A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as an NFS B if there is a polynomial P with nonnegative integer coefficients such that the complexity of any Boolean function f with respect to A is at most the value of P in the complexity of f with respect to B. In this paper we study monotonic NFSs, i.e., NFSs whose connectives are increasing or decreasing in each argument. We describe the monotonic NFSs that are optimal, i.e., that are minimal with respect to the latter preorder. We show that these minimal monotonic NFSs are all equivalent. Moreover, we address some natural questions, e.g.: does optimality depend on the arity of connectives? Does it depend on the number of connectives used? We show that optimal monotonic NFSs are exactly those that use a single connective or one connective and the negation. Finally, we show that optimality does not depend on the arity of the connectives.

表示布尔函数的范式系统（NFS）被认为是固定连接词集上的一组分层术语。对于固定的NFS A，布尔函数f相对于A的复杂度是A中表示f的项的大小的最小值。这引起一个preordering NFSS的：一个NFS阿是多项式作为有效作为NFS乙如果有一个多项式P与非负整数系数，使得任何布尔函数的复杂性˚F相对于甲至多值P在f的复杂度相对于乙。在本文中，我们研究了单调NFS，即在每个参数中其连接词增加或减少的NFS。我们描述了最优的单调NFS，即相对于后序而言最小。我们证明这些最小单调NFS都是等效的。此外，我们解决了一些自然的问题，例如：最优性是否取决于连接词的多样性？它取决于所使用的连接词的数量吗？我们表明，最佳单调NFS恰好是使用单个连接词或一个连接词与否定的NFS。最后，我们证明了最优性并不取决于连接词的多样性。