A symmetric pseudo-Boolean function is a map from Boolean tuples to real numbers which is invariant under input variable interchange. We prove that any such function can be equivalently expressed as a power series or factorized. The kernel of a pseudo-Boolean function is the set of all inputs that cause the function to vanish identically. Any $n$-variable symmetric pseudo-Boolean function $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one $n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^n x_l = \lambda$ for $\lambda\in \mathbb{C}$ constant. We use these results to analyze symmetric pseudo-Boolean functions appearing in the literature of spin glass energy functions (Ising models), quantum information and tensor networks.
翻译:匹配伪Boolean 函数是从 Boulean tuples 到在输入变量交换中无法变换的真实数字的地图。 我们证明任何这样的函数都可以以电源序列或因数化表示。 伪Boolean 函数的内核是导致函数完全消失的所有输入的一组。 任何美元可变的伪Boolean 函数 $f( x_ 1, x_ 2,\ dots, x_n) 都有至少相当于1 $n- affine 超平面的内核, 每个超平面都受一个限制 $\ sumäl=1 ⁇ n x_l =\lambda$ in\ mathbb{C} 恒定值。 我们用这些结果来分析在旋转玻璃能量函数( 发源模型)、 量信息 和 高压网络 的文献中出现的对准伪 Boolean 函数 $f( x_ 1, x_ 2,\ x_n) 。