We study the query complexity of slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e. when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that -- somewhat surprisingly -- have not been studied before.

翻译：我们研究了布尔函数切片上的查询复杂度。其中一项结果表明：存在一个布尔函数，即使我们事先知道输入中0和1的数量相等（即输入来自中间层），仍需查询除7个输入比特之外的所有输入位才能计算其值。这回答了Byramji提出的一个问题。我们的证明是非构造性的，但同时提出了一个可能具有上述性质的具体候选函数。我们的结果与某些自然的差异类问题相关——令人惊讶的是——这些问题此前未被研究过。