Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs

Nijesh Upreti; Vaishak Belle

cs.AIAug 21, 2026

Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs

Authors: Nijesh Upreti, Vaishak Belle

Organizations: The University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB, UK

Abstract

Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) inject background knowledge by grounding each logical axiom as a predicate and training the generator to raise its satisfaction, a fuzzy truth value in $[0,1]$ . Previous LTN-GAN work grounded every constraint this way, at the predicate level, and improved constraint satisfaction. A predicate, however, only scores a sample, so it cannot embed hard structural constraints, rules such as orderings, positivity, and definitional identities that must hold in every generated sample. In this work, we investigate grounding each axiom as a function symbol inside the LTN framework. We compare against the state-of-the-art alternative, a constraint layer that clamps each violating sample onto the feasible boundary and so produces outputs that are always valid. Our investigation shows that a valid sample is not always a realistic one. An inequality is not merely satisfied or violated. It holds by a margin, and a faithful generator should also reproduce the margin's real distribution. We find that the resolution ratio $R$ , the data's scale over the margin's spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn. When $R$ is large, the predicate receives no learning signal, the clamp pushes every sample onto the boundary, and the margin distribution is lost while every standard metric still looks fine. A function symbol avoids both failures, computing the constrained variable rather than scoring it. Together the function symbols form a chart, a coordinate system inside the feasible region, where every sample is valid by construction and the margin is learned like any other quantity.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the FSG-LTN-GAN. The grounding map $\varphi$ assembles valid samples from the generator’s free coordinates $z$ within the Fourier-Motzkin bounds of $\Pi$ , and the discriminator operates in the chart coordinates. The hybrid instead clamps low- $R$ and discrete variables in the loop (Appendix B ).

		margin KS $\downarrow$		per-prop. moment $\downarrow$
Dataset	$R$	C-DGM	FSG-LTN-GAN	C-DGM	FSG-LTN-GAN
Alchemy	$3{\times}10^{4}$ to $7{\times}10^{6}$	$1.000\pm 0.000$	$\mathbf{0.040\pm 0.007}^{\dagger}$	$0.472\pm 0.012$	$0.327\pm 0.058$
tmQM	$2.3\times 10^{4}$	$1.000\pm 0.000$	$\mathbf{0.045\pm 0.010}^{\dagger}$	$0.183\pm 0.014$	$0.192\pm 0.025$
Transition1x	$10^{3}$	$0.999\pm 0.000$	$\mathbf{0.065\pm 0.010}^{\dagger}$	$0.115\pm 0.036$	$0.196\pm 0.038$
Taxi	$3.6$ to $1.1{\times}10^{3}$	$0.574\pm 0.005$	$\mathbf{0.099\pm 0.010}^{\dagger}$	$0.492\pm 0.024$	$0.282\pm 0.032$

Table 1: Validity does not capture the distribution of the constraint margin (RQ1). Four real high-resolution datasets,

n{=}10

, mean

\pm

std. Both methods are

100\%

valid. Only FSG-LTN-GAN recovers the distribution of the constraint margin, while per-property moment error (lower better) is comparable, so the gap is invisible to it. † paired Wilcoxon

p<0.01

Figure 2: The distribution of the constraint margin on all four datasets. Real margin (grey) against C-DGM (orange, the constraint layer) and FSG-LTN-GAN (blue). All are $100\%$ valid. The clamp collapses every margin to a boundary point mass. FSG-LTN-GAN reproduces the real distribution. The annotated KS is for the displayed high- $R$ margin. Table 1 averages each dataset’s constraints, so its Taxi value also counts the low- $R$ $\mathrm{total}>\mathrm{fare}$ margin, which the clamp handles well.

Dataset	CTGAN $+$ CL	TVAE $+$ CL	C-DGM (MLP)	FSG-LTN-GAN
Alchemy	$0.674\pm 0.036$	$0.651\pm 0.015$	$1.000\pm 0.000$	$\mathbf{0.040\pm 0.007}$
tmQM	$0.579\pm 0.052$	$0.517\pm 0.011$	$1.000\pm 0.000$	$\mathbf{0.045\pm 0.010}$
Transition1x	$0.592\pm 0.040$	$0.510\pm 0.053$	$0.999\pm 0.000$	$\mathbf{0.065\pm 0.010}$
Taxi	$0.419\pm 0.024$	$0.348\pm 0.029$	$0.574\pm 0.005$	$\mathbf{0.099\pm 0.010}$

Table 2: Architecture generality (RQ2). Margin KS

\downarrow

(

n{=}10

, mean

\pm

std) for CTGAN

+

CL, TVAE

+

CL, the constraint layer on our MLP generator (C-DGM), and FSG-LTN-GAN. All are

100\%

valid. The constraint layer misses the distribution of the constraint margin on all three generator families.

Dataset	$R_{\max}$	C-DGM	FSG-all	hybrid
faults	$1.8\times 10^{3}$	$0.745\pm 0.014$	$0.117\pm 0.008$	$\mathbf{0.117\pm 0.008}$
heloc	$2$	$0.156\pm 0.012$	$0.208\pm 0.005$	$0.158\pm 0.014$
lcld	$1$	$0.131\pm 0.008$	$0.657\pm 0.004$	$0.133\pm 0.007$
url	$1$	$0.159\pm 0.010$	$0.196\pm 0.018$	$\mathbf{0.144\pm 0.021}$
news	$2$	$0.294\pm 0.008$	$0.318\pm 0.007$	$0.297\pm 0.007$
wids	$4$	$0.140\pm 0.007$	$0.319\pm 0.004$	$\mathbf{0.133\pm 0.008}$

Table 3: The constraint-layer benchmark of 35 (RQ3). Margin KS

\downarrow

n{=}10

, mean

\pm

std, all methods

100\%

valid.

R_{\max}

is the dataset’s largest per-constraint

R

. The hybrid outperforms the constraint layer (paired Wilcoxon) on faults (

p{=}0.002

, the high-

R

case), url (

p{=}0.03

), and wids (

p{=}0.004

), ties on heloc, lcld, news, and never underperforms it. FSG-all is worse than the constraint layer everywhere but faults, showing the selective in-loop clamp is necessary.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: High versus low resolution ratio $R$ (schematic). The same ordering $b>a$ in two regimes. Top: the data $(a,b)$ and the boundary $b=a$ . At low $R$ (left) the feasible margin $b-a$ is a visible fraction of the scale, so a free generator resolves it and the constraint layer’s clamp rarely acts. At high $R$ (right) the margin is tiny relative to the scale and invisible at the data’s magnitude, so every sample lies close to the boundary. Bottom: the margin distribution. At low $R$ the clamp (orange) preserves the distribution of the real constraint margin (grey); at high $R$ the free generator cannot place the sub-resolution margin, so the clamp collapses every sample onto the boundary, a point mass at $0$ disjoint from the real distribution. Function-symbol grounding repairs the high- $R$ case by making the margin a unit-scale coordinate (Figure 4 ).

Figure 4: How function-symbol grounding repairs the high- $R$ collapse (schematic). The same high- $R$ ordering margin as in Figure 3 ; top row shows the margin distribution at each stage, bottom row the corresponding $(a,b)$ geometry . (1) In ambient space the margin $b-a$ is sub-resolution, so the constraint layer clamps it to a point mass at $0$ . (2) The chart coordinate $z=\varphi^{-1}(x)$ standardizes the margin to unit scale, where the discriminator resolves it and a standard GAN learns the distribution of the real constraint margin (generator in blue matches real in grey). (3) Decoding $x=\varphi(z)=\ell+\mathrm{softplus}(z)$ carries that distribution back to the ambient margin. (4) Geometrically the clamp moves every violating sample onto the boundary $b=a$ (margin $0$ , the point mass of panel 1). (5) The function symbol $b=a+\mathrm{softplus}(z)$ is positive for every generator output $z$ , so $b>a$ holds by construction ( $\mathrm{Sat(KB)}=1$ ); it is smooth and monotone, so matching the distribution of $z$ matches the margin’s distribution, and asymptotically linear, so generator tails do not diverge. (6) Our decoded samples therefore land inside the feasible region $b>a$ , matching the real distribution with exact validity.

Dataset	$N$	$D$	Evaluated margin(s)	$R$
Alchemy	202,579	12	$U-U_{0}$ ; $H-U$ ; $H-G$	$9{\times}10^{4}$ ; $7{\times}10^{6}$ ; $3{\times}10^{4}$
tmQM	108,541	8	$E_{\mathrm{elec}}-E_{\mathrm{tot}}$	$2.3{\times}10^{4}$
Transition1x	10,073	3	$E_{\mathrm{TS}}-E_{\mathrm{react}}$ ; $E_{\mathrm{TS}}-E_{\mathrm{prod}}$	$1.0{\times}10^{3}$ ; $8.1{\times}10^{2}$
Taxi	50,000	6	duration; $\mathrm{total}-\mathrm{fare}$	$1.1{\times}10^{3}$ ; $3.6$

Appendix

Table 4: The four real high-resolution datasets.

N

rows,

D

properties;

R

is the resolution ratio of each evaluated ordering margin.

Dataset	Method	Validity	Margin KS $\downarrow$
Alchemy	unconstrained GAN	$0.053\pm 0.021$	$0.539\pm 0.014$
	projection	$1.000\pm 0.000$	$0.664\pm 0.026$
	C-DGM (clamp)	$1.000\pm 0.000$	$1.000\pm 0.000$
	FSG-LTN-GAN	$1.000\pm 0.000$	$\mathbf{0.040\pm 0.007}$
tmQM	unconstrained GAN	$0.229\pm 0.046$	$0.575\pm 0.037$
	projection	$1.000\pm 0.000$	$0.575\pm 0.037$

Appendix

Table 5: Full RQ1 results,

n=10

seeds, mean

\pm

std. Validity is the fraction satisfying all constraints. Margin KS is the two-sample KS between the generated margins and a fixed real reference sample (lower is better). C-DGM and FSG-LTN-GAN are both exactly valid. Only FSG-LTN-GAN recovers the distribution of the constraint margin.

Method	Alchemy	tmQM	Transition1x	Taxi
CTGAN (raw)	$0.596\pm 0.029$	$0.579\pm 0.052$	$0.569\pm 0.057$	$0.419\pm 0.024$
CTGAN + CL	$0.674\pm 0.036$	$0.579\pm 0.052$	$0.592\pm 0.040$	$0.419\pm 0.024$
TVAE (raw)	$0.528\pm 0.012$	$0.517\pm 0.011$	$0.478\pm 0.047$	$0.348\pm 0.029$
TVAE + CL	$0.651\pm 0.015$	$0.517\pm 0.011$	$0.510\pm 0.053$	$0.348\pm 0.029$
C-DGM (CL, MLP)	$1.000\pm 0.000$	$1.000\pm 0.000$	$0.999\pm 0.000$	$0.574\pm 0.005$
FSG-LTN-GAN	$\mathbf{0.040\pm 0.007}$	$\mathbf{0.045\pm 0.010}$	$\mathbf{0.065\pm 0.010}$	$\mathbf{0.099\pm 0.010}$

Appendix

Table 6: Full RQ2 (margin KS by architecture,

n=10

, mean

\pm

std). Raw CTGAN/TVAE validity is below

3\%

on the chemistry sets and

30

46\%

on Transition1x and Taxi. The

+

CL rows, C-DGM, and FSG-LTN-GAN all reach

100\%

validity. Only FSG-LTN-GAN recovers the distribution of the constraint margin across the architectures we test. On tmQM and Taxi the raw and

+

CL margin-KS entries coincide exactly. This is forced rather than copied, because the clamp maps the violating mass to the boundary point mass at

\varepsilon

while the reference margin lies above

\varepsilon

, so the KS supremum, attained at

\varepsilon

, has the same value before and after. Validity and moment error do move. On Alchemy and Transition1x the chained orderings share endpoints, so clamping shifts the evaluated margins and the KS.

Dataset	$\|\Pi\|$	charted	$R_{\max}$	C-DGM	FSG-all	hybrid
faults	4	4	$1.8{\times}10^{3}$	$0.745\pm 0.014$	$0.117\pm 0.008$	$\mathbf{0.117\pm 0.008}$
heloc	7	0	$2$	$0.156\pm 0.012$	$0.208\pm 0.005$	$0.158\pm 0.014$
lcld	4	0	$1$	$0.131\pm 0.008$	$0.657\pm 0.004$	$0.133\pm 0.007$
url	8	1	$1$	$0.159\pm 0.010$	$0.196\pm 0.018$	$\mathbf{0.144\pm 0.021}$
news	5	2	$2$	$0.294\pm 0.008$	$0.318\pm 0.007$	$0.297\pm 0.007$
wids	31	1	$4$	$0.140\pm 0.007$	$0.319\pm 0.004$	$\mathbf{0.133\pm 0.008}$

Appendix

Table 7: Full RQ3 results on the benchmark of 35 ,

n=10

seeds, mean

\pm

std, margin KS. “charted” is the number of variables the hybrid charts as function symbols; bold marks where the hybrid significantly outperforms the constraint layer (paired Wilcoxon

p<0.05

Dataset	Method	Density	Coverage	Margin KS $\downarrow$
Alchemy	C-DGM	$\mathbf{0.923\pm 0.031}$	$0.619\pm 0.014$	$1.000\pm 0.000$
	FSG-LTN-GAN	$0.780\pm 0.034$	$\mathbf{0.636\pm 0.017}$	$\mathbf{0.040\pm 0.007}$
tmQM	C-DGM	$\mathbf{0.839\pm 0.015}$	$\mathbf{0.752\pm 0.009}$	$1.000\pm 0.000$
	FSG-LTN-GAN	$0.801\pm 0.022$	$0.735\pm 0.011$	$\mathbf{0.045\pm 0.010}$
Transition1x	C-DGM	$0.200\pm 0.039$	$0.050\pm 0.005$	$0.999\pm 0.000$
	FSG-LTN-GAN	$\mathbf{0.248\pm 0.030}$	$\mathbf{0.203\pm 0.009}$	$\mathbf{0.065\pm 0.010}$

Appendix

Table 8: Density and coverage ( 27 ) (

n=10

, mean

\pm

std), with margin KS for reference. Higher is better for density and coverage, lower for KS. The better method per column is in bold.

Configuration	Margin KS $\downarrow$
C-DGM, natural order $[0,1,2]$	$0.999\pm 0.000$
C-DGM, apex-first $[1,0,2]$	$0.999\pm 0.001$
C-DGM, apex-first $[1,2,0]$	$0.999\pm 0.001$
C-DGM, apex-last $[0,2,1]$	$0.903\pm 0.049$
FSG-LTN-GAN	$\mathbf{0.065\pm 0.010}$

Appendix

Table 9: Constraint-layer robustness to variable ordering (Transition1x,

n=10

, mean

\pm

std, margin KS). Bracketed lists are the Fourier-Motzkin variable elimination orders.

Dataset	CTGAN $+$ CL	CTGAN in chart	TVAE $+$ CL	TVAE in chart
Alchemy	$0.674\pm 0.036$	$\mathbf{0.105\pm 0.041}$	$0.651\pm 0.015$	$\mathbf{0.127\pm 0.005}$
tmQM	$0.579\pm 0.052$	$\mathbf{0.109\pm 0.061}$	$0.517\pm 0.011$	$\mathbf{0.060\pm 0.014}$
Transition1x	$0.592\pm 0.040$	$\mathbf{0.089\pm 0.019}$	$0.510\pm 0.053$	$\mathbf{0.141\pm 0.037}$
Taxi	$0.419\pm 0.024$	$\mathbf{0.086\pm 0.016}$	$0.348\pm 0.029$	$\mathbf{0.064\pm 0.014}$

Appendix

Table 10: The method transfers to other generator families (margin KS

\downarrow

n{=}10

, mean

\pm

std). “in chart” trains the unmodified backbone on

\varphi^{-1}

-encoded data and decodes through

\varphi

. Both chart columns are exactly

100\%

valid. “

+

CL” columns from Table 2 .

Method	Validity	Margin KS $\downarrow$
unconstrained GAN	$0.000\pm 0.000$	$0.581\pm 0.044$
projection	$0.530\pm 0.068$	$0.541\pm 0.041$
C-DGM (clamp)	$1.000\pm 0.000$	$1.000\pm 0.000$
FSG-LTN-GAN	$1.000\pm 0.000$	$\mathbf{0.086\pm 0.011}$

Appendix

Table 11: Flight records (nycflights13) (

n{=}10

, mean

\pm

std).

R

and the predicted outcome were recorded before training.

Dataset	FSG (D in chart)	FSG-ambient-D	C-DGM (clamp)
Alchemy	$\mathbf{0.040\pm 0.007}$	$0.838\pm 0.154$	$1.000\pm 0.000$
tmQM	$\mathbf{0.045\pm 0.010}$	$0.737\pm 0.339$	$1.000\pm 0.000$
Transition1x	$\mathbf{0.065\pm 0.010}$	$0.267\pm 0.106$	$0.999\pm 0.000$
Taxi	$\mathbf{0.099\pm 0.010}$	$0.340\pm 0.126$	$0.574\pm 0.005$

Appendix

Table 12: Validity by construction versus discriminating in the chart. Margin KS (

n{=}10

, mean

\pm

std). FSG-ambient-D keeps the chart generator (validity exact) but shows the discriminator the decoded ambient sample; C-DGM from Table 5 for reference.

Method	Validity	Margin KS $\downarrow$	Duration KS $\downarrow$
C-DGM, ambient (Table 5 )	$1.000\pm 0.000$	$0.574\pm 0.005$	$1.000\pm 0.000$
margin-form $+$ C-DGM	$1.000\pm 0.000$	$0.134\pm 0.020$	$0.118\pm 0.033$
margin-form $+$ unconstrained GAN	$0.521\pm 0.055$	$0.121\pm 0.015$	$0.111\pm 0.024$
FSG-LTN-GAN	$1.000\pm 0.000$	$\mathbf{0.099\pm 0.010}$	$\mathbf{0.081\pm 0.024}$

Appendix

Table 13: Margin-form preprocessing on Taxi (

n{=}10

, mean

\pm

std). Preprocessing the data into margin form is a manual, single-constraint chart: it recovers the clamp’s margin, though not validity (without a constraint mechanism) and not the untransformed margins. Duration KS is the KS of the high-

R

duration margin alone.

Dataset	Mechanism	Validity	Target err. $\downarrow$	Margin KS $\downarrow$
Alchemy	unconstrained	$0.089\pm 0.050$	$0.067\pm 0.003$	$0.629\pm 0.072$
(target: gap)	penalty	$0.019\pm 0.009$	$0.349\pm 0.091$	$0.624\pm 0.035$
	projection	$0.999\pm 0.001$	$0.067\pm 0.003$	$0.652\pm 0.050$
	chart (FSG)	$\mathbf{1.000\pm 0.000}$	$0.073\pm 0.004$	$\mathbf{0.049\pm 0.008}$
Transition1x	unconstrained	$0.662\pm 0.235$	$0.053\pm 0.004$	$0.477\pm 0.170$
(target: $E_{\mathrm{react}}$ )	penalty	$0.797\pm 0.043$	$0.124\pm 0.009$	$0.851\pm 0.031$

Appendix

Table 14: Conditional inverse design (

n{=}10

seeds, mean

\pm

std, 1500 steps). Target error is the mean absolute deviation of the designable property from its conditioning target (standardized units). Margin KS is as in the main text. Only the chart is simultaneously valid, on target, and realistic in its margins.

Dataset	Paper’s charted set (cells)	KS	Alternative sets (cells)	KS
faults	4 charted (20)	$0.117\pm 0.008$	2 charted (5)	$0.548\pm 0.006$
heloc	none (15)	$0.158\pm 0.014$	1 charted (10)	$0.132\pm 0.008$
lcld	none (20)	$0.133\pm 0.007$	1 charted (5)	$0.134\pm 0.007$
url	1 charted (9)	$0.144\pm 0.021$	none (16)	$0.150\pm 0.011$
news	2 charted (6)	$0.297\pm 0.007$	none or 1 charted (19)	$0.292$ to $0.296$
wids	1 charted (6)	$0.133\pm 0.008$	11 distinct sets (19)	$0.128$ to $0.152$

Appendix

Table 15: Threshold sensitivity on the RQ3 benchmark (margin KS,

n{=}10

, mean

\pm

std). “cells” is how many of the

25

grid cells select each charted set. The paper’s cell is

(\tau_{s},\tau_{d}){=}(0.9,0.2)

Dataset	chance	G-LTN-GAN	D re-weight	D feature aug.	FSG-LTN-GAN
Alchemy	$0.49\pm 0.03$	$0.58\pm 0.02$	$0.51\pm 0.03$	$0.14\pm 0.08$	$\mathbf{1.000\pm 0.000}$
tmQM	$0.53\pm 0.08$	$0.59\pm 0.03$	$0.55\pm 0.08$	$0.29\pm 0.07$	$\mathbf{1.000\pm 0.000}$

Appendix

Table 16: RQ4: predicate placements on the high-

R

orderings (ordering satisfaction, the per-ordering satisfaction fraction averaged over the dataset’s orderings,

n{=}10

, mean

\pm

std). “chance” is a free generator with no constraint mechanism. “G-LTN-GAN” places the predicate in the generator loss. “D re-weight” re-weights each sample’s discriminator loss by its predicate satisfaction. “D feature aug.” appends the predicate value to the discriminator’s input. FSG-LTN-GAN is function-symbol grounding. No placement of the predicate moves the ordering more than

0.09

above chance, and the discriminator-feature placement falls well below it. Function-symbol grounding satisfies the ordering for every sample.

Logic Tensor Networks

See all 9 papers

Aug 11, 2026

sLTN: Structural Logic Tensor Networks

Logic Tensor Networks (LTN) provide a neurosymbolic framework in which first-order logic is interpreted through tensor operations, enabling logical constraints to be integrated with differentiable learning. However, the original formulation of LTN is primarily suited to data represented as flat collections of individuals, and does not explicitly capture structural organization such as temporal order, sequential position, or graph connectivity. We introduce sLTN, an extension of LTN that makes structural dimensions first-class elements of the language. Structural dimensions represent named tensor axes associated with domain-specific organization, such as time steps, sequence positions, or graph nodes. They can be quantified explicitly, related through structural relations, and used to express temporal, sequential, and relational constraints directly at the logical level. We formalize the syntax and fuzzy tensor semantics of sLTN and show that, in the absence of structural dimensions, the framework recovers the original LTN semantics as a special case. We further describe a PyTorch implementation based on a declarative signature, formula parsing, and tensorial interpretation. The framework is illustrated on representative temporal and sequential reasoning examples. This paper serves as a companion to the sltn library, available at https://github.com/logictensornetworks/sltn.

Davide Rinaldi, Luciano Serafini

Nokia Bell Labs · Fondazione Bruno Kessler

Tensor Networks Logic Tensor Networks Structured Reasoning Neuro-Symbolic Reasoning Differentiable Logic

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May 8, 2026

MathConstraint: Automated Generation of Verified Combinatorial Reasoning Instances for LLMs

We introduce MathConstraint, a hard, adaptive benchmark for evaluating the combinatorial reasoning capabilities of LLMs. We combine constraint satisfaction problems with rigorous solver-based verification and design an adaptive generator to create instances that remain challenging as the LLMs improve in their reasoning capabilities. Unlike existing benchmarks that quickly saturate on fixed datasets or use LLM-as-a-judge for checking solutions,MathConstraint uses parameterized problem types that enable scalable generation of arbitrarily difficult and automatically verifiable instances. We release MathConstraint-Easy (

266

instances), on which frontier models achieve between

72.6\%

(gemini-3.1-flash-lite) and

87.6\%

(gpt-5.5) accuracy, and MathConstraint (

329

instances) on which the same models drop to between

18.5\%

(claude-4.6-sonnet) and

66.9\%

(gpt-5.5) accuracy, demonstrating the resilience of our benchmark generator against rapid progress in LLM reasoning capabilities. We evaluate 12 frontier and open-weight models with and without access to a sandboxed Python environment that includes generic SAT/SMT solvers. Tool access roughly doubles frontier accuracy on MathConstraint (mean

+28

pp; up to

+52

pp for claude-4.6-sonnet). Further, halving the tool-call budget from

8

4

rounds erases up to

37

points -- a sensitivity that most single-budget benchmarks miss. We release the generator, dataset, and evaluation harness as a robust environment for studying combinatorial reasoning and tool-use behavior under adversarially-tunable difficulty.

Viresh Pati, Zhengyu Li, Piyush Jha +3

Georgia Institute of Technology, USA · KNOWIDEA Technologies

Mathematical Reasoning Benchmarks Synthetic Benchmark Generation Mathematical Reasoning LLM Tool Use Constraint Programming Mathematical Problem Generation

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Mar 6, 2026

Improved Constrained Generation by Bridging Pretrained Generative Models

Constrained generative modeling is fundamental to applications such as robotic control and autonomous driving, where models must respect physical laws and safety-critical constraints. In real-world settings, these constraints rarely take the form of simple linear inequalities, but instead complex feasible regions that resemble road maps or other structured spatial domains. We propose a constrained generation framework that generates samples directly within such feasible regions while preserving realism. Our method fine-tunes a pretrained generative model to enforce constraints while maintaining generative fidelity. Experimentally, our method exhibits characteristics distinct from existing fine-tuning and training-free constrained baselines, revealing a new compromise between constraint satisfaction and sampling quality.

Xiaoxuan Liang, Saeid Naderiparizi, Yunpeng Liu +2

University of British Columbia · Inverted AI

Constrained Generative Modeling Generative Modeling Constrained Sampling

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Abstract

Logic Tensor Networks

Explore similar work

sLTN: Structural Logic Tensor Networks

MathConstraint: Automated Generation of Verified Combinatorial Reasoning Instances for LLMs

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