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

Olga Mashkova; Asaad Mohammedsaleh; Fernando Zhapa-Camacho; Robert Hoehndorf

cs.AIAug 18, 2026

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

Authors: Olga Mashkova, Asaad Mohammedsaleh, Fernando Zhapa-Camacho, Robert Hoehndorf

Organizations: Computer Science Program, Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia

Abstract

OWL 2 DL ontologies, grounded in the description logic $\mathcal{SROIQ}$ , express large knowledge bases in biomedicine and the Semantic Web. Neuro-symbolic (NeSy) learners over description logics either embed the ontology in a continuous space, abandoning classical entailment, or restrict to the Horn fragment $\mathcal{EL}^{++}$ , which has a single canonical model. We present Baobab, which compiles a $\mathcal{SROIQ}$ ontology with a finite ABox into a Sentential Decision Diagram (SDD): it saturates a propositional core under a consequence-based calculus and instantiates the remaining $\mathcal{SROIQ}$ features (nominals, number restrictions, and the role axioms) over the active domain. The SDD's evidence-conditioned weighted model count then trains a perception network to recognize real images under partial ABox supervision: on an ontology that exercises every distinctive $\mathcal{SROIQ}$ feature, a CNN learns to read MNIST digits coupled by a successor relation and recovers latent ontology concepts that an independent perception leaves at chance. When the supervision admits several ontology-consistent completions, an independent perception collapses onto one, a reasoning shortcut: we show that a mixture indexed by the query's justifications can represent the calibrated posterior no independent perception can, and that seeding it from the circuit's enumerated completions attains the Bayes-optimal posterior on a real-image MNIST task where single-WMC and learned mixtures (the BEARS-ensemble hypothesis class) do not: to our knowledge the first to characterize and mitigate reasoning shortcuts in a non-Horn description logic. Soundness of the compiler and the representation result are machine-checked in Lean 4. Code is available at https://github.com/bio-ontology-research-group/baobab.

Figures & tables

regime	method	digit	latent	viol $\downarrow$	ECE $\downarrow$
grounded ( $\#\mathrm{RS}{=}1$ )	Independent	$0.25$ $\pm.09$	$0.49$ $\pm.14$	$0.80$ $\pm.12$	$0.33$ $\pm.14$
grounded ( $\#\mathrm{RS}{=}1$ )	WMC	$\mathbf{0.99}$ $\pm.00$	$\mathbf{1.00}$ $\pm.00$	$\mathbf{0.02}$ $\pm.01$	$\mathbf{0.00}$ $\pm.00$
under-det. ( $\#\mathrm{RS}{=}5$ )	Independent	$0.20$ $\pm.03$	$0.46$ $\pm.10$	$0.93$ $\pm.10$	$0.08$ $\pm.08$
under-det. ( $\#\mathrm{RS}{=}5$ )	WMC	$0.18$ $\pm.07$	$0.67$ $\pm.03$	$0.78$ $\pm.05$	$0.15$ $\pm.02$

Table 1: MNIST-

\mathcal{SROIQ}

, mean

\pm

std over ten seeds (metrics in § 4 ; digit identities never supervised). Grounding the concept evidence collapses

\#\mathrm{RS}

from

5

1

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

feature	emitted ground clauses	count
nominal $\mathsf{Nom}_{a}$	$\mathsf{Nom}_{a}(a)$ ; $\neg\mathsf{Nom}_{a}(b)$ , $b\neq a$	$d$
${\geq}\,n\,R.C$ at $x$	$Q(a)\leftrightarrow\bigvee_{\|S\|=n}\bigwedge_{y\in S}R(a,y)\wedge C(y)$	$d\,(\binom{d}{n}{+}1)$
${\leq}\,n\,R.C$ at $x$	$Q(a)\leftrightarrow\bigwedge_{\|T\|=n+1}\big(\bigwedge_{y\in T}R(a,y)\wedge C(y)\to\bigvee_{i<j}y_{i}\approx y_{j}\big)$	$O(d^{\,n+1})$
inverse $R\equiv S^{-}$	$R(a,b)\leftrightarrow S(b,a)$	$2d^{2}$
$\mathsf{Trans}(R)$	$R(a,b)\wedge R(b,c)\to R(a,c)$	$d^{3}{-}d$
chain $R_{1}{\circ}\dots{\circ}R_{k}\sqsubseteq S$	$R_{1}(a_{0},a_{1})\wedge\dots\wedge R_{k}(a_{k-1},a_{k})\to S(a_{0},a_{k})$	$d^{\,k+1}$

Appendix

Table 2: Grounding schemata over the active domain

\Delta

d=|\Delta|

a,b,c

range over

\Delta

;

\binom{d}{n}

is the

n

-subset count.

refinement	method	topping	property	viol $\downarrow$
closed	Independent	$0.95$ $\pm.01$	$0.45$ $\pm.15$	$0.89$ $\pm.18$
closed	WMC	$0.93$ $\pm.01$	$\mathbf{0.92}$ $\pm.02$	$0.06$ $\pm.02$
open	Independent	$0.95$ $\pm.01$	$0.45$ $\pm.16$	$0.86$ $\pm.28$
open	WMC	$0.95$ $\pm.01$	$0.78$ $\pm.01$	$0.00$ $\pm.00$

Appendix

Table 3: Closed-world ablation on the Pizzaiolo fragment (mean

\pm

std over ten seeds, five epochs; same fragment and metrics as Table 7 ). Closed materializes the exact biconditional

X(a)\Leftrightarrow\bigsqcup_{t}\mathsf{hasTopping}(a,t)\sqcap C(t)

; open keeps the forward direction

X(a)\!\to\!\exists\mathsf{hasTopping}.C

only, dropping the closure clause

\exists\mathsf{hasTopping}.C\!\to\!X(a)

. Removing the closure clause leaves supervised toppings untouched but lowers latent property recovery from

0.92

0.78

; the reported violation falls to

0.00

because the removed clause is the constraint whose violation was counted (§ 4.1 ).

	clauses	fixed vtree	minimized
ontology	unsat $\to$ sat	unsat $=$ sat	unsat	sat
horn-chain	$3\to 6$	$10$	$8$	$8$
disjunction	$7\to 12$	$23$	$17$	$\mathbf{12}$
two-disjunctions	$11\to 29$	$38$	$22$	$\mathbf{18}$
exists+disjunction	$8\to 73$	$73$	$41$	$\mathbf{36}$
bird-penguin	$11\to 44$	$1011$	$139$	$143$

Appendix

Table 4: Effect of saturation on SDD node count, measured with the accompanying library. Saturation adds entailed clauses (column clauses ) without changing the atom set. Over a fixed vtree the size is identical with and without saturation (Theorem 0.F.1 ); under dynamic vtree minimization it usually shrinks but is not guaranteed to ( bird-penguin ).

toppings ( $\|\Delta\|$ )	atoms	clauses	ground (s)	SDD nodes	peak mem
$1$ ( $\|\Delta\|{=}2$ )	$201$	$638$	$0.04$	$3{,}182$	$0.7$ GB
$2$ ( $\|\Delta\|{=}3$ )	$302$	$957$	$0.19$	$12{,}064$	$5.7$ GB
$3$ ( $\|\Delta\|{=}4$ )	$403$	$1{,}276$	$0.01$	—	$>22$ GB †

Appendix

Table 5: Compiling the full Pizzaïolo

\mathcal{SROIQ}

ontology (

66

named classes,

90

existential and

44

universal restrictions, right-hand-side disjunctions on

\mathsf{Pizza}

and

\mathsf{SpicyPizza}

) over a growing active domain, with a fixed vtree throughout. Grounding stays sub-second at every size; the compiled SDD grows super-linearly and exceeds a

22

GB /

600

s budget at three toppings. The

59

-atom two-topping fragment of § 4.1 is a separately extracted role-based sub-signature, not a row here.

experiment	feat. dim	hidden	epochs	$\lambda$	$\kappa$	$K$
MNIST (CNN)	$28^{2}$	CNN	12	0.5	–	–
Pizzaiolo frag. (ResNet-18)	$224^{2}$	linear head	5	0.5	–	–
Pizza (synthetic)	12	64	10	0.3	–	–
MNIST-Disj. (CNN)	$28^{2}$	CNN	20	0.5	2.0	4

Appendix

Table 6: Hyperparameters (Adam; learning rates and batch sizes in the text).

\lambda

: semantic-loss weight;

\kappa

: diversity weight;

K

: mixture / ensemble size.

method	topping	property	viol $\downarrow$
Independent	$0.95$ $\pm.01$	$0.45$ $\pm.15$	$0.89$ $\pm.18$
WMC	$0.93$ $\pm.01$	$\mathbf{0.92}$ $\pm.02$	$\mathbf{0.06}$ $\pm.02$

Appendix

Table 7: Pizzaiolo two-topping role-based fragment (mean

\pm

std over ten seeds; metrics in § 4 ). topping is supervised; property are the latent Vegetarian/NonVegetarian/Spicy classes recovered through

\mathsf{hasTopping}

Method	digit	NLL $\downarrow$	ECE $\downarrow$	mode-cov. TV $\downarrow$
Independent (UCI)	$0.99$ $\pm.00$	$6.93$ $\pm.00$	$0.00$ $\pm.00$	$1.00$ $\pm.00$
Single-WMC (SL / DPL)	$0.99$ $\pm.00$	$6.13$ $\pm.00$	$0.00$ $\pm.00$	$0.99$ $\pm.00$
JustWMC (learned mix)	$0.99$ $\pm.00$	$6.69$ $\pm.43$	$0.11$ $\pm.03$	$0.99$ $\pm.01$
BEARS	$-$	$6.36$ $\pm 2.36$	$0.41$ $\pm.12$	$0.44$ $\pm.11$
JustWMC (anchored)	$0.99$ $\pm.00$	$\mathbf{4.17}$ $\pm.00$	$\mathbf{0.00}$ $\pm.00$	$\mathbf{0.02}$ $\pm.00$

Appendix

Table 8: MNIST-Disjunction (

\#\mathrm{RS}=4

; mean

\pm

std over ten seeds; metrics in § 4 ). NLL is over the gender atoms (Bayes-optimal

4.16

); ECE and mode-coverage TV are the reasoning-shortcut diagnostics. BEARS is a

K{=}4

deep-ensemble mitigation [ 24 ] trained on the same circuit; its per-digit accuracy is not separately evaluated (

-

method	topping	pizza	latent	violation
	(observed)	(4 latent)	(11 latent)	rate
Independent	$0.99$ $\pm.00$	$0.44$ $\pm.12$	$0.43$ $\pm.08$	$1.00$
WMC	$0.76$ $\pm.01$	$\mathbf{0.75}$ $\pm.00$	$\mathbf{0.58}$ $\pm.05$	$\mathbf{0.00}$

Appendix

Table 9: Pizza-

\mathcal{SROIQ}

, 64 held-out examples, mean

\pm

std over ten seeds (higher is better except for violation). The pizza (4 latent) and latent (11 atoms) columns are the headline: WMC infers them from topping evidence though those atoms receive no direct supervision.

OWL kind	axioms	ours	DPL (Horn)	DPL (manual)	DPL clauses
SubClassOf	23	46	15	8	30
DisjointClasses	14	28	14	0	14
EquivalentClasses	3	12	0	3	11
RoleInclusion	2	4	2	0	2
AsymmetricRole	2	4	2	0	2
IrreflexiveRole	2	4	2	0	2

Appendix

Table 10: Per-OWL-axiom-kind encoding cost for the MNIST-

\mathcal{SROIQ}

ontology. ours : normalized DL-clauses produced by the structural transformation. DPL Horn / DPL manual : DPL admits / requires user-written disjunctive / cardinality / negation encoding.

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Moose: Latent concept learning with reasoning-shortcut awareness in $\mathcal{EL}^{++}$

Olga Mashkova, Asaad Mohammedsaleh, Fernando Zhapa-Camacho, Robert Hoehndorf

Computer Science Program, Computer, Electrical, and Mathematical Sciences & Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia

Neurosymbolic Learning Description Logic

Open paper AskPDF arXiv

The OWL 2 EL profile is used in some of the largest production ontologies, including the Gene Ontology and SNOMED CT. Existing neuro-symbolic (NeSy) learning methods accept propositional theories or Datalog, and reasoning-shortcut (RS) awareness has not been investigated in ontology settings. We present Moose, a method that compiles an

\mathcal{EL}^{++}

TBox and finite ABox to a Sentential Decision Diagram (SDD). The SDD acts as a differentiable weighted-model-counting layer, and we add closure clauses outside the

\mathcal{EL}^{++}

profile on declared exhaustive families to overcome the limited expressivity of

\mathcal{EL}^{++}

under partial supervision. We show termination, soundness, completeness, and polynomial intermediate sizes, and validate the proofs in Lean. We then define the first formal partial-supervision latent-concept-learning task over an OWL EL ontology, i.e., learning per-individual classifiers for latent concepts from observed ABox literals, and evaluate Moose on MNIST-with-ontology and Pizzaïolo. Moose improves over propositional-NeSy, fuzzy-logic, and ontology embedding baselines, and presents the first reasoning-shortcut analysis in an OWL EL setting.

Abstract

Neurosymbolic Learning

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