Monotone Boolean Functions Are Universal Approximators

The employment of proper codings, such as the base-2 coding, has allowed to establish the universal approximation property of Boolean functions: if a sufficient number b of inputs (bits) is taken, they are able to approximate arbitrarily well any real Borel measurable mapping. However, if the reduced set of monotone Boolean functions, whose expression involves only and and or operators, is considered, the standard approach points out significant limitations in their approximation capability. These limitations can be overcome by introducing new specific codings, called lattice coding and frame coding, which permit to show that also monotone Boolean functions possess the universal approximation property. The characteristics of these codings are analyzed in details, focusing on the ability of preserving metric and ordering. In particular, a comparison with classical base-2 coding shows that the lattice and the frame coding require an increase of O(log b) and of O( √ b), respectively, in the number of bits.