From synapses to rules

We present an integrated subsymbolic-symbolic procedure for extracting symbolically explained classification rules from data. A neural network maps features into propositional variables and a PAC-like algorithm learns Boolean expressions on these variables. The peculiarities of the whole procedure are: i. we do not know a priori the class of formulas these expression belong to, rather we get time to time some information about the class and reduce the uncertainty on the current hypothesis, and ii. the mapping from feature to variables also changes with time to improve the suitability of the wanted classification rules. The theoretical tools supporting the procedure are: 1. a new statistical framework that we call algorithmic inference, 2. a special functionality of the sampled points in respect to the formulas, denoted sentinelling, and 3. a proper fitness function at the basis of the feedback actions between the symbolic and subsymbolic stages of our procedure. Preliminary numerical results higlight the value of the procedure.