Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.

中文翻译：

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行列式对称电路的下限

Dawar 和 Wilsenach (ICALP 2020) 介绍了对称算术电路的模型，并展示了用于计算行列式和永久的对称电路的大小之间的指数分离。对称性限制是采用矩阵输入的电路不会因同时应用于矩阵行和列的排列而改变。在这样的限制下，我们有计算行列式的多项式大小的电路，但没有用于永久项的次指数大小的电路。在这里，我们考虑了更严格的对称性要求，即电路不会被分别应用于行和列的任意偶数排列所改变，并且即使对于计算行列式的电路也证明了指数下界。结果需要大量的新机器。基于新的支持定理和新的两人受限双射博弈，我们开发了一个通用框架，用于证明具有受限对称性的对称电路的下界。这些应用于行列式问题的矩阵的新构造，这些矩阵是基于 CFI 构造的图的双邻接矩阵。我们的通用框架为探索各种对称性限制和研究对称性与算术电路使用的其他资源之间的权衡开辟了道路。