The aim of this paper is to completely classify all aggregation functions based on the notion of arity gap. We first establish explicit descriptions of the arity gap of the Lovász extensions of pseudo-Boolean functions and, in particular, of the Choquet integrals. Then we consider the wider class of order-preserving functions between arbitrary, possibly different, posets, and show that similar ...

in this paper the concept of an $omega$- almost boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $omega$- almost boolean ring over a locally boolean space.

The minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f . In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log 2 (n+ 1)⌉ NOT gates are sufficient to compute any Boolean function on n variables. In this paper, we consider circuits computing non-deterministically and dete...

Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define fast point as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a n-variable Boolean function fo...

Alfredo Luis Departamento de Óptica, Facultad de Ciencias F́ısicas, Universidad Complutense, 28040 Madrid, Spain Abstract We develop a complete geometrical picture of paraxial light propagation including coherence phenomena. This approach applies both for scalar and vectorial waves via the introduction of a suitable Wigner function and can be formulated in terms of an inverted Huygens principle....