Separate BPP0 from P0^{SUPP}

Determine whether there exists a communication problem with constant bounded-error randomized cost (in BPP0) that cannot be computed by a constant-cost deterministic protocol with oracle access to any problem of constant support-rank (i.e., not in P0^{SUPP}).

Background

Introducing support-rank yields the class SUPP and the oracle class P^{SUPP}. The authors’ results show that many known constant-margin problems fall into P^{SUPP}, motivating the question of whether this containment is strict relative to BPP0.

A separation would reveal intrinsic limits of the support-rank method and identify problems whose constant-margin behavior cannot be captured via constant-support-rank oracles.