Artur d’Avila Garcez

Ouns El Harzli, Bernardo Cuenca Grau, Artur d'Avila Garcez et al.

Fibring of modal logics is a well-established formalism for combining countable families of modal logics into a single fibred language with common semantics, characterized by fibred models. Inspired by this formalism, fibring of neural networks was introduced as a neurosymbolic framework for combining learning and reasoning in neural networks. Fibring of neural networks uses the (pre-)activations of a trained network to evaluate a fibring function computing the weights of another network whose outputs are injected back into the original network. However, the exact correspondence between fibring of neural networks and fibring of modal logics was never formally established. In this paper, we close this gap by formalizing the idea of fibred models \emph{compatible} with fibred neural networks. Using this correspondence, we then derive non-uniform logical expressiveness results for Graph Neural Networks (GNNs), Graph Attention Networks (GATs) and Transformer encoders. Longer-term, the goal of this paper is to open the way for the use of fibring as a formalism for interpreting the logical theories learnt by neural networks with the tools of computational logic.

Daniel Sikar, Artur d'Avila Garcez, Tillman Weyde

Ensuring the reliability of automated decision-making based on neural networks will be crucial as Artificial Intelligence systems are deployed more widely in critical situations. This paper proposes a new approach for measuring confidence in the predictions of any neural network that relies on the predictions of a softmax layer. We identify that a high-accuracy trained network may have certain outputs for which there should be low confidence. In such cases, decisions should be deferred and it is more appropriate for the network to provide a \textit{not known} answer to a corresponding classification task. Our approach clusters the vectors in the softmax layer to measure distances between cluster centroids and network outputs. We show that a cluster with centroid calculated simply as the mean softmax output for all correct predictions can serve as a suitable proxy in the evaluation of confidence. Defining a distance threshold for a class as the smallest distance from an incorrect prediction to the given class centroid offers a simple approach to adding \textit{not known} answers to any network classification falling outside of the threshold. We evaluate the approach on the MNIST and CIFAR-10 datasets using a Convolutional Neural Network and a Vision Transformer, respectively. The results show that our approach is consistent across datasets and network models, and indicate that the proposed distance metric can offer an efficient way of determining when automated predictions are acceptable and when they should be deferred to human operators.

Son Tran, Edjard Mota, Artur d'Avila Garcez

Knowledge representation and reasoning in neural networks have been a long-standing endeavor which has attracted much attention recently. The principled integration of reasoning and learning in neural networks is a main objective of the area of neurosymbolic Artificial Intelligence (AI). In this chapter, a simple energy-based neurosymbolic AI system is described that can represent and reason formally about any propositional logic formula. This creates a powerful combination of learning from data and knowledge and logical reasoning. We start by positioning neurosymbolic AI in the context of the current AI landscape that is unsurprisingly dominated by Large Language Models (LLMs). We identify important challenges of data efficiency, fairness and safety of LLMs that might be addressed by neurosymbolic reasoning systems with formal reasoning capabilities. We then discuss the representation of logic by the specific energy-based system, including illustrative examples and empirical evaluation of the correspondence between logical reasoning and energy minimization using Restricted Boltzmann Machines (RBM). Learning from data and knowledge is also evaluated empirically and compared with a symbolic, neural and a neurosymbolic system. Results reported in this chapter in an accessible way are expected to reignite the research on the use of neural networks as massively-parallel models for logical reasoning and promote the principled integration of reasoning and learning in deep networks. We conclude the chapter with a discussion of the importance of positioning neurosymbolic AI within a broader framework of formal reasoning and accountability in AI, discussing the challenges for neurosynbolic AI to tackle the various known problems of reliability of deep learning.