We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

翻译：我们引入了对称算术电路,即具有自然对称限制的算术电路。在计算变量矩阵中定义的电路中,例如决定因素或永久变量,限制相当于要求电路的形状在矩阵的同步行和列间曲下是无变的。我们在计算永久值时,对任何对称电路的大小设定无条件的指数下下限。相反,我们显示,在计算特性零的字段的决定因素时,存在多数值对称电路。