Question

Give a regular expression generating the following languages over the alphabet {a,b}:

{w | w is any string except aa and bbb}

Answer 1

--> Since we need to remove two strings specifically, First simply add all the strings

      till length 3 except those not needed.

              ε + (a+b) + (ab+ba+ab) + (aaa+aab+aba+abb+bab+bba+bba)

--> Notice that 'aa' and 'bbb' are not there.

--> Now add expression for all strings of length 4 and above.

              (a+b)(a+b)(a+b)(a+b)(a+b)^*

--> Join both these expressions.

            ε + (a+b) + (ab+ba+ab) + (aaa+aab+aba+abb+bab+bba+bba) +

              (a+b)(a+b)(a+b)(a+b)(a+b)^*

--> We can further simply it by taking some terms common, But this from is also ok.

Answer 2

To generate the language {w | w is any string except "aa" and "bbb"} over the alphabet {a, b}, we can use a regular expression that covers all possible strings while excluding the specific patterns "aa" and "bbb". Here's the regular expression:

^(?!(aa|bbb))[ab]*$

Explanation:

• ^: Denotes the start of the string.

• (?!(aa|bbb)): A negative lookahead assertion. This ensures that the string does not start with "aa" or "bbb". If it does, the match fails.

• [ab]*: Matches any combination of "a" and "b", including empty string (i.e., zero occurrences of "a" and "b").

• $: Denotes the end of the string.

The regular expression will generate all possible strings over {a, b} except for "aa" and "bbb". For example, it will match strings like "a", "b", "aba", "babab", "baaab", "aabab", etc., but it will not match "aa" or "bbb".