Time-Space Lower Bounds for Bounded-Error Computation in the Random-Query Model

The random-query model was introduced by Raz and Zhan at ITCS 2020 as a new model of space-bounded computation. In this model, a branching program of length T and width 2^S attempts to compute a function f : {0, 1}ⁿ → {0, 1}. However, instead of receiving direct access to the input bits (x1, ..., xn), the input is given in pairs of the form (ij, xi_j) ∈ {1, ..., n} × {0, 1} for j = 1, 2, ..., T, where the indices i1, ..., iT are chosen at random from a pre-fixed distribution. Raz and Zhan proved that any branching program in the random-query model with the independent distribution (where {ij}j=1, ...,T are uniform and independent) that computes a function f with sensitivity k satisfies T · (S + log n) ≥ Ω(n · k). This gives a quadratic time-space lower bound for many natural functions which have sensitivity Ω(n), such as XOR and Majority. The bound was proved in the zero-error regime, where for each input, the branching program is required to output a value with high probability, and given that a value is output, it must be correct with probability 1. Furthermore, Raz and Zhan conjectured that (up to logarithmic factors in n) a quadratic time-space lower bound still holds for the XOR function in the more conventional bounded-error regime, where for each input, the output must be correct with high probability. In this paper, we prove this conjecture. More generally, let f : {0, 1}ⁿ → {0, 1} have average sensitivity (or total influence) I[f]. We prove that any branching program in the random-query model with the independent distribution that computes f in the bounded-error regime satisfies T·S ≥ Ω̃(n)·I[f] (where Ω̃ hides logarithmic factors in n). Moreover, we prove a quadratic time-space lower bound for the Majority function, even though its total influence is Θ(√n). Our proof is based on a reduction from a communication complexity problem.