We investigate the maximum number C(P) of arguments of P that must be tested in order to compute P, a Boolean function of d Boolean arguments. We present evidence for the general conjecture that C(P)&equil;d whenever P(0d) @@@@ P(1d) and P is left invariant by a transitive permutation group acting on the arguments. A non-constructive argument (not based on the construction of an "oracle") proves the generalized conjecture for d a prime power. We use this result to prove the Aanderaa-Rosenberg conjecture by showing that at least v2/9 entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property.
展开▼
机译：
我们调查了必须测试的 P 的最大数目 C（P），以计算 P， d 布尔参数的布尔函数。我们提供一般猜想的证据，即 C（P）＆equil; d 无论何时 P（0 d ）@@@@ P（1 d ）和 P 由作用在自变量上的可传递置换组保持不变。一个非构造性的论点（不是基于“ oracle”的构造）证明了对素数幂的广义猜想。我们通过显示至少 v- 的邻接矩阵的 v 2 / 9 项来证明Aanderaa-Rosenberg猜想顶点无向图 G 必须在最坏的情况下进行检查，以确定 G 是否具有给定的非平凡单调图属性。
