The regular expression for the complement of the language L = {anbm|n ≥ 4, m ≤ 3} is :
(λ + a + aa + aaa)b* + a* bbbbb* + (a + b)* ba(a + b)*
L contains exactly the strings of the form \(a^n b^m\) with at least four a's followed by at most three b's. A string is outside L in one of three disjoint ways.
Step 1 — It is not of the form a*b* at all, i.e. some b is followed by an a. These strings contain the substring ba: \((a + b)^* ba (a + b)^*\).
Step 2 — It is of the form a*b* but has at most three a's (any number of b's): \((\lambda + a + aa + aaa)\,b^*\).
Step 3 — It is of the form a*b* with at least four b's (any number of a's): \(a^*\,bbbb\,b^* = a^*\,bbbbb^*\).
Step 4 — The union of the three parts is the complement: \((\lambda + a + aa + aaa)b^* + a^*bbbbb^* + (a + b)^*ba(a + b)^*\).
The expression with a*bbbb* accepts aaaabbb (three b's), which belongs to L. The expressions that use the substring ab miss strings like ba and wrongly include aaaab, which is in L; the one without b* after the first group also drops strings such as b.
(λ + a + aa + aaa)b* + a* bbbbb* + (a + b)* ba(a + b)* option is correct.
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