  This paper describes a novel approach to synthesize molecular reactions to
train a perceptron, i.e., a single-layered neural network, with sigmoidal
activation function. The approach is based on fractional coding where a
variable is represented by two molecules. The synergy between fractional coding
in molecular computing and stochastic logic implementations in electronic
computing is key to translating known stochastic logic circuits to molecular
computing. In prior work, a DNA perceptron with bipolar inputs and unipolar
output was proposed for inference. The focus of this paper is on synthesis of
molecular reactions for training of the DNA perceptron. A new molecular scaler
that performs multiplication by a factor greater than 1 is proposed based on
fractional coding. The training of the perceptron proposed in this paper is
based on a modified backpropagation equation as the exact equation cannot be
easily mapped to molecular reactions using fractional coding.
