' Let Σ = {a, b}. Write a regular expression for the language L…

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Manu · Accepted Answer

Step 1 The regular expression for the portion of the string that occurs before the substring "aaa" and does not contain "aaa" is: ( b + a b + a a b ) ∗ (b + ab + aab)^* Explanation: This part of the regular expression accounts for any possible sequence of "a"s and "b"s that does not contain the substring "aaa." b b  matches a single "b." a b ab  matches a single "b" preceded by an "a." a a b aab  matches two "a"s followed by a "b." The entire group is enclosed in parentheses and repeated zero or more times using the *  operator. Step 2 The regular expression for the portion of the string that occurs after the substring "aaa" and does not contain "aaa" is: ( b a a + b a + b ) ∗ (baa + ba + b)^* Explanation: This part of the regular expression accounts for any possible sequence of "a"s and "b"s that comes after the occurrence of "aaa" and does not contain the substring "aaa." b a a baa  matches a single "b" followed by two "a"s. b a ba  matches a single "b" followed by an "a." b b  matches a single "b." The entire group is enclosed in parentheses and repeated zero or more times using the * operator. Example String: Consider the string: "bbaabaabaaab" Before the substring "aaa": The portion of the string before the first "aaa" is "bbaab". Let's check if it matches the first part of our regular expression, ( b + a b + a a b ) ∗ (b + ab + aab)^* : "b" (matches b b ) "b" (matches b b ) "aab" (matches a a b aab ) The substring "bbaab" can indeed be formed using the expressions "b", "b", and "aab", so it matches the first part ( b + a b + a a b ) ∗ (b + ab + aab)^* . Exact occurrence of "aaa": The next part of the string is "aaa". This directly matches the middle part of the regular expression, which is "aaa". After the substring "aaa": The portion of the string after "aaa" is "baaab". Let's check if it matches the last part of our regular expression, ( b a a + b a + b ) ∗ (baa + ba + b)^* : "baa" (matches b a a baa ) "ab" (matches b a ba  followed by a "b") The substring "baaab" can be formed using the expressions "baa" and "ab", so it matches the last part ( b a a + b a + b ) ∗ (baa + ba + b)^* . The string "bbaabaabaaab" is a valid string in the language described by the regular expression ( b + a b + a a b ) ∗ a a a ( b a a + b a + b ) ∗ (b + ab + aab)^*aaa(baa + ba + b)^* . It has exactly one occurrence of "aaa," with valid substrings before and after that match the respective parts of the regular expression. Final Answer The complete regular expression for the language  L  that contains strings with exactly one occurrence of "aaa" is: ( b + a b + a a b ) ∗ a a a ( b a a + b a + b ) ∗ (b + ab + aab)^*aaa(baa + ba + b)^*

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