Extracting Formulae in Many-Valued Logic From Deep Neural Networks

We propose a new perspective on deep rectified linear unit (ReLU) networks, namely as circuit counterparts of Łukasiewicz infinite-valued logic—a many-valued (MV) generalization of Boolean logic. An algorithm<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> for extracting formulae in MV logic from (trained) deep ReLU networks is presented. The algorithm respects the network topology, in particular compositionality, thereby honoring algebraic information present in the training data. While the two existing methods for turning truth functions in MV logic into formulae are for the univariate case only, the algorithm we propose applies to multivariate functions. Moreover, it is demonstrated—through numerical results and in one specific case analytically—that our algorithm, in the univariate case, can deliver shorter formulae than the other two methods. We also establish the representation benefits of deep networks from a mathematical logic perspective.