  Combining neural networks with continuous logic and multicriteria decision
making tools can reduce the black box nature of neural models. In this study,
we show that nilpotent logical systems offer an appropriate mathematical
framework for a hybridization of continuous nilpotent logic and neural models,
helping to improve the interpretability and safety of machine learning. In our
concept, perceptrons model soft inequalities; namely membership functions and
continuous logical operators. We design the network architecture before
training, using continuous logical operators and multicriteria decision tools
with given weights working in the hidden layers. Designing the structure
appropriately leads to a drastic reduction in the number of parameters to be
learned. The theoretical basis offers a straightforward choice of activation
functions (the cutting function or its differentiable approximation, the
squashing function), and also suggests an explanation to the great success of
the rectified linear unit (ReLU). In this study, we focus on the architecture
of a hybrid model and introduce the building blocks for future application in
deep neural networks. The concept is illustrated with some toy examples taken
from an extended version of the tensorflow playground.
