LogicCBMs: Logic-Enhanced Concept-Based Learning

TL;DR

LogicCBMs introduce differentiable fuzzy logic gates to concept-based models, enabling expressive predicates that connect intermediate concepts to classes while preserving end-to-end learnability. The approach yields improved accuracy, better concept alignment, and more effective interventions, validated on standard CBM benchmarks and synthetic datasets like XOR, CLEVR-Logic, and SUN-scale scenes. A new worst-case interventional metric, Concept Correction Gain (CCG), assesses robustness to misleading cues, and results demonstrate that diverse logic gates enhance predictive capacity without sacrificing interpretability. The work additionally provides extensive datasets, analyses, and ablations to support the viability and scalability of logic-enhanced concept-based learning. Overall, LogicCBMs offer a principled neurosymbolic pathway for integrating logic reasoning with semantic concepts in vision tasks.

Abstract

Concept Bottleneck Models (CBMs) provide a basis for semantic abstractions within a neural network architecture. Such models have primarily been seen through the lens of interpretability so far, wherein they offer transparency by inferring predictions as a linear combination of semantic concepts. However, a linear combination is inherently limiting. So we propose the enhancement of concept-based learning models through propositional logic. We introduce a logic module that is carefully designed to connect the learned concepts from CBMs through differentiable logic operations, such that our proposed LogicCBM can go beyond simple weighted combinations of concepts to leverage various logical operations to yield the final predictions, while maintaining end-to-end learnability. Composing concepts using a set of logic operators enables the model to capture inter-concept relations, while simultaneously improving the expressivity of the model in terms of logic operations. Our empirical studies on well-known benchmarks and synthetic datasets demonstrate that these models have better accuracy, perform effective interventions and are highly interpretable.

Figures (19)

• Figure 1: LogicCBMs: Overview of Our Approach. We enhance concept-based learning models by including differentiable logic gates in the network. The model now forms logical compositions of concepts while predicting the class label for a given input image. (Dark shades indicate higher strength in the figure; e.g. Furry is the strongest concept and Cat is the predicted class.)
• Figure 2: Logical predicates capture intra-class variability. Examples of predicates (bottom) captured by our method for classes (top) in the CUB dataset (test set). (a) A Ringed Kingfisher has black upper parts if it has a spotted wing pattern. This is captured by an IMPLIES operation. (b) A Grasshopper Sparrow has its primary color as grey or belly color as olive (not both, not neither), captured by an XOR operation in our method.
• Figure 3: LogicCBM Architecture. The proposed logic module/layer is added to a CBM after the concept layer $g(\cdot)$. To implement the logic module, we use two matrices: $CP$ (Concept Pairs) and $G$ (Logic Gates) after the concept layer to learn predicates using differentiable fuzzy logic operations. Our framework is end-to-end differentiable with a subsequent linear layer $f(\cdot)$ that learns the final predicate-class mapping to output the class label prediction.
• Figure 4: Qualitative results showing examples of the class-level logic captured by LogicCBM on the CUB dataset
• Figure 5: Sample images from our CLEVR-Logic dataset along with corresponding logic used to generate them.