$Π$-NeSy: A Possibilistic Neuro-Symbolic Approach
Ismaïl Baaj, Pierre Marquis
TL;DR
Pi-NeSy presents a neuro-symbolic framework that integrates neural perception with a possibilistic rule-based reasoning system to produce degrees of possibility for meta-concepts. By converting neural probability outputs into possibility distributions, it enables joint inference and learning via a structured matrix relation and a min–max equation system. The work introduces practical partitioning methods to construct these data structures and a learning approach that handles inconsistency, supports cascades, and leverages multi-sample data. Experimental results on MNIST Addition-k and Sudoku demonstrate competitive accuracy and solid scalability, with enhanced explainability through rule-based reasoning. The approach offers a principled, interpretable, and scalable path for combining neural and symbolic components under epistemic uncertainty.
Abstract
In this article, we introduce a neuro-symbolic approach that combines a low-level perception task performed by a neural network with a high-level reasoning task performed by a possibilistic rule-based system. The goal is to be able to derive for each input instance the degree of possibility that it belongs to a target (meta-)concept. This (meta-)concept is connected to intermediate concepts by a possibilistic rule-based system. The probability of each intermediate concept for the input instance is inferred using a neural network. The connection between the low-level perception task and the high-level reasoning task lies in the transformation of neural network outputs modeled by probability distributions (through softmax activation) into possibility distributions. The use of intermediate concepts is valuable for the explanation purpose: using the rule-based system, the classification of an input instance as an element of the (meta-)concept can be justified by the fact that intermediate concepts have been recognized. From the technical side, our contribution consists of the design of efficient methods for defining the matrix relation and the equation system associated with a possibilistic rule-based system. The corresponding matrix and equation are key data structures used to perform inferences from a possibilistic rule-based system and to learn the values of the rule parameters in such a system according to a training data sample. Furthermore, leveraging recent results on the handling of inconsistent systems of fuzzy relational equations, an approach for learning rule parameters according to multiple training data samples is presented. Experiments carried out on the MNIST addition problems and the MNIST Sudoku puzzles problems highlight the effectiveness of our approach compared with state-of-the-art neuro-symbolic ones.