On the weightwise nonlinearity of weightwise perfectly balanced functions

in Discrete Applied Mathematics (2022), 322

In this article we perform a general study on the criterion of weightwise nonlinearity for the functions which are weightwise perfectly balanced (WPB). First, we investigate the minimal value this ... [more ▼]

In this article we perform a general study on the criterion of weightwise nonlinearity for the functions which are weightwise perfectly balanced (WPB). First, we investigate the minimal value this criterion can take over WPB functions, deriving theoretic bounds, and exhibiting the first values. We emphasize the link between this minimum and weightwise affine functions, and we prove that for n≥8 no n-variable WPB function can have this property. Then, we focus on the distribution and the maximum of this criterion over the set of WPB functions. We provide theoretic bounds on the latter and algorithms to either compute or estimate the former, together with the results of our experimental studies for n up to 8. Finally, we present two new constructions of WPB functions obtained by modifying the support of linear functions for each set of fixed Hamming weight. This provides a large corpus of WPB function with proven weightwise nonlinearity, and we compare the weightwise nonlinearity of these constructions to the average value, and to the parameters of former constructions in 8 and 16 variables. [less ▲]

Set-reconstructibility of Post classes

in Discrete Applied Mathematics (2015), 187

Pivotal decompositions of functions

in Discrete Applied Mathematics (2014), 174

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary ... [more ▼]

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary section of a function only depends upon its values at two given elements. Pivotal decompositions appear to hold for various function classes, such as the class of lattice polynomial functions or the class of multilinear polynomial functions. We also define function classes characterized by pivotal decompositions and function classes characterized by their unary members and investigate links between these two concepts. [less ▲]

The arity gap of order-preserving functions and extensions of pseudo-Boolean functions

in Discrete Applied Mathematics (2012), 160(4-5), 383-390

The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are the so-called aggregation functions. We first explicitly ... [more ▼]

The aim of this paper is to classify order-preserving functions according to their arity gap. Noteworthy examples of order-preserving functions are the so-called aggregation functions. We first explicitly classify the Lovász extensions of pseudo-Boolean functions according to their arity gap. Then we consider the class of order-preserving functions between partially ordered sets, and establish a similar explicit classification for this function class. [less ▲]

Approximations of Lovász extensions and their induced interaction index

in Discrete Applied Mathematics (2008), 156(1), 11-24

The Lovász extension of a pseudo-Boolean function f : {0,1}^n --> R is defined on each simplex of the standard triangulation of [0,1]^n as the unique affine function \hat f : [0,1]^n --> R that ... [more ▼]

The Lovász extension of a pseudo-Boolean function f : {0,1}^n --> R is defined on each simplex of the standard triangulation of [0,1]^n as the unique affine function \hat f : [0,1]^n --> R that interpolates f at the n+1 vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses f. In this paper we investigate the least squares approximation problem of an arbitrary Lovász extension \hat f by Lovász extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman and then solved explicitly by Grabisch, Marichal, and Roubens, giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of \hat f and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche. [less ▲]

The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making

in Discrete Applied Mathematics (2000), 107(1-3), 139-164

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the ... [more ▼]

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the influence of variables on Boolean functions has been defined and studied. We extend this concept to pseudo-Boolean functions, thus making it possible to appraise the degree of influence of any coalition of players in cooperative games. In the case of monotone games, we also point out the links with the concept of interaction among players. Although they do not have the same meaning at all, both influence and interaction functions coincide on singletons with the so-called Banzhaf power index. We also define the influence of variables on continuous extensions of pseudo-Boolean functions. In particular, the Lovász extension, also called discrete Choquet integral, is used in multicriteria decision making problems as an aggregation operator. In such problems, the degree of influence of decision criteria on the aggregation process can then be quite relevant information. We give the explicit form of this influence for the Choquet integral and its classical particular cases. [less ▲]