Let L be the language over the alphabet = {a, b, c} that contains exactly those strings which satisfy all of the following properties: 1. the three symbols a, b, c occur in the string in the alphabetic order; 2. the length of the substring which consists of as is positive and even; 3. the length of the substring which consists of bs is equal to twice the length of the substring which consists of as; 4. the length of the substring which consists of cs is equal to twice the length of the substring which consists of bs;

Here is the answer of this question:::

Observe that all words of L satisfy the following characteristic property:

template: a^2n+2 b^4n+4 c^8n+8 Assume the opposite, that L is context free. Let pi be the constant as in the Pumping Lemma for L. Let w0 2 L be a string defined as follows: w0 = a^2n+2 b^4n+4 c^8n+8 where n > pi

w0 belongs to L because it satisfies the definition (which is equivalent to the template.)

w0 must pump because |w0| = 14n + 14 > n > pi

In any pumping decomposition of w0, the pumping window satisfies the following property:

spans no more than two adjacent segments

because it is too short to extend through all three segments, since it is shorter than by the Lemma, but each segment is longer than by construction.

By pumping 1 times, we obtain a string which violates the stated characteristic property because

the number of occurrences of at least one of the letters has increased, but the number of occurrences of at least one other letter has remained unchanged and thus does not belong to L.

Since L violates the Pumping Lemma, L is not context free.

Here is my question:

1) For the template why we have to +2 +4 and +8 ?

2) spans no more than two adjacent segments , it is too short to extend through all three segments, since it is shorter than by the Lemma, but each segment is longer than by construction. What this mean?

3) After we pumping 1 times, how its violates the stated characteristic property? Please explain clearly. I have no idea what this trying to prove.