对语言和CFG的一些限制

Question

I see one note about automaton theory:

Consider the following language:

L={xy : x,y in {a,b}*}

and consider following constraint:

1) x=y

2) x != y

3) x=(y)reverse

4) number of x is not equal to number of y

i read a language with constraint 2,3,4 is context free. any hint or tutorial for 1 to 3 is highly appreciated.

Answer 1

As you say, L 1 : {xy | x,y ∈ {a,b} * ∧ x=y} L 1 : {xy | x,y ∈ {a,b} * ∧ x=y} is not a context-free language.

However, the very similar L 3 : {xy | x,y ∈ {a,b} * ∧ x=y -1 } L 3 : {xy | x,y ∈ {a,b} * ∧ x=y -1 } is context free, as follows:

S → Ø
S → a S a
S → b S b

Technically speaking, the two other languages in your question are trivial:

L 2 : {xy | x,y ∈ {a,b} * ∧ x≠y}
L 4 : {xy | x,y ∈ {a,b} * ∧ |x|≠|y|}

because the only string which does not satisfy the constraint is the empty string. Any non-empty string S can be trivially decomposed into the two non-equal strings Ø and S ; it's clear that the concatenation of these two strings is S and that unless S is empty, the two strings are unequal and have different lengths. The language consisting of all non-empty strings is not only context free, but also regular: (a|b) + .

Since that was so easy, I suspect that it is not what you were actually asking, but I cannot guess what you really meant. I suggest you edit the question to be more precise.