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Abstract

We solve some fundamental problems in the number-on-forehead (NOF) kplayer communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided falsepositives error probability of 1/3, but which has linear communication complexity for deterministic protocols and, in fact, even for the more powerful nondeterministic protocols. The result holds for every ε > 0 and every k ≤ 2 (1−ε)n players, where n is the number of bits on each player's forehead. As a consequence, we obtain the NOF communication class separation coRP ⊂ NP. This in particular implies that P = RP and NP = coNP. We also show that for every ε > 0 and every k ≤ n 1−ε , there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential; no explicit function is known to satisfy nontrivial lower bounds for k ≥ log n players. However, for every ε > 0 and every k ≤ (1 − ε) • log n players, the NP = coNP separation (and even the coNP ⊂ MA separation) was obtained independently by Gavinsky and Sherstov (2010) using an explicit construction. In this work, for k ≤ (1/9) • log n players, we exhibit an explicit function which has communication

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“…The problem was open for k 3. As a corollary, we obtain an explicit separation of NP and coNP for up to k = (1 − ε) log n players, complementing an independent result by Beame et al (2010) who separate these classes nonconstructively for up to k = 2 (1−ε)n players. …”

“…For k = 2 players, the relations among these models are well understood: Papadimitriou and Sipser [20] showed that coNP cc 2 = NP cc 2 , and Klauck [16] proved that additionally coNP Independently of our work, Beame, David, Pitassi, and Woelfel [3] proved nonconstructively that NP cc k = coNP cc k for k 2 (1−ε)n . An advantage of Theorem 1.1 is that it gives an explicit separation and additionally applies to Merlin-Arthur complexity.…”

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“…The surveyed work on communication complexity with unbounded and weakly unbounded error considered the two-party model. The past few years saw a resurgence of interest in multiparty communication complexity classes, with numerous separations established over the past decade [19,11,14,3,15]. Recently, Chattopadhyay and Mande [12] revisited the unbounded versus weakly unbounded question in the multiparty setting.…”

“…THEORY OF COMPUTING, Volume 14(22), 2018, pp [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. where the maximum is over cylinder intersections χ.…”

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“…However, for k ≥ 3, things become more delicate. While for k ≥ 3 Beame et al [5] separated P cc k from BPP cc k not too long ago, it is still outstanding to find an explicit function witnessing this separation even for k = 3. A recent line of work [24,13,12,33,31,26] showed that Set-Disjointness also separates BPP cc k and PP cc k for k ≤ δ • log n for some constant δ < 1.…”