Zero-Information Protocols and Unambiguity in Arthur–Merlin Communication

Abstract

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur---Merlin ($${{\textsf {AM}}}$$AM) communication protocols. Our starting point is to show that--in contrast to plain randomized communication complexity--every boolean function admits an $${{\textsf {AM}}}$$AM communication protocol where on each yes-input, the distribution of Merlin's proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur's randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward $${{\textsf {AM}}}$$AM. Zero-information protocols ($${{\textsf {ZAM}}}$$ZAM): Our basic $${{\textsf {ZAM}}}$$ZAM protocol uses exponential communication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes $${{\textsf {NL}}}$$NL and $${\oplus }{{{\textsf {L}}}}$$źL have polynomial-communication $${{\textsf {ZAM}}}$$ZAM protocols. We also prove that $${{\textsf {ZAM}}}$$ZAM complexity is lower bounded by conondeterministic communication complexity. Unambiguous protocols ($${{\textsf {UAM}}}$$UAM): Our most technically substantial result is a $$\Omega (n)$$Ω(n) lower bound on the $${{\textsf {UAM}}}$$UAM complexity of the $${{\textsf {NP}}}$$NP-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the "zero-information barrier" described above. We also prove that in general, $${{\textsf {UAM}}}$$UAM complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound.