Regular Expression for a binary string that accepts even number of zeros

Question

As stated in the title, follows the DFA that accepts the proposed language:

My question is about the regular expression of that DFA which is as follows:(1*01*01*)*. Conforms to regular expression, it does not mean that any string coming from there starts with binary 1?

Answer 1

No, the string doesn’t necessarily have to start with 1.

After all, 1* that is to say "zero or more occurrences of 1". The quantifier * admits zero occurrences, which means that the 1 is not mandatory (see some examples).

Just one detail: regex has a quantifier around the parentheses: (...)*, which means that everything between parentheses can also occur zero times. That is, regex also accepts empty strings (or "nothing" - see here how she finds a match).

Although zero is an even number, then "zero occurrences" of digit 0 is also an "even number of zeros" (but if the string is not empty, there must be at least two zeroes).

---

If you do not want to consider an empty string valid, you can change the regex to (1*01*01*)+, for the quantifier + means "one or more occurrences", meaning that what is in parentheses must occur at least once. Thus, the empty string is no longer considered valid.