No free reduction from MIP* to MIP

Unconditional · against free reductions · BFL90, NW19

Statement

No reduction of any class from MIP* to MIP exists, since $\mathbf{MI}{\mathbf{P}}^{\ast }⊈\mathbf{MIP}$: $\mathbf{MIP}=\mathbf{NEXP}$ — BFL90; $\mathbf{NEEXP}\subseteq \mathbf{MI}{\mathbf{P}}^{\ast }$ — NW19; and $\mathbf{NEXP}⊊\mathbf{NEEXP}$ by the nondeterministic time hierarchy theorem — standard.

Notes

• The converse $\mathbf{MIP}\subseteq \mathbf{MI}{\mathbf{P}}^{\ast }$ holds: $\mathbf{MIP}=\mathbf{NEXP}\subseteq \mathbf{NEEXP}\subseteq \mathbf{MI}{\mathbf{P}}^{\ast }$ — BFL90, NW19.