语言的CFG

Question

I'm trying to create a cfg generating following language:

Is This language context-free and could be generated by a cfg? if yes , how could the grammar generating this language be created?

I'm not experienced so much in creating cfg's for cfl's. I would be glad if any help or solution is given

Answer 1

To start you off, do you know how to create a CFG for the language {a^nd^t | n = t} {a^nd^t | n = t} ? This will be your starting point.

S -> aSd | epsilon

From here, you need to extend what happens when there are more a s than d s, or more d s than a s. The goal will always be to make sure that for every a or b consumed on the left, a c or d is consumed on the right.

S -> aSd | aAc | bDd | epsilon

A is the case where there are more a s than d s, D is the other case. In each of these, you you need to consume a left character for every right character.

There's an additional case, which is the case that either or both of n and t are 0 . I'll leave it to you to determine how to handle that.

Then you'll have to deal with running out of one side's character. For example, if you're in state A , you expect to match a s with c s until the a s run out and turn into b s. Something like,

A -> aAc | bCc

I'll leave the rest to you to figure out.