Equational characterizations of Boolean function classes

Abstract

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.

Join-Irreducible Boolean Functions

“…This quasi-order can be described as follows: for f, g ∈ Ω, g ≤ f if g can be obtained from f by identification of variables, permutation of variables, and addition or deletion of dummy variables. As shown in [11], equational classes of Boolean functions coincide exactly with the initial segments ↓K = {g ∈ Ω : g ≤ f, for some f ∈ K} of this quasi-order, or equivalently, they correspond to antichains A of Boolean functions in the sense that they constitute sets of the form Ω \ ↑A. Moreover, those equational classes definable by finitely many equations correspond to finite antichains of Boolean functions.…”

“…As it turned out, these two approaches have the same expressive power in the sense that they specify exactly the same classes (or properties) of Boolean functions. The characterization of these classes was first obtained by Ekin, Foldes, Hammer and Hellerstein [11] who showed that equational classes of Boolean functions can be completely described in terms of a quasi-ordering ≤ of the set Ω of all Boolean functions, called identification minor in [11,16], simple minor in [20,8,5,6], subfunction in [25], and simple variable substitution in [3]. This quasi-order can be described as follows: for f, g ∈ Ω, g ≤ f if g can be obtained from f by identification of variables, permutation of variables, and addition or deletion of dummy variables.…”

Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions

“…Theorem 3.1 (Ekin et al 2000) The recognition problem for the quadraticity property of a Boolean function given by a DNF is coNP-complete.…”

On the Effect of Variable Identification on the Essential Arity of Functions on Finite Sets

“…Identification of variables together with permutation of variables and cylindrification induces a quasi-order on operations whose relevance has been made apparent by several authors [3,7,8,9,10,12,14]. In the case of Boolean functions, this quasi-order was studied in [4] where Theorem 2 was fundamental in deriving certain bounds on the essential arity of functions.…”