Question

Give a regular expression for the complement of the re 111 with respect to the alphabet {0, 1}. that is, the re should allow every possible binary string except for the string 111.

Answer 1

A regular expression for the complement of the binary string '111' can be constructed by combining strings that do not contain '111' as a substring, and allows all binary strings except for '111'.

Step-by-step explanation:

A regular expression (RE) that represents the complement of the binary string '111' with respect to the alphabet {0, 1} is one that accepts every binary string except for '111'. This regular expression can be quite complex, as it has to account for all possible strings, and exclude exactly the string '111'. A possible expression is this: (0|1)*0(0|1)(0|1)(0|1)*|1(0|1)0(0|1)(0|1)*|11(0|1)(0|100)(0|1)*|111(0|1)+|1(0|1)(0|1)(0|1)*0(0|1)*. This regular expression is basically the combination of all strings that don't have '111' as a substring.

Answer 2

To obtain the complement of the regular expression (re) 111 with respect to the alphabet {0, 1}, use the regular expression (?!111)^.{3}$. This expression allows every possible binary string except for 111.

Step-by-step explanation:

To find the complement of the regular expression (re) 111 with respect to the alphabet {0, 1}, we need to find a regular expression that matches every possible binary string except for the string 111. One way to do this is by using negative lookahead. The regular expression for the complement of 111 is (?!111)^.{3}$.

This regular expression uses a negative lookahead assertion (?!111) to ensure that the string is not followed by 111. The ^.{3}$ part matches any 3-character string, which covers all possible binary strings except for 111. Therefore, this regular expression allows every possible binary string except for 111.