Regular expression for the complement of language L = {an bm | n ≥ 4, m ≤ 3} is

We want the complement of L = { a^n b^m | n ≥ 4, m ≤ 3 } over the alphabet {a,b}.

Strings not in L fall into three mutually covering categories:

• Strings that are not of the form a* b* (i.e., there is at least one occurrence of the substring "ba"). Regex: (a+b)* b a (a+b)*

• Strings of the form a^n b^m with too few a's (n ≤ 3). Regex: (λ + a + aa + aaa) b*

• Strings of the form a^n b^m with too many b's (m ≥ 4). Regex: a* bbbb b*

Combine these by union to get a full regular expression for the complement:

(λ + a + aa + aaa) b* + a* bbbb b* + (a + b)* b a (a + b)*

Remarks: overlaps between the three parts are allowed (the union handles duplicates). This expression accepts exactly those strings that are not of the form a^n b^m with n ≥ 4 and m ≤ 3.