What can be expected from a Boolean derivative?

Several concepts of a Boolean derivative have been investigated in the literature. In this paper we find out whether a Boolean operator can satisfy all of the three basic derivative-like properties: additivity, homogeneity and the Leibniz rule. In a certain sense, the answer is negative. The attempts to establish Boolean analogues of several concepts and results from Calculus begun in 1917 with a paper by Daniell [6], which sketched a theory of convergence for sequences and series in a Boolean algebra. Some forty years later Reed [11], Huffman [9] and Akers Jr. [1] introduced (partial) derivatives of Boolean functions and pointed out their applicability to switching theory. Ever since then the theory of Boolean derivatives has developed tremendously, both in view of applications and for its own algebraic interest; see e.g. [3], [4], [5], [7], [10], [13], [15], [16], [17], [18], [19]. There are several Boolean analogues of the conventional concept of a derivative; of course, these Boolean derivatives share some, but not all of the properties of their conventional model. A paper by Bazsó and Lábos [2] states that a “good” concept of a derivative should be additive, i.e., (f + g) = f ′ + g, homogeneous, i.e., (kf) = kf , and should satisfy the Leibniz identity, i.e., (fg) = fg + gf . Bazsó and Lábos are concerned with algebras of Boolean functions. They remark that the well-known sensitivity function does not