Integer Linear Programming (ILP) has been proposed as a formalism for encoding precise structural and semantic constraints for Natural Language Inference (NLI). However, traditional ILP frameworks are non-differentiable, posing critical challenges for the integration of continuous language representations based on deep learning. In this paper, we introduce a novel approach, named Diff-Comb Explainer, a neuro-symbolic architecture for explanation-based NLI based on Differentiable BlackBox Combinatorial Solvers (DBCS). Differently from existing neuro-symbolic solvers, Diff-Comb Explainer does not necessitate a continuous relaxation of the semantic constraints, enabling a direct, more precise, and efficient incorporation of neural representations into the ILP formulation. Our experiments demonstrate that Diff-Comb Explainer achieves superior performance when compared to conventional ILP solvers, neuro-symbolic black-box solvers, and Transformer-based encoders. Moreover, a deeper analysis reveals that Diff-Comb Explainer can significantly improve the precision, consistency, and faithfulness of the constructed explanations, opening new opportunities for research on neuro-symbolic architectures for explainable and transparent NLI in complex domains.

Integer Linear Programming (ILP) provides a viable mechanism to encode explicit and controllable assumptions about explainable multi-hop inference with natural language. However, an ILP formulation is non-differentiable and cannot be integrated into broader deep learning architectures. Recently, Thayaparan et al. (2021a) proposed a novel methodology to integrate ILP with Transformers to achieve end-to-end differentiability for complex multi-hop inference. While this hybrid framework has been demonstrated to deliver better answer and explanation selection than transformer-based and existing ILP solvers, the neuro-symbolic integration still relies on a convex relaxation of the ILP formulation, which can produce sub-optimal solutions. To improve these limitations, we propose Diff-Comb Explainer, a novel neuro-symbolic architecture based on Differentiable BlackBox Combinatorial solvers (DBCS) (Pogan\v{c}i\'c et al., 2019). Unlike existing differentiable solvers, the presented model does not require the transformation and relaxation of the explicit semantic constraints, allowing for direct and more efficient integration of ILP formulations. Diff-Comb Explainer demonstrates improved accuracy and explainability over non-differentiable solvers, Transformers and existing differentiable constraint-based multi-hop inference frameworks.