recognized by any one$-$state PDA that accepts via final state

$2.$ Construct a PDA for the following languages

$($a$)L_5=\{a^n{b}^{3n}|n1\}$

$($b$)L_6=\{a^n{b}^n{c}^m{d}^m|n,m1\}$

$($c$)$ The set of all non$-$palindromes over $\{a,b\}$

$($d$)$ The set of all strings of balanced parentheses $($each left parenthesis has a matching right parentheses and pairs of matching parentheses are properly nested$)$

$($e$)L_7=\{x0y1|x,$yin$\{0,1{\}}^{*}||x|=|y|\}$

$3.$ Construct a PDA for the following languages

$($a$)L_8=\{a^n{b}^m{c}^{m+n}|n,m0\}$

$($b$)L_9=\{a^n{b}^m|nm3n\}$

$4.$ Find an npda with no more than two internal states that accepts the language $L(a{a}^{*}b{a}^{*})$

$5.$ Find an npda for the following language $L_{10}=\{a^n{b}^n|n0\}\{a\}$