Learning Proposition-Level Links from Sentence-Level Supervision for Neuro-Symbolic NLI

Xuyao Feng; Antonis Bikakis

cs.CLOct 10, 2026

Learning Proposition-Level Links from Sentence-Level Supervision for Neuro-Symbolic NLI

Authors: Xuyao Feng, Antonis Bikakis

Organizations: University College London, United Kingdom

Abstract

We study neuro-symbolic natural language inference (NLI) with independently translated propositional representations of the premise and hypothesis. Before symbolic reasoning, the system must infer semantic links between their proposition-level units (atoms). Existing systems based on Abstract Meaning Representation (AMR) and Boolean satisfiability (SAT) identify these links using fixed embedding-based source selection and pretrained sentence-level NLI models not trained for atom-level linking. We introduce a trainable atom linker that jointly predicts link existence, source selection, and relation type. Without atom-pair training annotations, we treat cross-formula link structures as latent and train the linker from sentence-level NLI labels, keeping the translator and SAT reasoner fixed. We evaluate on the full public test splits of Stanford Natural Language Inference (SNLI) and Sentences Involving Compositional Knowledge (SICK), and the matched and mismatched validation splits of Multi-Genre Natural Language Inference (MNLI). Our system improves accuracy by 11.40-27.09 percentage points over a baseline applying sentence-trained NLI models unchanged to atom pairs within the same SAT pipeline. It also improves over the prior fixed-linker AMR-to-SAT system by 8.14-26.32 percentage points on SNLI and both MNLI splits, with a smaller gain on SICK.

Figures & tables

Panel of Figure 1 — Figure 1: A held-out SNLI example (gold $\mathrm{Con}$ ) with independently translated formulae and atom texts. (a) Training is illustrated using the MAP structure in (b) as the pool center; this example is not used for training. (b) Colored arrows show the inference MAP links, which yield SAT label $\mathrm{Con}$ .

Data	System	Acc. $\uparrow$	M-F1 $\uparrow$	$R_{E}\uparrow$	$R_{C}\uparrow$	$R_{N}\uparrow$	Inv. $\downarrow$
SNLI	Direct Sentence	$85.23\pm 0.36$	$85.14\pm 0.41$	$87.72\pm 1.22$	$88.91\pm 0.45$	$78.94\pm 2.54$	0.00
	Sentence-NLI Transfer	$51.13\pm 2.26$	$45.08\pm 2.76$	$48.39\pm 5.43$	$95.45\pm 0.71$	$9.42\pm 1.97$	0.00
	FH-SAT	51.90	48.65	74.67	16.19	64.00	1.85
	Learned Linker	$\mathbf{78.22\pm 0.30}$	$\mathbf{78.17\pm 0.28}$	$82.41\pm 1.59$	$79.13\pm 0.72$	$72.92\pm 0.42$	0.00
MNLI-m	Direct Sentence	$79.11\pm 0.45$	$79.03\pm 0.41$	$81.21\pm 1.83$	$78.14\pm 0.84$	$77.76\pm 1.66$	0.00
	Sentence-NLI Transfer	$39.41\pm 1.04$	$37.62\pm 1.34$	$28.60\pm 1.40$	$66.11\pm 0.55$	$23.98\pm 2.88$	0.06

Table 1: Results on full public splits (%). Neural values are three-seed mean

\pm

sample standard deviation; FH-SAT is deterministic. Bold marks the highest Accuracy and Macro-F1 among Transfer, FH-SAT, and Learned Linker.

R_{E}/R_{C}/R_{N}

: class recalls; Inv.: Invalid rate.

Data	System	Acc. $\uparrow$	M-F1 $\uparrow$	$R_{E}\uparrow$	$R_{C}\uparrow$	$R_{N}\uparrow$
SNLI	Sentence-NLI Transfer	$64.52\pm 1.41$	$58.88\pm 1.84$	$97.33\pm 0.88$	$77.33\pm 3.84$	$18.89\pm 2.71$
	Learned Linker	$\mathbf{74.26\pm 1.94}$	$\mathbf{72.92\pm 1.89}$	$98.67\pm 0.33$	$78.78\pm 5.32$	$45.33\pm 1.67$
MNLI-m	Sentence-NLI Transfer	$66.44\pm 0.22$	$62.68\pm 0.61$	$85.89\pm 1.90$	$86.67\pm 3.21$	$26.78\pm 2.78$
	Learned Linker	$\mathbf{67.78\pm 1.20}$	$\mathbf{64.59\pm 1.33}$	$97.11\pm 0.69$	$76.89\pm 2.50$	$29.33\pm 2.03$
MNLI-mm	Sentence-NLI Transfer	$68.44\pm 0.78$	$64.49\pm 1.69$	$86.89\pm 1.39$	$91.11\pm 1.64$	$27.33\pm 4.58$
	Learned Linker	$\mathbf{68.81\pm 1.12}$	$\mathbf{65.50\pm 0.98}$	$96.67\pm 0.00$	$79.78\pm 3.53$	$30.00\pm 0.88$

Table 2: Raw relation classification on 3,600 atom pairs: 900 per split, 300 per class (%). Values are three-seed mean

\pm

sample standard deviation. Labels generated by a large language model (LLM) were reviewed by an author.

Data	Representation	Multi-source	Local action	Other/uncertain
	issue	requirement	disagreement
SNLI	269 (53.8)	59 (11.8)	287 (57.4)	90 (18.0)
MNLI-m	434 (86.8)	52 (10.4)	328 (65.6)	30 (6.0)
MNLI-mm	443 (88.6)	55 (11.0)	348 (69.6)	55 (11.0)
SICK	178 (35.6)	55 (11.0)	300 (60.0)	88 (17.6)
Pooled	1,324 (66.2)	221 (11.1)	1,263 (63.2)	263 (13.2)

Table 3: Issues in 2,000 Linker errors (seed 2026; 500 per split). Entries are sentence-pair counts (%); categories overlap. Pooled percentages weight the four splits equally. LLM-generated annotations were reviewed by an author.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

	Direct Sentence			Learned Linker
Setting	SNLI	MNLI	SICK	SNLI	MNLI	SICK
Encoder LR candidates	$\{5\times 10^{-6},2\times 10^{-5}\}$
Effective batch candidates	$\{64,128\}$
Encoder LR	$2\times 10^{-5}$
Head LR ( $10\times$ encoder)	$2\times 10^{-4}$
Effective batch size	64	64	128	128	64	128

Appendix

Table 4: Selected training settings and search ranges. All six searches use seed 2026. Effective batch size counts examples per optimizer update.

Data	System	2026			2027			2028
		Epoch	Acc.	M-F1	Epoch	Acc.	M-F1	Epoch	Acc.	M-F1
SNLI	Direct Sentence	7	85.87	85.83	6	86.80	86.83	7	86.70	86.68
	Learned Linker	8	80.07	80.06	8	80.03	80.06	6	79.83	79.88
MNLI	Direct Sentence	4	78.97	79.00	5	79.07	79.00	4	79.50	79.48
	Learned Linker	8	60.57	60.79	10	60.80	61.08	8	60.00	60.32
SICK	Direct Sentence	11	87.68	87.13	11	87.27	86.92	17	87.47	87.32

Appendix

Table 5: Selected epochs and development Accuracy and Macro-F1 (%) for the 18 neural runs in the main experiment. Transfer reuses the corresponding Direct Sentence checkpoint; MNLI-m and MNLI-mm share one MNLI checkpoint per system and seed.

Dataset	Dev examples	$\tau_{m}$	$\tau_{c}$ (%)
SNLI	3,000	0.75	80
MNLI	3,000	0.65	90
SICK	495	0.85	100

Appendix

Table 6: FH-SAT development sizes and selected thresholds.

Semantic family	SNLI	MNLI-m	MNLI-mm	SICK
Entailment
Argument projection	60	68	63	60
Event entailment	60	68	67	60
Modifier generalization	60	40	34	60
Nominal entailment	60	59	65	60
Paraphrase	60	65	71	60

Appendix

Table 7: Atom-pair counts by relation, semantic family, and split. Within each split and relation, the family cap is 75, except for color and exact-quantity conflicts (60 each) and clock/date-time conflicts (15).

Data	System	Metric	2026	2027	2028
SNLI	Transfer	Acc.	63.00	65.78	64.78
		M-F1	57.15	60.82	58.67
	Linker	Acc.	72.33	76.22	74.22
		M-F1	70.95	74.73	73.08
MNLI-m	Transfer	Acc.	66.22	66.67	66.44
		M-F1	62.99	63.08	61.97

Appendix

Table 8: Per-seed Accuracy and Macro-F1 on the 3,600-pair probe (%); each split contains 900 pairs.

Data	System	2026		2027		2028
		Acc.	M-F1	Acc.	M-F1	Acc.	M-F1
SNLI	Direct Sentence	84.82	84.68	85.49	85.46	85.38	85.28
	Transfer	48.81	42.12	51.24	45.52	53.33	47.59
	Learned Linker	78.38	78.29	78.41	78.36	77.87	77.85
MNLI-m	Direct Sentence	78.79	78.76	79.62	79.51	78.91	78.82
	Transfer	39.08	37.30	40.57	39.09	38.57	36.46

Appendix

Table 9: Per-seed Accuracy and Macro-F1 on the full public evaluation splits (%). Transfer denotes Sentence-NLI Transfer. Deterministic FH-SAT results appear in Table 1 .

Comp.	Data	Acc. $\Delta$	95% CI	M-F1 $\Delta$	95% CI
L–T	SNLI	$27.09\pm 2.52$	$[24.53,29.61]$	$33.09\pm 2.96$	$[30.28,36.07]$
L–T	MNLI-m	$21.99\pm 0.90$	$[20.62,23.42]$	$23.99\pm 1.22$	$[22.47,25.64]$
L–T	MNLI-mm	$22.91\pm 1.23$	$[21.32,24.54]$	$25.05\pm 1.69$	$[23.15,26.96]$
L–T	SICK	$11.40\pm 10.02$	$[4.97,22.73]$	$13.65\pm 10.31$	$[6.97,25.29]$
L–FH	SNLI	$26.32\pm 0.30$	$[25.15,27.49]$	$29.51\pm 0.28$	$[28.36,30.69]$
L–FH	MNLI-m	$8.14\pm 0.60$	$[6.95,9.33]$	$6.75\pm 0.63$	$[5.55,7.94]$

Appendix

Table 10: Paired differences in percentage points (three-seed mean

\pm

sample standard deviation), with crossed paired bootstrap 95% confidence intervals. L/T/FH/D: Linker, Transfer, FH-SAT, Direct.

Data	2026	2027	2028
SNLI	$<10^{-300}$	$<10^{-300}$	$<10^{-300}$
MNLI-m	$7.67\times 10^{-264}$	$4.41\times 10^{-260}$	$4.67\times 10^{-275}$
MNLI-mm	$5.40\times 10^{-290}$	$3.29\times 10^{-260}$	$3.01\times 10^{-293}$
SICK	$8.46\times 10^{-222}$	$1.82\times 10^{-28}$	$1.34\times 10^{-29}$

Appendix

Table 11: Exact McNemar

p

-values for Linker versus Transfer, Holm-corrected across four splits within each seed. Underflowed values are reported as bounds.

Data	$N$	T	L	FH
SNLI	9,824	0	0	182
MNLI-m	9,815	6	6	701
MNLI-mm	9,832	10	10	796
SICK	4,906	1	1	0

Appendix

Table 12: Invalid counts on full splits. D/T/L: Direct, Transfer, Linker. Counts are identical across neural seeds; FH-SAT is deterministic. Invalid outputs count as errors.

Method	SNLI	SICK
Differentiable Natural Logic ( Feng et al., 2020 )	81.2	–
Neuro-symbolic Natural Logic + IR ( Feng et al., 2022 )	87.5	–
SLR-NLI ( Stacey et al., 2022 )	90.33	85.43
NeuralLog ( Chen et al., 2021 )	–	90.3
Learned Linker	$78.22\pm 0.30$	$84.17\pm 0.24$

Appendix

Table 13: Published Accuracy (%) under different training and evaluation settings. IR: introspective revision. Our SICK evaluation contains 4,906 examples; NeuralLog reports results on 4,927. The SNLI result for Feng et al. (2020) is reported by Feng et al. (2022) . Our results are three-seed means and standard deviations; a dash denotes an unreported result.

Accuracy
Data	30k	60k	$\Delta$
SNLI	$78.22\pm 0.30$	$79.60\pm 0.78$	$1.38\pm 0.48$
MNLI-m	$61.40\pm 0.60$	$62.35\pm 1.18$	$0.96\pm 1.65$
MNLI-mm	$62.00\pm 0.39$	$62.59\pm 0.93$	$0.59\pm 1.28$
Macro-F1
SNLI	$78.17\pm 0.28$	$79.54\pm 0.72$	$1.37\pm 0.45$

Appendix

Table 14: Learned Linker scaling (%). SNLI and MNLI increase from 30,000 to 60,000 training examples. Values are three-seed means and sample standard deviations.

\Delta

: paired 60k minus 30k difference in percentage points.

Data	Hyp. atoms	Examples	Acc.	M-F1	H1 coverage	Invalid count
SNLI	1	3,386	$83.19\pm 0.10$	$80.66\pm 0.04$	$99.79\pm 0.00$	0
	2	2,847	$77.87\pm 0.37$	$77.65\pm 0.37$	$98.22\pm 0.32$	0
	3-4	2,506	$75.01\pm 0.09$	$75.03\pm 0.05$	$96.21\pm 0.60$	0
	5-8	1,005	$71.91\pm 1.70$	$70.13\pm 1.75$	$91.71\pm 0.85$	0
	$\geq 9$	80	$60.42\pm 1.91$	$55.27\pm 1.47$	$79.17\pm 1.44$	0
MNLI-m	1	817	$65.03\pm 0.14$	$61.57\pm 0.20$	$93.15\pm 0.00$	0

Appendix

Table 15: Learned Linker performance and complete Hamming-1 gold-label coverage by hypothesis atom count (%). Accuracy, Macro-F1 and coverage are three-seed means and sample standard deviations; Invalid counts are identical across seeds. Invalid outputs count as errors; unavailable representations form a separate row.

Data	Epoch 1	Selected epoch
SNLI	$74.78\pm 21.56$	$97.24\pm 0.74$
MNLI	$58.11\pm 28.67$	$89.37\pm 0.18$
SICK	$62.75\pm 30.44$	$97.14\pm 0.27$

Appendix

Table 16: Training-set Hamming-1 gold-label coverage (%) before updates at epoch 1 and each run’s development-selected epoch. Values are three-seed means and sample standard deviations.

Quantity	SNLI	MNLI-m	MNLI-mm	SICK
Premise atoms $\|\mathcal{A}_{P}\|$	5.7 / 5.0 / 12.0	9.2 / 8.0 / 21.0	9.6 / 8.0 / 22.0	2.8 / 2.0 / 6.0
Hypothesis atoms $\|\mathcal{A}_{H}\|$	2.5 / 2.0 / 6.0	4.3 / 4.0 / 9.0	4.9 / 4.0 / 10.0	2.7 / 2.0 / 6.0
$\|\mathcal{A}_{P}\|\|\mathcal{A}_{H}\|$	15.0 / 10.0 / 45.0	45.0 / 30.0 / 143.0	52.9 / 36.0 / 154.0	9.8 / 6.0 / 32.0
Candidate pairs	12.4 / 8.0 / 36.0	35.0 / 21.0 / 112.0	40.6 / 27.0 / 120.0	6.5 / 4.0 / 24.0
$\|\mathcal{W}_{1}\|$	25.9 / 17.0 / 73.0	71.1 / 43.0 / 225.0	82.3 / 55.0 / 241.0	14.0 / 9.0 / 49.0
$\log_{10}\|\mathcal{W}\|$	2.2 / 1.7 / 5.7	4.2 / 3.5 / 10.2	4.8 / 4.2 / 11.1	1.6 / 1.0 / 4.8

Appendix

Table 17: Structural workload for available representations: mean / median / 95th percentile (nearest-rank). Values are independent of the training seed.

Data	System	Completed epochs	Minutes
SNLI	Direct Sentence	9/8/9	$12.52\pm 0.81$
	Learned Linker	10/10/8	$55.62\pm 6.16$
MNLI	Direct Sentence	6/7/6	$9.30\pm 0.83$
	Learned Linker	10/10/10	$160.72\pm 4.78$
SICK	Direct Sentence	16/16/20	$3.07\pm 0.40$
	Learned Linker	20/20/20	$11.58\pm 0.34$

Appendix

Table 18: Recorded training times for the main experiments; resource contention was not controlled. Completed training epochs follow seed order 2026/2027/2028; minutes are mean

\pm

sample standard deviation of recorded run totals.

Split	System	Seconds	Examples/s
SNLI	Direct	$9.04\pm 0.07$	$1087.26\pm 8.08$
	Transfer	$109.85\pm 0.12$	$89.43\pm 0.10$
	Linker	$23.15\pm 0.83$	$424.65\pm 15.04$
MNLI-m	Direct	$9.16\pm 0.01$	$1071.24\pm 1.45$
	Transfer	$306.04\pm 0.52$	$32.07\pm 0.05$
	Linker	$61.65\pm 1.25$	$159.25\pm 3.25$

Appendix

Table 19: Seed-2026 inference times on one RTX 4090 (three-run mean

\pm

sample SD). Timing boundaries and padding are specified in Appendix G .

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