特殊句法多线性ABPs的下界
Lower Bounds for Special Cases of Syntactic Multilinear ABPs
摘要：
代数分支程序（ABP）是计算多项式的标准模型。句法多线性ABP（smABP）是ABP的限制，其中每个变量被允许在从开始到结束节点的每条路径中最多出现一次。在代数复杂性理论中，证明句法多线性ABPs的下界一直是一个具有挑战性的问题。目前最知名的界限只有二次[Alon-Kumar-Volk，ECCC 2017]。在这篇文章中，我们开发了一种新的方法上界的语法多线性ABP的偏导矩阵的秩：转换ABP的语法多线性公式与超多项式的大小爆破，然后利用所得到的公式的结构限制，以获得一个秩上界。使用这种方法，我们证明指数下界的特殊情况下的smABP和电路-即不经意的只读一次ABP，r-通过多线性ABP和稀疏ROABP的总和。同时，我们还证明了一类特殊的句法多线性算术电路的超多项式下界。
英文摘要：
Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the terminal node. Proving lower bounds against syntactic multilinear ABPs remains a challenging open question in Algebraic Complexity Theory. The current best known bound is only quadratic [Alon-Kumar-Volk, ECCC 2017]. In this article we develop a new approach upper bounding the rank of the partial derivative matrix of syntactic multlinear ABPs: Convert the ABP to a syntactic mulilinear formula with a super polynomial blow up in the size and then exploit the structural limitations of resulting formula to obtain a rank upper bound. Using this approach, we prove exponential lower bounds for special cases of smABPs and circuits - namely sum of Oblivious Read-Once ABPs, r-pass mulitlinear ABPs and sparse ROABPs. En route, we also prove super-polynomial lower bound for a special class of syntactic multilinear arithmetic circuits.
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