solve it urgently..

Let ( Sigma={mathrm{a}, mathrm{b}} ). Note that we define ( c(w, x) ) to be the count of ( x ) in the string ( w ). [ egin{array}{c} L_{1}={w mid w ext { is a palindrome and the length of } w ext { is even. }} \ L_{2}=left{w_{1} # w_{2} # w_{3} mid cleft(w_{1}, mathrm{~b} ight)=cleft(w_{3}, mathrm{~b} ight) ext { or } cleft(w_{2}, mathrm{a} ight)=cleft(w_{3}, mathrm{a} ight) ight} end{array} ] Now solve the following problems. (a) Give the state diagram of a pushdown automaton that recognizes ( L_{1} ). (4 points) (b) Find all strings ( w in L_{2} ) such that ( w ) starts with babbab#abba# and has a length of 15. (1 point) (c) Give the state diagram of a pushdown automaton that recognizes ( L_{2} ). (5 points)

Let ={a,b}. Note that we define c(w,x) to be the count of x in the string w. L1={wwisapalindromeandthelengthofwiseven.}L2={w1#w2#w3c(w1,b)=c(w3,b)orc(w2,a)=c(w3,a)} Now solve the following problems. (a) Give the state diagram of a pushdown automaton that recognizes L1. (4 points) (b) Find all strings wL2 such that w starts with babbabuabball and has a length of 15. (1 point) (c) Give the state diagram of a pushdown automaton that recognizes L2. (5 points)