Adding Real Coefficients to Łukasiewicz Logic: An Application to Neural Networks

Łukasiewicz Equivalent Neural Networks

“…In an analogous way, in [3] the authors describe the connection between (a particular class of) neural networks and RMV-algebras; for instance the authors define the one-layer neural networks which encode min(x, y) and max(x, y) as follows:…”

“…There exist many typologies of neural networks used in specific fields. We will focus on feedforward neural networks, in particular multilayer perceptrons, as in [3], which have applications in different fields, such as speech or image recognition. This class of networks consists of multiple layers of neurons, where each neuron in one layer has directed connections to the neurons of the subsequent layer.…”

“…In [3] the authors showed that multilayer perceptrons, whose activation functions are the identity truncated to zero and one, can be fully interpreted as logical objects, since they are equivalent to (equivalence classes of) formulas of an extension of Łukasiewicz propositional logic obtained by considering scalar multiplication with real numbers (corresponding to Riesz MV-algebras, defined in [4,5]). Now we propose more general multilayer perceptrons which describe not necessarily linear events.…”

A Constraint-Based Approach to Learning and Explanation

“…Symbolic interpretation of neural networks (rule extraction) has been the subject of many researches by several authors, especially in the nineties. Some approaches are about Fuzzy Logic (Kasabov 1996;Huang and Xing 2002;Castro and Trillas 1998;Di Nola, Gerla, and Leustean 2013), but are generally less straightforward in terms of explainability than the ones based on Boolean Logic (Fu 1991;Towell and Shavlik 1993;Tsukimoto 2000;Sato and Tsukimoto 2001). In the latter case, it is pretty common to rely on a discretization of the input and output values of the neurons, pruning the network to keep it simple.…”