Neuro-symbolic artificial intelligence (AI) systems typically couple a neural perception module to a discrete symbolic solver through a non-differentiable boundary, preventing constraint-satisfaction feedback from reaching the perception encoder during training. We introduce AS2 (Attention-Based Soft Answer Sets), a fully differentiable neuro-symbolic architecture that replaces the discrete solver with a soft, continuous approximation of the Answer Set Programming (ASP) immediate consequence operator TP. 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 TP, thereby differentiating through the constraint check without invoking an external solver at either training or inference time. The architecture is entirely free of conventional positional embeddings. Instead, it encodes problem structure through constraint-group membership embeddings that directly reflect 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 (verified by Clingo) across 1,000 test boards, using a greedy constrained decoding procedure that requires no external solver. On MNIST Addition with N∈{2,4,8} addends, AS2 achieves digit accuracy above 99.7% across all scales. 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.
Figures & tables
Figure 1: AS 2 architecture. Raw perceptual signals are encoded cell-by-cell by a shared-weight convolutional neural network (CNN) (§ 3.4 ). A concept bottleneck head produces per-position pre-reasoning logits; clue cells are evidence-clamped before the soft distributions are projected to Rdmodel and enriched by constraint-group membership embeddings (row, column, and box for Sudoku), replacing conventional positional encodings. A multi-layer Transformer encoder propagates information across all positions jointly. The output head produces post-reasoning logits supervised end-to-end by the differentiable TP fixed-point loss (Equation 3 ), which enforces constraint satisfaction without calling an external solver.
Benchmark
Train
Val
Test
Input size
Constraint type
#Groups
MNIST Add ( N=2 )
30,000
N/A
5,000
2×28×28
Arithmetic
1
MNIST Add ( N=4 )
30,000
N/A
5,000
4×28×28
Arithmetic
1
MNIST Add ( N=8 )
30,000
N/A
5,000
8×28×28
Arithmetic
1
Visual Sudoku
9,000
1,000
1,000
252×252
Latin square
27
Table 1: Dataset statistics. Constraint type classifies the symbolic constraint family. #Groups lists the number of constraint groups per instance.
N=2
N=4
N=8
Method
Digit-Acc ↑
Sum-Acc ↑
Digit-Acc ↑
Sum-Acc ↑
Digit-Acc ↑
Sum-Acc ↑
DeepProbLog † ( Manhaeve et al., 2021a )
N/A
97.20 ± 0.50
N/A
N/A
N/A
N/A
NeurASP † ( Yang et al., 2020 )
N/A
97.30 ± 0.30
N/A
N/A
N/A
N/A
Scallop † ( Li et al., 2023 )
N/A
98.22
N/A
97.00
N/A
N/A
A-NeSI † ( van Krieken et al., 2023 )
N/A
97.66 ± 0.21
N/A
95.96 ± 0.38
N/A
N/A
Perception-Only
99.69 ± 0.05
99.37 ± 0.10
99.91
99.64
99.98
99.86 ± 0.03
Table 2: Results on MNIST Add . Digit-Acc = individual digit accuracy (%); Sum-Acc = sum accuracy (%). Results for prior work are taken from the respective publications. Bold = best in column. “ † ” indicates the result uses only sum-level supervision (no per-digit labels); Digit-Acc is not reported for these methods because per-digit predictions are implicit. DeepProbLog and A-NeSI results are taken from van Krieken et al. (2023) . NeurASP result at N=2 is taken from Manhaeve et al. (2021b) . Scallop results are reported by Li et al. (2023) (best variant). All AS 2 and ablation results use both digit and sum supervision.
Method
Sym-Acc ↑
Board-Acc ↑
CSR ↑
VCSR ↑
Perception-Only §
60.51
0.00
100.00
N/A
Pipeline §
60.51
0.00
100.00
N/A
PBCS † ( Mulamba et al., 2024 )
N/A
99.4
N/A
N/A
AS 2 (ours) ‡
99.89
100.0
100.0
100.0
Table 3: Results on Visual Sudoku (our primary benchmark). Bold = best in column; underline = second-best. § Perception-Only and Pipeline use Clingo to complete blank cells at inference; their CSR of 100% reflects solver post-processing, not a learned constraint mechanism. Board-Acc of 0% indicates the solver produces a valid but incorrect completion. † PBCS uses a pipeline architecture with CP-SAT at inference and does not publish Sym-Acc , CSR , or VCSR . Its Board-Acc of 99.4% is taken from Mulamba et al. (2024) . ‡ AS 2 CSR and VCSR are measured after greedy constrained decoding (Section 3.7 ). No external solver is invoked during inference. “Raw” Board-Acc (argmax only, no post-processing) is 95.60%.
We present a general neurosymbolic reasoning and learning methodology based on a modular integration of answer set programming with an energy based model substrate. Key contributions are: (1) supporting joint optimisation in the continuous latent space through explicit ASP-based declarative semantics fully incorporating background knowledge, constraints, non-monotonic inference; and (2) advancing recent works at the interface of answer sets, probabilistic logic, and answer set modulo theories by providing a generalised model and practical platform for ASP-centric robust, end-to-end training for applications in dynamic domains (e.g., involving perception and interaction). We provide a practical implementation, and demonstrate basic use and application (with MNIST), and evaluate with the visual question-answering benchmark Clevr and the multi-object tracking benchmark MOT.
Jakob Suchan, Julius Monsen, Salim Baloch +1
Constructor University Bremen, Bremen, Germany · CoDesign Lab > Cognitive Vision · Örebro University, Örebro, Sweden
Integration of Answer Set Programming (ASP) with neural networks has emerged as a promising tool in Neuro-symbolic AI. While existing approaches extend the capabilities of ASP to real world domains, their reasoning pipelines depend on classical solvers, which is a bottleneck for scalability. To tackle this problem, we propose a new method to compute stable models, called decision-propagation (DProp), which alternates falsity decisions and truth propagations. Successful DProp computations are shown to capture the stable model semantics. We then develop Neural DProp (NDProp), a differentiable extension of DProp with neural computation for decisions and fuzzy evaluation for propagations. We evaluate the capabilities of NDProp for learning decision heuristics as well as neuro-symbolic integration, and compare it with existing neuro-symbolic approaches. The results show that NDProp can learn to efficiently compute stable models, and it improves accuracy and scalability on neuro-symbolic benchmarks.
Thomas Eiter, Katsumi Inoue, Sota Moriyama
1Vienna University of Technology (TU Wien), Austria · 2National Institute of Informatics, Japan · 3The Graduate University for Advanced Studies, SOKENDAI, Japan
Neural networks excel at pattern recognition but struggle with constraint reasoning -- determining whether configurations satisfy logical or physical constraints. We introduce Differentiable Symbolic Planning (DSP), a neural architecture that performs discrete symbolic reasoning while remaining fully differentiable. DSP maintains a feasibility channel (phi) that tracks constraint satisfaction evidence at each node, aggregates this into a global feasibility signal (Phi) through learned rule-weighted combination, and uses sparsemax attention to achieve exact-zero discrete rule selection. We integrate DSP into a Universal Cognitive Kernel (UCK) that combines graph attention with iterative constraint propagation. Evaluated on three constraint reasoning benchmarks -- graph reachability, Boolean satisfiability, and planning feasibility -- UCK+DSP achieves 97.4% accuracy on planning under 4x size generalization (vs. 59.7% for ablated baselines), 96.4% on SAT under 2x generalization, and maintains balanced performance on both positive and negative classes where standard neural approaches collapse. Ablation studies reveal that global phi aggregation is critical: removing it causes accuracy to drop from 98% to 64%. The learned phi signal exhibits interpretable semantics, with values of +18 for feasible cases and -13 for infeasible cases emerging without supervision.