Final answer:
To demonstrate that ADD is not a regular language, one can use the Pumping Lemma for regular languages. By assuming ADD is regular and using a string from ADD that represents a valid binary addition, we deduce that pumping any substring of that string would invalidate the original equality, contradicting the conditions of the Pumping Lemma and proving that ADD is not regular.
Step-by-step explanation:
To show that the language ADD, which consists of strings representing the addition of binary integers, is not regular, we will use the Pumping Lemma for regular languages. The Pumping Lemma states that for any regular language, there exists a pumping length p such that any string s in the language with length at least p can be divided into three parts, xyz, satisfying the following conditions:
- The string xynz is in the language for every n ≥ 0.
- The length of xy is at most p (|xy| ≤ p).
- The length of y is at least 1 (|y| ≥ 1).
Assume for the sake of contradiction that ADD is regular and let p be the pumping length given by the Pumping Lemma.
Consider the string s = '0p + 1 = 1p0', where s is in ADD and represents the addition of '0p + 1' which equals '1p0'. This string has a length greater than p, therefore it should be pumpable according to the Pumping Lemma.
However, no matter how the string is divided into xyz, pumping y will disrupt the equality because it will either add more zeros to the first part or more zeros to the result, without appropriately increasing the other. Hence, for some value of n, the string xynz will not represent a valid addition, contradicting the Pumping Lemma's conditions. Therefore, we can conclude that ADD is not regular.