T-Norm Operators for EU AI Act Compliance Classification: An Empirical Comparison of Lukasiewicz, Product, and Gödel Semantics in a Neuro-Symbolic Reasoning System

Adam Laabs

cs.AIMar 30, 2026

T-Norm Operators for EU AI Act Compliance Classification: An Empirical Comparison of Lukasiewicz, Product, and Gödel Semantics in a Neuro-Symbolic Reasoning System

Authors: Adam Laabs

Organizations: TriStiX S.L., Altea, Alicante, Spain

Abstract

We present a first comparative pilot study of three t-norm operators -- Lukasiewicz (T_L), Product (T_P), and Gödel (T_G) - as logical conjunction mechanisms in a neuro-symbolic reasoning system for EU AI Act compliance classification. Using the LGGT+ (Logic-Guided Graph Transformers Plus) engine and a benchmark of 1035 annotated AI system descriptions spanning four risk categories (prohibited, high_risk, limited_risk, minimal_risk), we evaluate classification accuracy, false positive and false negative rates, and operator behaviour on ambiguous cases. At n=1035, all three operators differ significantly (McNemar p<0.001). T_G achieves highest accuracy (84.5%) and best borderline recall (85%), but introduces 8 false positives (0.8%) via min-semantics over-classification. T_L and T_P maintain zero false positives, with T_P outperforming T_L (81.2% vs. 78.5%). Our principal findings are: (1) operator choice is secondary to rule base completeness; (2) T_L and T_P maintain zero false positives but miss borderline cases; (3) T_G's min-semantics achieves higher recall at cost of 0.8% false positive rate; (4) a mixed-semantics classifier is the productive next step. We release the LGGT+ core engine (201/201 tests passing) and benchmark dataset (n=1035) under Apache 2.0.

Figures & tables

T-norm	Formula	Dead zone	Gradient
Łukasiewicz $T_{L}$	$\max(0,\,a+b-1)$	$a+b\leq 1$	Zero in dead zone
Product $T_{P}$	$a\cdot b$	None	Smooth: $\partial T/\partial a=b$
Gödel $T_{G}$	$\min(a,b)$	None	Subgradient

Table 1: Properties of the three canonical t-norms. The “dead zone” is the region where

T=0

and gradient is zero — relevant for training, not for deterministic inference.

Legal practitioner’s question	Conjunction intuition	Practical implication
Does the rule require strong, joint confirmation of all conditions?	Strong conjunction	One weaker condition substantially reduces the overall score
Is it sufficient that each condition individually reaches a reasonable threshold?	Bottleneck conjunction (weakest link)	The lowest-scored condition determines the outcome
Are some conditions cumulatively required and others only minimally required?	Hybrid semantics (rule-specific)	Different parts of a rule may warrant different aggregation methods

Table 2: Mapping legal interpretive questions to formal conjunction semantics. The first two rows correspond to

T_{L}

T_{P}

(strong conjunction) and

T_{G}

(bottleneck conjunction) respectively. The third row motivates the mixed-semantics hypothesis (RH4).

System	$T_{L}$ (runtime)	Graph transf.	Proof trail	Legal domain
LTNtorch [ 1 ]	$\checkmark$	$\times$	$\times$	$\times$
IBM/LNN [ 8 ]	$\checkmark$	$\times$	$\times$	$\times$
GNN-QE [ 10 ]	$\times$	$\checkmark$	$\times$	$\times$
SGAT-MS [ 9 ]	$\checkmark$	$\checkmark$	$\times$	$\times$
LGGT+ (ours)	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

Table 3: Comparison of LGGT+ with related neuro-symbolic systems.

Rule ID	Cat.	Conditions	Article
prohibited_rt_biometric	prohib.	real_time_processing ,	Art. 5(1)(h)
		public_space ,
		biometric_identification
prohibited_social_scoring	prohib.	public_authority ,	Art. 5(1)(c)
		evaluates_social_behavior ,
		detrimental_treatment

Table 4: Selected EU AI Act rules formalised in LGGT+. All rules use ALL-conjunction.

T-norm	Overall	Clear	Marginal	Borderline
Łukasiewicz $T_{L}$	78.5%	96.3%	56.9%	25.0%
Product $T_{P}$	81.2%	99.5%	56.9%	35.0%
Gödel $T_{G}$	84.5%	100.0%	54.5%	85.0%

Table 5: Classification accuracy by t-norm operator and case type (

n=1035

T_{G}

achieves 85% on borderline cases (

n=80

) vs. 25%/35% for

T_{L}

T_{P}

. All pairwise differences are statistically significant (

p<0.001

). Bold: highest value per column.

T-norm	FP	FN	FP Rate	FN Rate
$T_{L}$	0	223	0.0%	21.5%
$T_{P}$	0	195	0.0%	18.8%
$T_{G}$	8	152	0.8%	14.7%

Table 6: Error breakdown. Zero false positives is a structural property of multi-condition conjunctive classification at

\theta=0.5

(not an empirical finding): any system requiring all conditions to exceed threshold will default conservatively. The diagnostically relevant metric is false negatives.

Comparison	$b$	$c$	$n$	$p$ -value	Sig. $\alpha=0.05$
$T_{L}$ vs $T_{P}$	0	28	28	$<0.001$	Yes
$T_{L}$ vs $T_{G}$	8	71	79	$<0.001$	Yes
$T_{P}$ vs $T_{G}$	3	43	46	$<0.001$	Yes

Table 7: McNemar’s exact binomial test (

n=1035

). All pairwise comparisons reach statistical significance at

\alpha=0.05

T_{G}

significantly outperforms

T_{L}

(

p<0.001

n=79

discordant pairs).

T_{L}

and

T_{P}

are no longer decision-equivalent at this scale (

p<0.001

, 28 discordant pairs).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

ID	Description (abbrev.)	Expert	Type	$T_{L}$	$T_{G}$
P01	Realtime face recog. in metro	prohibited	clear	$\checkmark$	$\checkmark$
P02	Government social credit system	prohibited	clear	$\checkmark$	$\checkmark$
PM01	Nudge recommender (not subliminal)	limited	marginal	$\times$	$\times$
PM02	Post-event biometric search	high	marginal	$\times$	$\times$
PM03	Private loyalty scoring	limited	marginal	$\times$	$\times$
HR01	AI CV screening system	high	clear	$\checkmark$	$\checkmark$

Appendix

Table 8: Representative benchmark cases with

T_{L}

and

T_{G}

predictions (

\checkmark

correct,

\times

incorrect).

T_{P}

identical to

T_{L}

in all cases shown. HRM04 and HRM05 are the borderline divergence cases.

Fuzzy Logic

Fuzzy logic represents degrees of truth and uncertainty with graded membership functions and rule-based reasoning.

See all 48 papers

Jun 27, 2026

Statistically Indistinguishable, Operationally Distinct: A Formal Barrier for Tabular Foundation Models

Tabular foundation models cannot reason about data produced by running systems without access to the rules that govern them. We make this statement falsifiable. The Operational Turing Test (OTT) constructs pairs of legal and rule-violating database states whose

1

- and

2

-way column-value marginals match to a total variation of

<0.02

; Le~Cam's lemma then bounds any values-only classifier at

\geq0.49

Bayes error. Three values-only baselines (XGBoost, TabICL, TabPFN) hit the bound exactly (accuracy

0.50

, pre-registered two one-sided tests (TOST)

p<0.002

), raw row-level access does not help, exposing relational value consistency closes most of the gap, and only a classifier fed by seven executable rule-derived audits reaches

1.00

classification accuracy. In three matched

100

-state frontier large-language-model (LLM) runs, models given the schema, trigger source, rule tables, and state files classify at most

2/50

legal states as LEGAL; GPT-5.5 accepts

0/50

legal states even with higher reasoning effort and a Structured Query Language (SQL) executor. The access-ladder pattern also appears on a second schema with structurally distinct rule families (banking ledger: cross-row balance, cumulative aggregate). The barrier is identifiability, not capacity: scale, data, and richer features cannot cross it without operational grounding.

Tassilo Klein, Johannes Hoffart

SAP SE

Tabular Foundation Models

Ask PDF

Oct 5, 2026

Rule-Based Languages for Neurosymbolic AI

Logic programming is increasingly used as the symbolic component of neurosymbolic AI systems. We survey the main rule-based languages in this setting, namely Datalog, answer set, and probabilistic logic programs, along four axes: semantics, expressiveness, neural integration, and evaluation mechanism. We analyse over 50 recent systems and applications, comparing formalism usage across four research areas: databases and programming languages, machine learning, vision, and robotics. We provide a decision matrix mapping application scenarios to required features and close by outlining open problems.

Stefania Dumbrava, Efthymia Tsamoura

ENSIIE, SAMOVAR, Télécom SudParis, Inria, IRIF, France · Huawei Labs, United Kingdom

Neuro-Symbolic AI Logic Programming Answer Set Programming Probabilistic Logic

Ask PDF

Mar 19, 2026

Correct Chains, Wrong Answers: Dissociating Reasoning from Output in LLM Logic

LLMs can execute every step of chain-of-thought reasoning correctly and still produce wrong final answers. We introduce the Novel Operator Test, a benchmark that separates operator logic from operator name, enabling rigorous distinction between genuine reasoning and pattern retrieval. By evaluating Boolean operators under unfamiliar names across depths 1-10 on five models (up to 8,100 problems each), we demonstrate a reasoning-output dissociation that existing benchmarks cannot detect. At Claude Sonnet 4's depth 7, all 31 errors have verifiably correct reasoning yet wrong declared answers; 17/19 errors in mixed-operator chains exhibit the same pattern. The benchmark reveals two failure types: strategy failures at depth 2, where models attempt terse retrieval (+62pp from scaffolding), and content failures at depth 7, where models reason fully but err systematically (+8-30pp, 0/300 errors post-intervention). A Trojan operator (XOR's truth table under a novel name) confirms name alone does not gate reasoning (p >= 0.49), while Llama's novelty gap widens to 28pp at depth 8-9 with the Trojan at 92-100%, isolating genuine difficulty with novel logic from name unfamiliarity.

Abinav Rao, Sujan Rachuri, Nikhil Vemuri

LLM Reasoning LLM Evaluation Reasoning Consistency in Language Models Structured CoT Reasoning

Ask PDF

Abstract

Fuzzy Logic

Explore similar work

Statistically Indistinguishable, Operationally Distinct: A Formal Barrier for Tabular Foundation Models

Rule-Based Languages for Neurosymbolic AI

Correct Chains, Wrong Answers: Dissociating Reasoning from Output in LLM Logic