Is this language recognizable?
To show that $L$&nbsp;is not recognizable, you can reduce the complement of the halting problem to it. $\overline{\mathrm{HALT}}$ is not recognizable because, if a language and its complement are both recognizable, then they're both decidable, and we know that $\mathrm{HALT}$ isn't recursive.

Given a Turing machine&nbsp;$M$ and input&nbsp;$w$, you want to accept iff $M(w)$ loops. So construct the machine&nbsp;$M'$ which halts if its input is $1100$ and simulates $M(w)$, otherwise. If $M'\in L$, then $M'$ cannot halt on $0011$ or $1000$,* which means that $M(w)$ doesn't halt.

* It seems there's a typo in the question, which mentions $0011$ twice. Obviously, this doesn't make any real difference.