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TR23-084 | 31st May 2023 01:56
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#### Time-Space Lower Bounds for Bounded-Error Computation in the Random-Query Model

**Abstract:**
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 $f:\{0,1\}^n \rightarrow \{0,1 \}$. However, instead of receiving direct access to the input bits $(x_1,\ldots,x_n)$, the input is given in pairs of the form $(i_j, x_{i_j}) \in \{1,\ldots,n\} \times \{0,1\}$ for $j = 1,2,\ldots,T$, where the indices $i_1,\ldots,i_T$ 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 $\{i_j\}_{j = 1,\ldots,T}$ are uniform and independent) that computes a function $f$ with sensitivity $k$ satisfies $T \cdot (S + \log n) \geq \Omega(n \cdot k)$.

This gives a quadratic time-space lower bound for many natural functions which have sensitivity $\Omega(n)$, 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 $f:\{0,1\}^n \rightarrow \{0,1 \}$ have average sensitivity (or total influence) $\mathrm{I}[f]$. We prove that any branching program in the random-query model with the independent distribution that computes $f$ in the bounded-error regime satisfies $T \cdot S \geq \tilde{\Omega}(n) \cdot \mathrm{I}[f]$ (where $\tilde{\Omega}$ hides logarithmic factors in $n$). Moreover, we prove a quadratic time-space lower bound for the Majority function, even though its total influence is $\Theta(\sqrt{n})$.

Our proof is based on a reduction from a communication complexity problem.