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Equality characterization of UPP ∩ BPP

Determine whether the intersection UPP ∩ BPP equals precisely the set of communication problems that reduce, via constant-cost deterministic oracle protocols, to the Equality function E(x,y)=1 iff x=y.

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Background

UPP denotes problems solvable by constant-cost unbounded-error randomized protocols and is equivalent to matrices of bounded sign-rank. Clarifying the relationship between UPP and BPP is tied to multiple structural conjectures; in particular, an Equality-based characterization of UPP ∩ BPP would rule out new BPP examples within UPP and imply several other conjectures.

References

The relationship between UPP and BPP is the subject of several open problems. [HZ24] ask if UPP ∩ BPP is exactly the class of problems which reduce to Equality, which would rule out any new examples in this class.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section 1.5 (Constant-Cost Communication: the Story so Far and Farther), paragraph “Sign-rank.”