Logical Information Cells I
  In this study we explore the spontaneous apparition of visible intelligible
reasoning in simple artificial networks, and we connect this experimental
observation with a notion of semantic information. We start with the
reproduction of a DNN model of natural neurons in monkeys, studied by
Neromyliotis and Moschovakis in 2017 and 2018, to explain how "motor equivalent
neurons", coding only for the action of pointing, are supplemented by other
neurons for specifying the actor of the action, the eye E, the hand H, or the
eye and the hand together EH. There appear inner neurons performing a logical
work, making intermediary proposition, for instance E V EH. Then, we remarked
that adding a second hidden layer and choosing a symmetric metric for learning,
the activities of the neurons become almost quantized and more informative.
Using the work of Carnap and Bar-Hillel 1952, we define a measure of the
logical value for collections of such cells. The logical score growths with the
depth of the layer, i.e. the information on the output decision increases,
which confirms a kind of bottleneck principle. Then we study a bit more complex
tasks, a priori involving predicate logic. We compare the logic and the
measured weights. This shows, for groups of neurons, a neat correlation between
the logical score and the size of the weights. It exhibits a form of sparsity
between the layers. The most spectacular result concerns the triples which can
conclude for all conditions: when applying their weight matrices to their
logical matrix, we recover the classification. This shows that weights
precisely perform the proofs.
