$($c$)$ Find out the CFL for: $($c$)$ Design a PDA that accepts the language

$L_{wwr}=\{\frac{w^R}{w}in(0+1{)}^{**}\}$

$($d$)$ Convert to a regular expression the following DFA :

$($a$)$ Define decidable and undecidable problem.

$($b$)$ Generate the transition diagram for the Turing Machine that

accepts the language $\{0^n{1}^n|n>0\}.$

$($c$)$ Define useless symbol and E$-$production with example.

$($d$)$ Remove null production from the following:

$S$ASA$|aB|b$

$AB$

$Bb|.$

$($d$)$ Minimize the following DFA by minimization algorithm :

$($a$)$ Define regular expression. Write down the operators of

regular expression.

$($b$)$ State and prove the pumping lemma for regular expressions.

$($c$)$ The following grammar generates prefix expressions with

operands $x$ and $y$ and binary operators,$+-,$ and $*$:

$E+EE{|}^{**}EE|-EE|x|y$

$(1)$ Find leftmost and rightmost derivations and a derivation

tree for the string $+**-xyxy.$

$($ii$)$ Prove that this grammer is unambiguous.

$($d$)$ Define homomorphism. Suppose $h$ is the homomorphism

from the alphabet $\{0,1,2\}$ to the alphabet $\{a,b,c\}$ defined

by $h(0)=a,h(1)=ab,h(2)=bc-$

$($i$)$ What is $h(0211)?$

ii$)$ If $L$ is the language $L(1+20)$; What is $h(L)?$