  Traditional neural networks represent everything as a vector, and are able to
approximate a subset of logical reasoning to a certain degree. As basic logic
relations are better represented by topological relations between regions, we
propose a novel neural network that represents everything as a ball and is able
to learn topological configuration as an Euler diagram. So comes the name Euler
Neural-Network (ENN). The central vector of a ball is a vector that can inherit
representation power of traditional neural network. ENN distinguishes four
spatial statuses between balls, namely, being disconnected, being partially
overlapped, being part of, being inverse part of. Within each status, ideal
values are defined for efficient reasoning. A novel back-propagation algorithm
with six Rectified Spatial Units (ReSU) can optimize an Euler diagram
representing logical premises, from which logical conclusion can be deduced. In
contrast to traditional neural network, ENN can precisely represent all 24
different structures of Syllogism. Two large datasets are created: one
extracted from WordNet-3.0 covers all types of Syllogism reasoning, the other
extracted all family relations from DBpedia. Experiment results approve the
superior power of ENN in logical representation and reasoning. Datasets and
source code are available upon request.
