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.

Topos and Stacks of Deep Neural Networks

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy (P.Baudot and D.B. 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators.