In this work we study oblivious complexity classes. These classes capture the power of interactive proofs where the prover(s) are only given the input size rather than the actual input. In particular, we study the connections between the symmetric polynomial time $\mathsf{S_2P}$ and its oblivious counterpart $\mathsf{O_2P}$. Among our results, we construct an explicit language in $\mathsf{O_2P}$ that cannot be computed by circuits of size $n^k$, and thus prove a hierarchy theorem for $\mathsf{O_2TIME}$. Along the way we also make partial progress towards the resolution of an open question posed by Goldreich and Meir (TOCT 2015) that relates the complexity of $\mathsf{NP}$ to its oblivious counterpart $\mathsf{ONP}$. To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for $\mathsf{O_2P}$. The smallest uniform complexity class for which such lower bounds were previously known was $\mathsf{S_2P}$, due to Cai (JCSS 2007). In addition, this is the first uniform hierarchy theorem for a semantic class. All previous results required some non-uniformity.

翻译：本文研究遗忘复杂性类。这些类别刻画了证明者仅获知输入规模而非实际输入时的交互式证明能力。我们特别关注对称多项式时间$\mathsf{S_2P}$与其遗忘对应类$\mathsf{O_2P}$之间的关联。研究成果包括：构造$\mathsf{O_2P}$中一个显式语言，该语言无法被规模为$n^k$的电路计算，由此证明$\mathsf{O_2TIME}$的层级定理。研究过程中，我们对Goldreich与Meir（TOCT 2015）提出的关于$\mathsf{NP}$与其遗忘对应类$\mathsf{ONP}$复杂度的开放问题取得了部分进展。据我们所知，这些成果首次为$\mathsf{O_2P}$建立了显式的固定多项式下界及层级定理。此前已知具有此类下界的最小均匀复杂性类是Cai（JCSS 2007）研究的$\mathsf{S_2P}$。此外，这是首个针对语义类建立的均匀层级定理，既往所有相关结果均需依赖某种非均匀性条件。