New Neurosymbolic Inference Generalization Accounts for Symmetries and Proofs.

Traditional neurosymbolic (NeSy) systems often compute a belief-weighted sum of logical quantities over a space of sigma-structures, encompassing methods like weighted model counting and probabilistic logic. However, these set-based approaches inherently lose crucial information regarding when two sigma-structures are equivalent due to theory symmetries, and how many distinct proofs might witness a particular query. This new research proposes a generalization of NeSy inference by replacing the underlying sets with types, drawing from homotopy type theory. This shift allows the system to preserve the previously lost information about symmetries and proof multiplicity. Consequently, the functional is transformed into a belief-weighted homotopy cardinality, a measure that inherently counts objects inversely proportional to their symmetries. The framework is developed from first principles for NeSy systems, proving a conservativity theorem that recovers classical functionals when symmetries are trivial. A significant practical outcome is that the shortcut-aware concept posterior, typically achieved by complex ensembling or expressive density estimation in recent methods, can now be computed in closed form by simply averaging a single model over the symmetry group. This single-model wrapper demonstrates improved calibration on MNIST reasoning-shortcut benchmarks compared to diversity-trained ensembles, without compromising label accuracy or identifiable concepts.