$O=1($if there is at least one $4-$bit bit string inside A that starts with $\text{'}0\text{'}$

and ends with $\text{'}1\text{'})$

$O=0($if there is no $4-$bit bit string inside A that starts with $\text{'}0\text{'}$

and ends with $\text{'}1\text{'})$

For example, if $A={?}^{\text{'}}$dots$\mathrm{.}0111$dots...

there is a $4-$bit string called ${?}^{\text{'}}0****1^{\text{'}}$ inside $A,$ so $0=1$