Soft Ontology Priors over Latent Concept Posteriors for Weakly Supervised Neuro-Symbolic Learning

Tengwei Song; Robert Hoehndorf

cs.AIOct 21, 2026

Soft Ontology Priors over Latent Concept Posteriors for Weakly Supervised Neuro-Symbolic Learning

Authors: Tengwei Song, Robert Hoehndorf

Organizations: CEMSE, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Abstract

Neuro-symbolic (NeSy) predictors that supervise only the task label suffer from reasoning shortcuts: distinct concept mappings achieve the same maximum likelihood, leaving the recovered concepts unidentifiable and unsafe for downstream reuse. We attack this failure by injecting a soft ontology prior over the latent concepts into the training objective and compiling it together with the task constraint into a smooth, decomposable probabilistic circuit. Under independent Bernoulli perception, the normalisers of the resulting structured concept posterior reduce to weighted model counts that are linear in the compiled circuit size, because the neural leaf weights fold into the circuit before evaluation. The soft compilation strategy places expert axioms in the circuit's weights rather than its support (task-structural axioms guaranteed by the data-generating process stay hard), so noisy expert axioms only redistribute prior mass and cannot empty the support. We characterise how the prior reweights label-restricted shortcut mass through differences in ontology violation energy, show that soft priors incur bounded log-density perturbations under bounded axiom noise, and prove that the hard-support counterpart can be made degenerate by a single corrupted axiom. We validate these predictions on an audited six-concept view of the real-world BDD-OIA driving benchmark and on controlled NeSy tasks.

Figures & tables

Figure 1: Method overview. (a) Forward + backward pass with leaf reweighting and two LogWMC evaluations. (b) Compiling one $\mathsf{Disjoint}(r,g)$ axiom: hard support removal (left) vs. soft edge reweighting (right).

	BDD-OIA-6c			MNLogic-3bit-CNF
Method	F1(C) $\uparrow$	Acc.(Y) $\uparrow$	ECE $\downarrow$	Acc. $\uparrow$	F1(C) $\uparrow$	ECE $\downarrow$
DPL ‡	$0.309\pm 0.033$	$0.766\pm 0.004$	$0.346\pm 0.024$	$0.9989\pm 0.0008$	$0.775\pm 0.202$	$\mathbf{0.105\pm 0.091}$
SPL †	$0.279\pm 0.049$	$0.766\pm 0.002$	$0.331\pm 0.019$	$\underline{0.9999\pm 0.0002}$	$\underline{0.776\pm 0.231}$	$0.111\pm 0.123$
HardPC ⋆	$0.330\pm 0.040$	$0.766\pm 0.004$	$0.340\pm 0.011$	$0.9997\pm 0.0001$	$0.520\pm 0.004$	$0.441\pm 0.009$
BEARS § (k=3)	$0.291\pm 0.017$	$\underline{0.767\pm 0.002}$	$\mathbf{0.184\pm 0.001}$	$0.9997\pm 0.0004$	$0.521\pm 0.058$	$0.177\pm 0.061$
BEARS § (k=1)	$\underline{0.330\pm 0.036}$	$0.765\pm 0.003$	$\underline{0.326\pm 0.012}$	$0.9997\pm 0.0006$	$0.728\pm 0.240$	$0.143\pm 0.127$

Table 1: Main benchmark results. Left: BDD-OIA-6c, the audited real-world concept view. Right: MNLogic-3bit-CNF, used to test benchmark-provided CNF priors. F1(C) and ECE are computed on the raw-

\theta

concept head. Bold / underline

=

best/second-best non-Oracle per column.

	MNIST-Add (full sums)			MNIST-Half (BEARS sparse training)
Method	Acc. $\uparrow$	F1(C) $\uparrow$	ECE $\downarrow$	Acc. $\uparrow$	F1(C) $\uparrow$	ECE $\downarrow$
DPL ‡	0.526 $\pm$ 0.124	0.215 $\pm$ 0.110	0.137 $\pm$ 0.024	1.000 $\pm$ 0.001	0.278 $\pm$ 0.054	0.233 $\pm$ 0.028
SPL †	0.985 $\pm$ 0.005	0.213 $\pm$ 0.076	0.169 $\pm$ 0.015	1.000 $\pm$ 0.000	0.272 $\pm$ 0.007	0.232 $\pm$ 0.006
HardPC ⋆	0.987 $\pm$ 0.002	0.267 $\pm$ 0.070	0.225 $\pm$ 0.029	1.000 $\pm$ 0.000	0.298 $\pm$ 0.042	0.286 $\pm$ 0.024
BEARS § (k=1)	0.985 $\pm$ 0.005	0.212 $\pm$ 0.131	0.170 $\pm$ 0.022	1.000 $\pm$ 0.000	0.271 $\pm$ 0.006	0.233 $\pm$ 0.005
BEARS § (k=3)	0.991 $\pm$ 0.002	0.240 $\pm$ 0.093	0.112 $\pm$ 0.043	1.000 $\pm$ 0.000	0.296 $\pm$ 0.076	0.118 $\pm$ 0.029

Table 2: Digit-sum results. Left: MNIST-Add, the controlled setting for testing informative priors. Right: MNIST-Half, following the BEARS sparse-training variant.

Method	F1(C) raw	F1(C) structured	near-oracle	$n_{\mathrm{modes}}$
DPL ‡	$0.215\pm 0.110$	$0.222\pm 0.108$	$0/10$	$3$
SPL †	$0.213\pm 0.076$	$0.208\pm 0.081$	$0/10$	$2$
HardPC ⋆	$0.267\pm 0.070$	$0.257\pm 0.074$	$0/10$	$2$
BEARS § (k=1)	$0.215\pm 0.112$	$0.210\pm 0.122$	$0/10$	$2$
Ours	$\mathbf{0.498\pm 0.089}$	$\mathbf{0.915\pm 0.199}$	$\mathbf{9/10}$	$2$

Table 3: Concept identifiability diagnostic on MNIST-Add.

Prior-richness (MNIST-Add)			Component (BDD-OIA-6c)
Variant	F1(Y) $\uparrow$	F1(C) $\uparrow$	Variant	Acc.(Y) $\uparrow$	F1(C) $\uparrow$
Level 3 ( $=$ Ours)	0.988 $\pm$ 0.001	0.498 $\pm$ 0.089	Ours	0.768 $\pm$ 0.002	0.362 $\pm$ 0.001
Level 2 ( $-$ ordering)	0.986 $\pm$ 0.003	0.260 $\pm$ 0.072	w/o soft ( $\beta\!=\!0$ )	0.767 $\pm$ 0.001	0.332 $\pm$ 0.041
Level 1 ( $-$ parity)	0.986 $\pm$ 0.004	0.211 $\pm$ 0.068	w/o normalisation	0.766 $\pm$ 0.001	0.113 $\pm$ 0.004
Level 0 (empty prior)	0.986 $\pm$ 0.005	0.222 $\pm$ 0.088	hard-support variant	0.767 $\pm$ 0.001	0.336 $\pm$ 0.027

Table 4: Ablation studies. Left: prior-richness ablation on MNIST-Add. Right: component ablation on BDD-OIA-6c.

Figure 2: Prior strength and ontology noise on MNIST-Add. Left: prior-strength sweep, reporting raw-

\theta

concept F1 and ECE as

\beta

varies. Middle: structured concept F1 under random deletion of soft prior axioms at corruption fraction

\rho

. Right: structured concept F1 under targeted shortcut injection as the injected axiom weight

\lambda

increases;

\lambda\!=\!0

reuses the clean-prior baseline of Table 3 .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Train	Val.	Test (in-dist.)	OOD test
BDD-OIA-6c	RSBench preprocessed release (no OOD)
MNIST-Add	$30{,}000$	$2{,}000$	$5{,}000$	$2{,}000$ (sums $\geq\!9$ )
MNIST-Half	$30{,}000$	$1{,}000$	$2{,}000$	$2{,}000$ (held-out pairs)
MNLogic-3bit-CNF	$30{,}000$	$5{,}000$	$5{,}000$	$5{,}000$ (held-out combinations)

Appendix

Table 5: Per-split sample counts.

Method	In-dist. Acc. $\uparrow$	OOD Acc. $\uparrow$	OOD F1(C) $\uparrow$	OOD ECE $\downarrow$
DPL ‡	0.627 $\pm$ 0.022	0.000 $\pm$ 0.000	0.026 $\pm$ 0.008	$\mathbf{0.175\pm 0.018}$
SPL †	0.974 $\pm$ 0.023	$\mathbf{0.029\pm 0.028}$	0.037 $\pm$ 0.026	0.216 $\pm$ 0.003
HardPC ⋆	0.990 $\pm$ 0.001	0.021 $\pm$ 0.018	0.030 $\pm$ 0.010	0.282 $\pm$ 0.011
BEARS §	$\mathbf{0.993\pm 0.001}$	0.000 $\pm$ 0.000	0.025 $\pm$ 0.006	0.180 $\pm$ 0.017
Ours	0.991 $\pm$ 0.001	0.014 $\pm$ 0.024	$\mathbf{0.077\pm 0.042}$	0.263 $\pm$ 0.067
Oracle	0.994 $\pm$ 0.001	0.259 $\pm$ 0.068	0.281 $\pm$ 0.084	0.063 $\pm$ 0.005

Appendix

Table 6: Out-of-distribution diagnostics on MNIST-Add: training restricted to sums

\leq 8

, evaluated on the held-out sums

\{9,\dots,18\}

$\beta$	$0$	$0.25$	$0.5$	$1$	$2$	$4$
F1(C) $\uparrow$	$0.332\pm 0.041$	$0.361\pm 0.000$	$0.361\pm 0.001$	$\mathbf{0.362\pm 0.001}$	$0.361\pm 0.001$	$0.362\pm 0.001$
Acc.(Y) $\uparrow$	$0.767\pm 0.001$	$0.766\pm 0.001$	$0.765\pm 0.002$	$\mathbf{0.768\pm 0.002}$	$0.765\pm 0.001$	$0.766\pm 0.001$
ECE $\downarrow$	$\mathbf{0.343\pm 0.007}$	$0.346\pm 0.007$	$0.347\pm 0.005$	$0.351\pm 0.001$	$0.351\pm 0.001$	$0.351\pm 0.003$

Appendix

Table 7: BDD-OIA-6c prior-strength sweep for Ours under the canonical traffic prior; best value per row in bold .

$\lambda_{\mathrm{sup}}$	F1(C) $\uparrow$	Acc. $\uparrow$
$0.0$	$0.211\pm 0.068$	$0.986$
$0.1$	$0.282\pm 0.067$	$0.976$
$1.0$	$\mathbf{0.895\pm 0.000}$	$\mathbf{0.994}$
$10.0$	$\mathbf{0.895\pm 0.001}$	$\mathbf{0.994}$

Appendix

Table 8: Concept-supervision sweep on MNIST-Add at

L\!=\!1,\beta\!=\!1

	BDD-OIA-6c		MNIST-Add
Method	raw- $\theta$	structured	raw- $\theta$	structured
DPL ‡	$0.309$	$0.309$	$0.215$	$0.222$
SPL †	$0.279$	$0.279$	$0.213$	$0.208$
HardPC ⋆	$0.330$	$0.330$	$0.267$	$0.257$
BEARS § (k=3)	$0.291$	$0.184$	$0.240$	$0.314$
Ours	$0.361$	$0.361$	$0.498$	$0.915$

Appendix

Table 9: Raw versus structured concept F1. The structured score uses the circuit MAP given the predicted label.

Neurosymbolic Learning

Neurosymbolic learning combines neural models with symbolic rules, knowledge, and reasoning to support structured, interpretable AI.

See all 62 papers

Soft Symbol Grounding for Prototypical Concepts
Sep 10, 2026Marcos Galván-López, Nijesh Upreti, Hiram Calvo +2Prototype-Based Classification Weakly Supervised Learning
Centro de Investigación en Computación, Instituto Politécnico Nacional. · School of Informatics, University of Edinburgh.
Neuro-symbolic learning over OWL 2 DL via consequence-based compilation to differentiable circuits
Aug 18, 2026Olga Mashkova, Asaad Mohammedsaleh, Fernando Zhapa-Camacho +1Neurosymbolic Learning Logical Reasoning
Computer Science Program, Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia
A homotopy-type-theoretic generalization of neurosymbolic inference
Jun 16, 2026Fernando Zhapa-Camacho, Robert HoehndorfSymbolic Reasoning Neuro-Symbolic Reasoning
Computer, Electrical and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, 4700 KAUST, 23955, Thuwal, Saudi Arabia · KAUST Center of Excellence for Smart Health (KCSH) · KAUST Center of Excellence for Generative AI

Sep 10, 2026J/K move · Enter open · S save

Soft Symbol Grounding for Prototypical Concepts

Marcos Galván-López, Nijesh Upreti, Hiram Calvo, Carlos Aguilar-Ibáñez+1

Centro de Investigación en Computación, Instituto Politécnico Nacional. · School of Informatics, University of Edinburgh.

Prototype-Based Classification Weakly Supervised Learning Neuro-Symbolic Reasoning

Open paper AskPDF arXiv

Neuro-symbolic models are usually trained with supervision only on final labels, leaving the intermediate concepts unobserved. Since many concept assignments are consistent with a given label, training can predict labels correctly while recovering the wrong concepts, a failure known as a reasoning shortcut. Prototypical networks reduce shortcuts by anchoring each concept to a few labeled examples, but existing methods still couple perception and reasoning through a hand-crafted, task-specific differentiable loss that must be redesigned for every task. We introduce Soft-PNet, which removes this loss: it reframes concept grounding as a Metropolis walk over a precomputed cache of feasible symbolic solutions, guided by a prototype distribution built from a single labeled anchor per concept, and trains against one KL objective between the prototype-weighted cache and the network's concept predictions. The objective is identical across tasks and remains applicable when the solution space cannot be enumerated. On MNIST-EvenOdd, Visual Sudoku, and Kand-Logic under scarce supervision, Soft-PNet matches loss-engineered prototypical networks at the concept and label levels and recovers concepts that soft-grounding baselines miss, with no loss engineering and lower training time.

Abstract

Neurosymbolic Learning

Explore similar work

Soft Symbol Grounding for Prototypical Concepts

Neuro-symbolic learning over OWL 2 DL via consequence-based compilation to differentiable circuits

A homotopy-type-theoretic generalization of neurosymbolic inference