Several noteworthy classes of Boolean functions can be characterized by algebraic identities (e.g. the class of positive functions consists of all functions f satisfying the identity f(x) ∨ f(y) ∨ f(x ∨ y) = f(x ∨ y)). We give algebraic identities for several of the most frequently analyzed classes of Boolean functions (including Horn, quadratic, supermodular, and submodular functions) and proceed then to the general question of which classes of Boolean functions can be characterized by algebraic identities. We answer this question for function classes closed under addition of inessential (irrelevant) variables. Nearly all classes of interest have this property. We show that a class with this property has a characterization by algebraic identities if and only if the class is closed under the operation of variable identiÿcation. Moreover, a single identity su ces to characterize a class if and only if the number of minimal forbidden identiÿcation minors is ÿnite. Finally, we consider characterizations by general ÿrst-order sentences, rather than just identities. We show that a class of Boolean functions can be described by an appropriate set of such ÿrst-order sentences if and only if it is closed under permutation of variables.
Given a Boolean formula in disjunctive normal form, the variable deletion control set problem consists in finding a minimum cardinality set of variables whose deletion from the formula results in a DNF satisfying some prescribed property. Similar problems can be defined with respect to the fixation of variables or the deletion of terms in a DNF. In this paper, we investigate the complexity of such problems for a broad class of DNF properties.
Cataloged from PDF version of article.A Boolean function is called k-convex if for any pair x,y of its true points at Hamming distance at most k, every point "between" x and y is also true. Given a set of true points and a set of false points, the central question of Logical Analysis of Data is the study of those Boolean functions whose values agree with those of the given points. In this paper we examine data sets which admit k-convex Boolean extensions. We provide polynomial algorithms for finding a k-convex extension, if any, and for finding the maximum k for which a k-convex extension exists. We study the problem of uniqueness, and provide a polynomial algorithm for checking whether all k-convex extensions agree in a point outside the given data set. We estimate the number of k-convex Boolean functions, and show that for small k this number is doubly exponential. On the other hand, we also show that for large k the class of k-convex Boolean functions is PAC-learnable. (C) 2000 Elsevier Science B.V. All rights reserved
Cataloged from PDF version of article.A Boolean function is called (co-)connected if the subgraph of the Boolean hypercube induced\ud
by its (false) true points is connected; it is called strongly connected if it is both connected and\ud
co-connected. The concept of (co-)geodetic Boolean functions is de ned in a similar way by\ud
requiring that at least one of the shortest paths connecting two (false) true points should consist\ud
only of (false) true points. This concept is further strengthened to that of convexity where every\ud
shortest path connecting two points of the same kind should consist of points of the same kind.\ud
This paper studies the relationships between these properties and the DNF representations of the\ud
associated Boolean functions. ? 1999 Elsevier Science B.V. All rights reserved
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