Question

Suppose that $L = { a ^{n}b^{n} : n \geq 0}$

a) show that $L^{2}$ is a context free language

b) show that $L^{k}$ for every $k \geq 1$ is also context free

Answer 1

CFG is a Formal Grammer is used to Generate Set of strings . It is also called as Type 2 Grammer

Syntax = G= {V,T,P,S}

Given L = anbn : n > = 0

It means for a derived string have  number of a's is equal to number of b's

if n=2 then L = {a2b2} String = aabb

=> aSb here  S = string(i.e Derive any number of a's and any number of b's)

=> aabb ( S-> ab)

Grammer G = { V.T,P,S}

V= { S }

T = { a,b,e} (here  e = epsilon)

P : S->aSb/ab/ e

S = Sta

a) Given that L2 is a CFG

as L = { anbn}

L2 is represented as L x L = { a2n b2n } if n = 1 ,

L2 = { a2 b2 } String = aabb ( in which derived string is same as L}

if n=0 L2 string derives (Epsilon e)

Therefore L2 is also a CFG.

b) Given Lk : k > = 1

Lk = {ak bk }

as L and L2 are CFG , Lk is also be CFG

ex : if k = 2

L2  derives the string aabb . It is also CFG