XORBased Boolean Matrix Decomposition Boolean matrix factorization (BMF) is a data summarizing and dimensionreduction technique. Existing BMF methods build on matrix properties defined by Boolean algebra, where the addition operator is the logical inclusive OR and the multiplication operator the logical AND. As a consequence, this leads to the lack of an additive inverse in all Boolean matrix operations, which produces an indelible type of approximation error. Previous research adopted various methods to address such an issue and produced reasonably accurate approximation. However, an exact factorization is rarely found in the literature. In this paper, we introduce a new algorithm named XBMaD (XORbased Boolean Matrix Decomposition) where the addition operator is defined as the exclusive OR (XOR). This change completely removes the errormitigation issue of ORbased BMF methods, and allows for an exact errorfree factorization. An evaluation comparing XBMaD and classic ORbased methods suggested that XBMAD performed equal or in most cases more accurately and faster.

A Hybrid Method for Spectral Translation Equivalent Boolean Functions The equivalence of Boolean functions with respect to five invariance (aka translation) operations has been well considered with respect to the RademacherWalsh spectral domain. In this paper, we introduce a hybrid approach that uses both the ReedMuller and the RademacherWalsh spectra. A novel hybrid algorithm that maps a Boolean function to a representative function for the equivalence class containing the original function is presented. The algorithm can be used to determine a sequence of translations that maps one function to an equivalent function. We present experimental results that show the hybrid algorithm can determine the equivalence classes for 5 variables much more efficiently than before. We also show that for 6 variables where there are 150,357 equivalence classes, 8 are very difficult, a further 58 are difficult and the remainder are straightforward in terms of the CPU time required by the hybrid algorithm.