Bi-tone and k-tone decompositions of Boolean functions

For a Boolean Matrix A a binary vector v is called tfrequent if Av has at least t entries of value supp (v). Given two parameters t1 < t2 the t1-frequent but t2-infrequent vectors of a matrix represent a Boolean function that has two domains of (opposite) monotonicity. These functions were studied for the purpose of data analysis and abstract concept discovery in (Eisenschmidt et al. 2010). In this paper we introduce as a generalization very structured Boolean functions partitioning their domain into k alternating regions of monotonicity. We show how given two sets W0 and W1 prescribing values 0 and 1 such a function with minimal k can be constructed. We believe that in data analysis or data fitting tasks such well-structured functions are advantageous, especially for small values of k. 1 Bitone Boolean functions A Boolean function is a function f mapping from {0, 1} into {0, 1}. A Boolean function is called (up-)monotone, if for x, y ∈ {0, 1} with x ≤ y also f(x) ≤ f(y) holds. A Boolean function is monotone if and only if it is representable as conjunction and disjunction of binary variables (i.e., without any negations). We will denote by |f | the number of maximal false points of f . For an exhaustive overview see (Crama and Hammer 2011). Let f1, f2 : {0, 1} → {0, 1} be monotone Boolean functions such that f1(x) ≤ f2(x) for all x ∈ {0, 1}. Let g := ¬f1∧f2. Then g is a Boolean function that is in general not monotone. We will call a function arising in this way a bitone Boolean function. The term bitonicity can be explained as follows: Let x1, . . . , xs ∈ {0, 1} be the maximal false points (MFPs) of f1, i.e. f1(xi) = 0 for all i = 1, . . . , s. Let y1, . . . , yt be MFPs of f2. Then the relation f1 ≤ f2 implies that for every yi there exists xj such that yi ≤ xj . Now we partition the set of Boolean vectors into three sets: M0 := {x ∈ {0, 1} : ∃i ∈ {1, . . . , t} such that x ≤ yi} M2 := {x ∈ {0, 1} : ∃j ∈ {1, . . . , s} such that xj < x} M1 := {0, 1} \ (M0 ∪M2). (1) Figure 1 illustrates the partition induced by a bitone function. The set M1 forms a band between the sets M0 and y1