Question

Its mostly in the title, what I mean by alternating is that there is a single 1 or 0 between two occurrences of 0’s. So no such occurrences of 010 or 000.

I am trying to understand the theoretical nature of the question so I would prefer the answer in terms of just Concatenation, Union, and Closure (10, 1|0, and 10*).

Please note this is NOT a homework question I am just interested in the question, so please no patronizing comments to that affect.

Edit: changed the wording in the first paragraph from “between the occurrences” to “between two occurrences”.

Answer 1

The first important observation is that we can reduce the problem: if S is allowed and is 1 or ends with two ones, S concatenated with any allowed sequence remains allowed. So what makes a valid S?

1. 1 is explicitly in S, and any sequence of more than one 1 will always end in 11 and so is in S.
2. A 01 may only be followed by another 1, as a 0 would give 010. This ends in 11, and so is a valid S.
3. A 00 may only be followed by 11, as 0000, 0001 and 0010 all have alternating zeros. This ends in 11, and so is a valid S.

So that tells us any S matches ^(1|011|0011)*$. But we’re not done, as there are a few other sequences that are not valid for S but are themselves allowed sequences and so could be concatenated to an S.

• 0
• 00
• 01
• 001

000 is not allowed, of course, and anything longer is either in S or is itself not allowed.

So the entire regex matching allowed sequences is:

^(1|011|0011)*(|0|00|01|001)$

That is zero or more S sequences, optionally followed by one of our allowed non-S sequences.