Formal Binary String Regular Expression (each pair of 00 must have 11 before it)

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Here is a way to describe a binary string in which every 00 must have at least two 1s before it.

It is a string that
$\quad$ starts with zero or one 0;
$\quad$ followed by zero or more blocks, where each block is
$\quad\quad$ either one or more 1s followed by a 0, such as 10, 110, 1110, etc,
$\quad\quad$ or two or more 1s followed by two 0s, such as 1100, 11100, 111100, etc;
$\quad$ followed by zero or more 1s.

Translated into regular expression, the description above becomes
$\quad$ (0|ε)(11*0|111*00)*1*

You might want to ask, how can we arrive that kind of description?

The observation here is that the number of consecutive 0s in the string is at most 2, a very small number. That indicates that we can break down the string by enumerate different possibilities of consecutive 0s. To paraphrase rici's comment, we could try using enumeration to simulate conditionals.

Of course, there is a mild stroke of luck involved in the approach above on that language. As a last resort, we can always use a heavier machinery such as constructing a DFA for the language (which can be done in one or several ways) and then converting the DFA to a regular expression.

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