Attention-Based Soft Answer Sets

Neuro-symbolic systems typically couple a neural perception module with a discrete symbolic solver, where exact constraint inference is intractable at training time and a discrete solver must be invoked at inference time, precluding end-to-end optimization of the model. We introduce AS2 (Attention-Based Soft Answer Sets), a fully differentiable neuro-symbolic architecture that replaces the discrete solver with a continuous approximation of the Answer Set Programming (ASP)immediate-consequence operator $T_P$. AS2 maintains per-position probability distributions over a finite symbol domain throughout the forward pass and trains end-to-end by minimizing the fixed-point residual of a probabilistic lift of $T_P$, thereby differentiating through the constraint check without invoking an external solver during either training or inference. On spatial constraint-satisfaction tasks, AS2 replaces conventional positional embeddings entirely, and problem structure is encoded through constraint-group membership embeddings derived directly from the declarative ASP specification, making the model agnostic to arbitrary position indexing. On Visual Sudoku, AS2 achieves 99.89\% cell accuracy and 100\%constraint satisfaction, without invoking any external solver. On CLEVR-Hans, AS2 achieves 99.94% on CLEVR-Hans3 and 87.77% on CLEVR-Hans7. These results demonstrate that a soft differentiable fixpoint operator, combined with constraint-aware attention and declarative constraint specification, can match or exceed pipeline and solver-based neuro-symbolic systems while maintaining full end-to-end differentiability.