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Which one of the following regular expressions represents the language: the set of all binary strings having two consecutive 0s and two consecutive 1s?

  • A.\((0+1)^{*}0011(0+1)^{*} + (0+1)^{*}1100(0+1)^{*}\)
  • B.\((0+1)^{*}(00(0+1)^{*}11 + 11(0+1)^{*}00)(0+1)^{*}\)
  • C.\((0+1)^{*}00(0+1)^{*} + (0+1)^{*}11(0+1)^{*}\)
  • D.\(00(0+1)^{*}11 + 11(0+1)^{*}00\)

Answer: B

The regular expression is \((0+1)^{*}(00(0+1)^{*}11 + 11(0+1)^{*}00)(0+1)^{*}\).

A string with both a 00 and a 11 has one of them occurring first. So it is: anything, then 00, then anything, then 11, then anything; or the same with 11 first. The middle \((0+1)^{*}\) lets the two pairs be separated, and the outer ones allow any prefix and suffix. That is exactly this expression.

The 0011/1100 option forces the pairs to be adjacent, missing strings such as 001011, where the pairs are separated. The option joined by '+' with \((0+1)^{*}00(0+1)^{*}\) accepts strings with only 00 or only 11. The last option forces the string to start and end with the pairs, missing 100110.

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