Direct sum theorems state that the cost of solving $k$ instances of a problem is at least $\Omega(k)$ times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.

翻译：直和定理指出，解决某个问题的$k$个实例所需成本至少是解决单个实例成本的$\Omega(k)$倍。我们在随机化奇偶决策树模型中首次证明了此类结果。我们证明，当满足以下任一条件时，直和定理成立：(1) 奇偶决策树的下界是通过差异法证明的；或 (2) 下界是相对于乘积分布证明的。