T-norm Selection for Object Detection in Autonomous Driving with Logical Constraints
Authors: Thomas Eiter
Katsumi Inoue
Nelson Higuera Ruiz
Sota Moriyama
Venue: NeurIPS 2025 poster ()
Keywords: Neurosymbolic
Computer Vision
Fuzzy Logic
Primary area: applications
TL;DR: We implement a framework for learning constraints in vision models and algorithms to support this. We showcase it in Autonomous Driving.

Abstract: Integrating logical constraints into object detection models for autonomous driving (AD) is a promising way to enhance their compliance with rules and thereby increase the safety of the system. T-norms have been utilized to calculate the constrained loss, i.e., the violations of logical constraints as losses. While prior works have statically selected a few t-norms, we conduct an extensive experimental study to identify the most effective choices, as suboptimal t-norms can lead to undesired model behavior. To this end, we present MOD-ECL, a neurosymbolic framework that implements a wide range of t-norms and applies them in an adaptive manner. It includes an algorithm that selects well-performing t-norms during training and a scheduler that regulates the impact of the constrained loss. We evaluate its effectiveness on the ROAD-R and ROAD-Waymo-R datasets for object detection in AD, using attached common-sense constraints. Our results show that careful selection of parameters is crucial for effective constrained loss behavior. Moreover, our framework not only reduces constraint violations but also, in some cases, improves detection performance. Additionally, our methods offer fine-grained control over the trade-off between accuracy and constraint violation.

T-norm Selection for Object Detection in Autonomous Driving with Logical Constraints
Thomas Eiter'
Nelson Higuera Ruiz! Katsumi Inoue
Sota Moriyama
VVienna University of Technology, Austria
NNational Institute of Informatics, Japan
The Graduate University for Advanced Studies, SOKENDAL, Japan (thomas. eiter, nelson. ruiz)atuwienn ac.at, {inoue, sotam) (onottmm@@nil t jc.jp
Abstract
Integrating logical constraints into object detection models for autonomous driving (AD) a promising way to enhance their compliance rules and thus increase the safety of the system. In this, t-norms have been utilized to calculate the constrained loss, i..., the violations of logical constraints as losses. While prior works have statically selected few t-norms, conduct extensive experimental study identify the most effective choices, as suboptimal t-norms can lead to undesired model behavior. For this, present MOD-ECI neurosymbolic framework that implements a wide range of t-norms and can use them in an adaptive manner, with algorithm that selects pelforerformnn t-norms during training and a scheduler that regulates the impact the constrained loss. We evaluate its effectiveness on the ROAD-R and ROAD MADDWW datasets for object detection in AD with attached common-sense constraints. Our results show that careful selection of parameters is crucial for good behavior of the constrained loss and that our framework allows us to obtain not only lower constraint violation but in some cases also an increase in detection performance. Furthermore, our methods allow fine control over the tradeoff between accuracy and violation
Introduction
Object detection models 33 have become essential for autonomous driving (AD) applications Liv due to their efficiency and accuracy (34], and a variety of datasets have been created evaluate these models [2, 18
27 9 However, standard data-driven models do not consider formal symbolic knowl- edge, and they often fail to satisfy logical constraints that reflect real-world rules and relationships between detected objects; for example, in an AD scenario pedestrians usually should not be detected on top of a moving vehicle. The consistent satisfaction of this and similar kinds of constraints crucial for creating safe and reliable autonomous driving systems.
Neurosymbolic methods [5] have emerged as promising solution to this problem. supporting different ways to integrate knowledge into deep learning models, cf. [4, [). An effective approach for autonomous driving to embed logical constraints into loss functions using t-norms (. with the constrained loss 7 as a representative. The latter is aggregated with a regularization term À to the loss functions of neural network, driving the model simultaneously learn satisfying the logical constraints and solving the detection task.
Recent research has considered the inclusion of requirements for AD scenarios expressed logical constraints in datasets 112 15 It appeared that using constrained loss can increase constraint satisfaction as wells model performance of the overall AD system. On the other hand, the results
our code contained an online https:/:/gggtttoo.ooo///aaaaaaap
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(a) w/o Constrained Loss, (b) w/ Constrained Loss, using Product t-norm,
(c) w/ Constrained Loss, using Product t-norm,
(d) w/ Constrained Loss, using Eukasiewicz t-norm, = 50.
i.e.,
50.
1000.
Figure I: Four images showing the inference results after 10 epochs of training the model MOD-ECL (see Section ). with labels associated to bounding boxes above the top-left corner. Bounding boxes in green indicate satisfaction of logical requirements and in red violations. Compared to the baseline in Fig. (regularization term À = 0), we highlight that using constrained loss with product t-norm can yield higher detection performance and lower requirement violations, shown in Fig. [b] However, improper À respectively t-norm selection can severely hinder the learning and constraint satisfaction, as shown in Figs. Tc and 1d
showed that the choice of t-norm and regularization term À may significantly affect constraint satisfaction as well as detection performance, leading to undesired results as in Figure Furthermore, there plethora of t-norms (28), but only few of them have been considered so far. This poses the challenge of finding a best t-norm efficiently, which should achieve high constraint satisfaction while degrading detection performance.
To overcome these issues, we make the following contributions:
First, to explore a possibly large number of t-norms, we introduce an adaptive algorithm to dynamically explore and select the one that reduces some score by the most during training. This adaptive approach also enables the user to measure which t-norms are used the most, and thus to focus on them for their particular application, with asserted theoretical properties.
Secondly, to control the effect of the constrained loss, we introduce a scheduler that dynamically adjusts X during training, attempting to separate stages where the model is focused on learning object detection or on constraint satisfaction, where intuitively, the former should happen in early stages.
Thirdly, we present MOD-ECL, a novel neurosymbolic framework that handles a wide range of t-norms, with functions that automatically search t-norms and allow for parameter tuning in order to optimize model performance, based on adaptive t-norm selection and À scheduling.
Finally, we conduct an extensive experimental evaluation of the approach detection models deployed to two AD datasets augmented with requirements as logical constraints, viz. ROAD-R* an extension of ROAD from the ROAD Challenge" with 243 requirements, and ROAD-Waymo-R, subset of GOAD-Wayme from the ROAD++ Challenge with 251 requirements. In the evaluation, we study the choice of t-norms as well as variation of the regularization A. The results reveal that the choice and variation can considerably affect the outcome. Some t-norms yield higher detection performance while failing to simultaneously increase constraint satisfaction, and vice versa. However, we found that there are certain scenarios where both metrics improve drastically or offer reasonable tradeoff (see Section 5.4 for detailed discussion).
backbone object
Overall, our results advance the integration of logical constraints in practical AD scenarios, providing a framework and tools to assist exploration of different options in controlled manner. Furthermore, they provide insight into the behavior of range of t-norms for this task.
2 Related Work
Neurosymbolic systems consist neural and logical modules that interact with each other to solve
some task 20
An interesting question is how use the logical part to enhance the learning
htttpe//sttes.ooglllcc//////////

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capabilities of the neural one by modifying the loss function. Many systems [32, 19 30 rely on model counting techniques for calculating the semantic loss [I], whose use proved be successful in reducing the amount of data needed to reach level of accuracy and making neural networks comply with a set of constraints. Unfortunately, due to the high complexity of model counting [3], this method has only been fruitfully applied to small scale problems such as arithmetic operations over MNIST [6] or sudoku.
A different approach called Semantic Based Regularization 7 (essentially the constrained loss) bypasses this issue by considering the single world with the highest predictions from the neural network and the use of t-norms to calculate the violations. This allows one to handle the scale of real world problems such as AD where integrating logical constraints is a promising way to improve compliance with formal background knowledge. Diligenti et al. considered the Product, Gödel and Eukasiewicz t-norms in their first approach and experimented on an object detection task [29], where they showed that adding rules can lead to gains in accuracy and F1 score. Their research was continued 10, where they expanded the theory of t-norms to include generators and considered also parameterized t-norms such as the Schweizer-Sklar and Frank t-norms. They showed that by solely using t-norm based learning, their model was able to learn the MNIST dataset with high accuracy. In a further paper MM the authors proposed Deep Fuzzy Logic (DFL), a theoretical framework for constrained learning maintaining the aforementioned t-norms. They compare with various baselines and show that using DFL improves x 5% over neural network over the CiteSeer (25) dataset. While these works developed the idea of integrating t-norms, they fall short of addressing practical real-worle use case such autonomous driving.
In the AD domain, the recent ROAD-R dataset 12 provides a structured framework to evaluate a model's 's ability to comply with logical constraints. The authors showed that existing state-of-the-an models frequently violate these constraints, with more than 90% of predictions being non-admissible, i.e. they violate some constraint. To address this, they proposed incorporating constrained loss, constrained output (filtering compliant predictions only), or both. A follow-up paper (26) addressed the computational challenges of integrating logical constraints via t-norms into deep learning models. The authors proposed a memory--fficient implementation of t-norm- t-norm-based loss functions that reduces the usually prohibitive memory requirements of these methods. They conducted experiments with the ROAD-R dataset to demonstrate that incorporating standard t-norms improves object detection performance, particularly in scenarios with limited labeled data, achieving up to 3.95% improvement when only 20% of labeled data is available. Furthermore, they showed that applying t-norm- based constraints to both labeled and unlabeled data yields further performance gains. The work is continued in ROAD-Waymo-R [15), where follow-up AD dataset with requirements, the experiments showed that using standard t-norms in the constrained loss improves performance over the baseline I3D model 3 The Gödel t-norm offered a performance boost highlighting the potential of domain knowledge integration during training to enhance complex perception tasks.
3 Background
3.1 ROAD-R and ROAD-Waymo-R Datasets
The ROAD-R dataset extends the ROAD dataset by incorporating common sense logical constraints in the AD context as requirements. The original ROAD dataset, built on top of the Oxford RobotCar Dataset [18], contains total of 41 classes and 122K frames from 22 annotated videos, where each of them is about 8 minutes long. The labels are in the format of tuples (o. a) where is an object class, /. is a location class, and a is an action class. For example, the detection (Pedestrian, Crosswalk, Walking, PushObj) represents a pedestrian located on a crosswalk, perform- ing the activities of walking and pushing an object. Note that each component can include multiple labels of the same type, making multi-label classification necessary.
ROAD-R introduces set of 243 logical requirements expressed in propositional logic. These constraints cover relationships between objects, locations and actions; for instance, the statement "vehicle lane cannot be a parking lot" says that two locations are different. The logical constraints disjunctions of literals which we represent as sets of literals, e.g., -VehileLLnne V -ParkingLot as { VehicleLLnnn, -ParkingLot).
The ROAD- Daymo [15] dataset, built on top of the Waymo Open Dataset 21, is considerably more comprehensive and challenging than any existing AD benchmark, including ROAD. It contains 198k
3

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Combé red
Loss
Tenrr choice
cholcee
Model
Surkd Inom
MODYOLO
Average
Sicwesldrr
Figure 2: MOD-ECL framework instantiated for AD with MODYOLO. The model generates predic- tions, which evaluated against a constraint using I-norms. A choice between static and adaptive t-norms can be made, with the latter dynamically adjusting during training. The regularization term (A) can be either static adjusted using a scheduler. The final combined loss incorporates both the neural loss and the constrained loss, balancing predictive accuracy with requirement consistency.
frames from 5 hours of footage and labels are in the same tuple format (o, a) as in ROAD. There 58 classes, and some of them differ from the ones in ROAD. It includes set of 251 constraints, modeled keeping the central common parts with the ROAD-R constraints, removing constraints that do not apply, and adding new ones that apply.
3.2 T-norms
T-norms (triangular norms) 16 are mathematical functions used fuzzy logic generalize the concept of logical conjunction (AND). Given two values a, b E [0, 1], t-norm T combines them to produce value T(a,b) b) that represents their joint degree of truth, obeying monotonicity, i.e., b) < T(a', b') a S a' and b b< b', and is viewed as a binary operator commutative, associative, and has as neutral element. T-norms enable the integration of logical constraints into neural networks, they allow for the relaxation of hard constraints into differentiable forms that can be optimized during training.
While t-norms are defined as binary functions, we sometimes want to apply them to constraints with more than two variables. The presented t-norms can be naturally extended to multiple variables. yet one should be careful when using more complex expressions. We apply t-norm 7 to a set fo = {a1.-
Gn ). n 2, fuzzy values iteratively, i.e., we define T(fv) by T(fu) T(a 02) if n = 2 and T(fv) T[T((( an-1), a,) otherwise; by commutativity and associativity of T. the enumeration fv is immaterial. An assignment of values a constraint can be defined as c/ul la,-, where .1,..., Fn are the variables in c, vI..... Un is a list of values, and J; + v; denotes substitution each variable v; by v;, resulting the modified formula c(v).
3.3 Constrained Loss
is a technique to enforce logical constraints during the training of neural networks. It modifies the traditional loss function by adding penalty term that measures the degree to which the model's predictions violate predefined constraints. This penalty can be expressed as
Constrained loss 7 [2]
Leomb Lan + XCe,,
where Lan is the standard loss (c.g.. cross-entropy), La is the constraint violation loss [12). and balances task performance and constraint satisfaction; Leumb their regularized combination. Considering that negations are computed using strong negation, which is formally defined as 1 given a E [0, 1], we define the constrained loss term inspired by [26)
Ca
E Σ T [c. bj]),
where T' is the t-norm for evaluating constraint satisfaction, C = {cI,..., en the set constraints, and B {bi..... bm.} } is the set of detections, where each bj contains predicted confidence scores per label and by denotes their strong negation.
Example (Constrained Loss Computation). Consider the rule: "Only pedestrians or cyclists can wait to cross" " This can be written in propositional logic as:
Pedestrian V Cyclist - WWai t2x.

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Framework
Datasets
Models
T-norm library
Adaptive T-norm selection
X-scheduler
MOD-CL
1
1
3 t-norms
MOD-ECL
2
2
12 t-norms
Table 1: Key differences between MOD-ECL and the prior MOD-CL framework.
Assume the detector assigns the following confidence values:
Pedestriar an
0.9, Cyclist
0.2, Vait2X 0.7.
Using the Product t-norm, Tp(r, y) = x y, the corresponding constrained loss is: Ta(a, b, c) 0.1 x 0.8 x 0.7 = 0.056
Hence, the model incurs loss of 0.056 for this rule under the current predictions.
4 MOD-ECL
Prior to introducing our novel algorithms, we present in this section our framework MOD-ECI, which stands for Multilabel Object Detection with Enhanced Constrained Loss. is framework for integrating constraint knowledge into multilabel object detection models through constrained loss, leveraging MOD-CL [22]. framework that produced the winning model for Task 2 and the 3rd-best model of Task of the NeurlPS 2023 ROAD-R Challenge y MOD-ECL comprises a collection MOD-ECL- (D, C, T) of dataset D, a multilabel object detection model M, set C' of constraints, and finally a set T of t-norms. It is assumed that M returns, given an input image, a set B of bounding boxes as output, where each bounding box by B can have multiple labels.
MOD-EC ECL integrates the background knowledge of the constraints for the dataset D into the neural network model M using the loss function shown in Equation (T) Moreover, for the calculation of the constrained loss in Equation (2). it uses the t-norms in the set T. The latter may consist of a single t-norm (static t-norm), of multiple t-norms that will be adaptively selected considered below (adaptive t-norm). The pipeline of MOD-ECL for AD is shown in Figure 2
Comparison to MOD-CL. MOD-ECL extends the previous MOD-CL CL framework for constraint- aware object detection through three key innovations, the differences highlighted in Table []] First, it introduces a significantly broader library of t-norms, cxpanding from 3 to 12, including both non-parameterised and parameterised families. Second, it proposes an adaptive t-norm selection mechanism, which frames the problem multi-armed bandit and dynamically identifies well- performing t-norms during training. Third, a novel X-scheduler is introduced to modulate the influence of the constrained loss over time, ensuring that the model initially focuses on detection performance before progressively enforcing constraint satisfaction. These additions result in a robust, modular, and fully automated training pipeline that improves constraint compliance with minimal degradation- occasional improvements in detection accuracy.
4.1 MODYOLO: Multi-label Object Detection with YOLO
In our work, specifically focus an instantiation of MOD-ECL for the ROAD-R and ROAD- Waymo-R datasets, using the constraints given in prior works. In this section, we particularly focus on the object detection model M that is used.
Specifically, we use M MODyaLo from the MOD-CL framework for the ROAD- R Challenge dataset. MODyolo is multilabel extension of YOLO models v8n and 11n [14 Both the training and the inference procedure of the YOLOv8 model were modified: In training. n-hot vector representations of all labels were used as ground truth labels to allow for training on multilabel instances. In inference, Non-Maximum Suppression (NMS) [24] is used in a specific manner: object specific labels were used. In the case of ROAD, each object is given only one agent label; we perform NMS based on agent labels.
https://yolov8.oom com https: htt.::::/ddossulubally......ooo hll.aalyyiicc..ccmmooaaal/////ld
5

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T
Definition
T
Definition
min(a, b),
Tp
a-b
Tis
max (0, (a? + 1))
otherwise
a.
Ta
min(a,6) 6)
Th
p p) (a+ + bb a-
a b
p=1
Ti
max(a + 1 1,
logp 14 (p" -1)(p" -
otherwise
p p-1
1, a 1orb=1
Tp
max 0,1 1 ((1 a)" b))))/1)
0, otherwise
b b
a. abb
a a b p.p.
Tue
a b a 0,
Tsw
otherwise
1+p
Tsm
min(a, b). 0,
a
TeA
exp 1-
otherwise
Table 2: T-norms implemented in MOD-ECL. Symbols correspond to: Tp (Product), Ta (Gödel), Ti. (tukasiewicc)) TD (Drastic), Tup (Hamacher Product), TNm (Nilpotent Minimum), Tss (Schweizer- Sklar), Tn (Hamacher), TF (Frank), Ty (Yager), Tsw (Sugeno-Weber), Txa (Aczél-Alsina).
For t-norm computation, further prune detections: In YOLOv8, models may output numerous detections, many of which may have very low confidence score for each possible class. Such detections are considered to be "background" classifications, and do not want compute violations from them. Thus, when calculating violations, we look at detections that have at least one label with confidence score above 0.5, value that is deemed as leaning towards in fuzzy logic.
4.2 Implemented T-norms
For the instantiation of T. we implement standard t-norms from related literature, namely Tp. TG. and Ti. alongside numerous ones, all shown in Table They are divided into non- non- non-parameterized and parameterized t-norms, where the latter offer a degree of flexibility regarding some parameter p 0. The semanties of t-norms may vary drastically, which motivates the datasets. In addition to Ta. TL, and Tp, we implement three more non- parammetrrizzed t-norms. To is extreme t-norm that returns if either value is I and returns 0 otherwise. THP produces smooth continuous curve, similar to Tp. TNM is a stricter version of Ta that returns whenever the sum of the inputs is at most 1. In addition, parameterized t-norms have parameter p 0 adjust the behavior, offering some degree of control between strict and relaxed conjunctions. Some of them admit also p < we disregard this for simplicity. Depending on p. some parameterized t-norms amount to non- parameterizeee ones; e.g., TH becomes THe whenever = 0 and Tss coincides with Ti for p and Tp with p
empirical evaluation
4.3 Adaptive Algorithm for T-norm Selection
To select the best t-norm 7 E T during training, we use an adaptive algorithm that balances exploration (random selection with probability (8) and exploitation (choosing the t-norm with the highest score). As shown in Algorithm [ the score of each t-norm reflects its recent performance in reducing the constrained loss in Equation (2). which measures the degree to which logical constraints are violated. At each iteration, it either explores by randomly selecting a t-norm with probability 3, exploits by choosing the t-norm with the highest accumulated score (line 5). After selecting t-norm, the algorithm computes its constrained loss (lines 6). If this is the first time the t-norm evaluated, the loss is stored without updating the score. Otherwise, the algorithm computes a normalized improvement score (lines 7-11) based on the reduction in constraint loss; this is scaled either by the previous loss or a small constant e to avoid instability when losses are zero. Finally, the score for the selected t-norm is updated using exponential discounting (lines 12-13), allowing the algorithm to gradually adapt to changing performance trends while preserving historical performance. At the end of the iteration the t-norm with the highest score is returned the best choice (line 14).

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Algorithm Training with Adaptive T-norm Selection
Require: Number of batches per epoch b, epochs e, set of t-norms T, exploration probability B,
discount factor 8, small constant
Ensure: Trained model model
losses, scores < (7:0 VT 7}. iterations b.
2: T" t random. choice(7)
Il Initialize best t-norm
3: for i to iterations
4: model <- train_model(T))
I/ Train on current best t-norm I/ Candidate evaluate
5: T. random.choice(()
if raaddm()) S
arg maXTET scores otherwise
6: de
Soapuutecoontrrintt ( (7ss(T:)
7:
dp f losses [7:). losses T3 t ce
8:
if cl xo then
9:
continue end if
11:
(dp -0- dee //////mxxcca/p clp) s + scores TT
// Normalized improvement
This
8
13: scores [T]]
8 (1 8) n4:
otherwise
14: T f- arg maXTET scores
II Update best I-norm
15:
end for
16: return model
We can prove for Algorithm 1 that every t-norm is selected infinitely often, ensuring sufficient exploration and furthermore Theorem Tunder common assumptions, such as stationary and bounded scores, guarantees that the adaptive t-norm selection regret, i.e., the number of times suboptimal t-norm is chosen is bounded. This comes from results multi armed bandit (MAB) (23) theory.
Theorem (Regret Bound), For any suboptimal t-norm T whose expected discounted normalized gain pr is strictly lower than that of the best t-norm T (i.e., pr < (TT)). the expected nunber of
times T is selected by Algorithm [ satisfies [[]r S logn/ (B AT) where AT pr- pr is the expected discounted performance gap.
In practice, not all assumptions may be fulfilled, but we see in our empirical results that the algorithm stabilizes and chooses one t-norm consistently. Proofs are included in the Appendix.
4.4 Scheduler
After calculating the constrained loss Las we must add it to the neural loss Con and tune the value (cf. (I). A similar approach called warm-up is presented in (26). but the À remains constant. In contrast, we study the effect of increasing À during training. Intuitively, if À is very high, the constraints will be "hard", meaning that the model will try satisfy them at the cost of proper object detection. On the other hand, if À is too low, the constraints become increasingly "soft", meaning the neural network can violate them and may do it often to prioritize learning object detection. To strike a good balance between proper object detection and constraint violation is key to our approach.
We propose scheduling mechanism that dynamically adjusts À during training. The scheduler follows an exponential function that starts at an initial value Ao that increased to a multiple at every epoch. Note that this is in contrast to conventional learning rate schedulers, which usually decrease the learning rate over time. allows the model to first learn the detection without receiving excessive influence, and to focus on constrained losses later in the training. Furthermore, it is beneficial to change A only after a warmup phase. This leads to the following Equation ( for the À scheduler:
X = do if S by else 7 A
Here, Xe is the regularization at epoch and lnp is the number of epochs we do not change the constrained loss; > 1 is the progression rate; for y = 1, we have conventional (static) scheduling.
7

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ROAD-R
KOAD-Waymon
T
f-mAP@0.5
MPBV
MPBV %
3DRN
Yv8n
YIIn
3DRN
Yv8n
Yl
I3D
Yv8n
Y1In
I3D
Yv8n
YlIn
B
33.39
57.55
61.43
40.45
39.52
16.38
34.88
34.33
47.85
46.04
Tp
33.34
61.00
58.75
35.66
36.57
35.15
33.74
43.04
41.88
To
32.16
50.23
55.87
41.75
35.70
17.10
32.54
30.33
34.17
40.48
TL
34,42
59.55
61.74
66.4
67.61
16.19
36.59
35.69
66.81
57.49
HP
57.92
59.43
34.01
35.20
33.04
31.93
39.00
41.36
TNM
54.86
61.53
45.88
38.76
35.32
33.91
46.08
44.18
Tb
50,48
60.08
50.55
36.11
34.33
33.21
46.74
42.36
TF
59.01
60.74
39.36
34.68
35.38
33.71
45.22
42.56
T
53.03
54.39
29.48
29.97
28.79
29.42
47.53
49.27
Tsw
60.87
61.35
60.67
62.21
37.08
35.54
68.27
59.12
TAA
59.41
56.50
37.72
33.91
33.35
32.08
43.24
41.17
T 59.21
59.59
36.46
40.32
36.10
34.13
49.27
44.65
T5S 59.95
60.04
61.32
57.90
35.57
35.89
63.96
49.78
Table 3: Combined accuracy (f-mAP@0.5) and constraint violation (MPBV %) results for individual t-norm training ROAD-R (A 50) and ROAD- MeoAD-Wymo-R (A 100), across evaluation setups: 3D-RetinaNet, I3D (baselines), YOLOv8n or YOLO! In (ours) as vision models for MOD-ECL.
5
Methodology and Experiments
5.1 Methodology
Our methodology evaluates and optimizes t-norm performance given dataset by first assessing multiple t-norms. Using an adaptive algorithm, we automated t-norm selection during training and analyzed the frequency of use for each t-norm. Based this analysis, we identified subset of frequently selected t-norms and further evaluated their effectiveness through individual t-norm training, as shown in Table (in our experiments, we included all t-norms). Subsequently, selected the best-perforiing t-norm in terms of both detection accuracy and violation rate, and tuned the parameter examine its impact detection performance and constraint satisfaction. Finally, scheduler controlled by a parameter y was applied to optimize the tradeoff between detection accuracy and constraint satisfaction.
5.2 Experimental Setup
We conduct our experiments on the available data ROAD-R and ROAD MAAD-Wymo-RR As the ground- truth labels for the test sets provided in ROAD and ROAD ROAD-Waymo videos into training and test sets (details are given the repository). ). The computational resources used specified in the Appendix. We MOD first produce a baseline without constrained loss, and then with each t-norm mentioned in Section individually. We then present results for the adaptive t-norm algorithm varying the parameters B and 6, which represent the probability of random exploration, and the discount factor respectively, We conclude by experimenting on by using extreme values and scheduler for that increases its value at every epoch.
not public, split the given
top of YOLOv8n and YOLO1 We
All results were calculated with frame-mAl with IOU 0.5 (f-mAP@0.5) in 12). The mAP for each score was calculated using the torchmetrics library. Furthermore, we calculated the percentage of violated bounding boxes (sets of labels that violate at least one constraint), which refer as Mean Per Box Violation (MPBV). The best results are shown bold and the worst in gray.
5.3 Experimental Results
The results for our baseline (B) and individual t-norms in Table presented Table 3 The baseline does not use constrained loss, i.e., À 0. For the experiments with t-norms, we used À = 50, 100 for ROAD-R, ROAD- Waymo- respectively it provides favorable tradeoffs. For comparison, we present the state-of-the-ar results given 26 21 ROAD-R (3D-RetinaNet) and ROAD-Waymo-F R (I3D) respectively, marked with superscript and
Individual T-Norms.
8

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Accuracy vs. Regularizatin Woight
NPCV vs. Reguiariiinnn Reguratzation lnation Welgght
100
VOLDxIN BANG WANTTO
VCUGZSN ROAD-
VOLSNES 10AD-WYTTT
ROADE
10
ioquloriation WeightA
Regubrizztion
(a) Accuracy (mAP@0.5) as a function of regulariza- tion weight À (log-scale).
(b) Mean Per Bounding Box Violation (MPBV) as function of regularization weight À (log-scale).
Comparison of accuracy and violation rate across YOLOv8n and YOLOIIn models on ROAD-R and ROAD-WaymoRR.
Figure 3:
Params (8,6)
ROAD-R
MPRV (5)
EmAP200.5 Yvšn
ROAD- Wayuso-R &
MPRV (%)
Params
F-AP@IIS
ROAD-K
MPRV (%)
[-AAP60.5
WOAD--Wymo--
MPRV (%)
Yv8n YiIa
YvSn
Y1In
Yvña
YIIn
Yvlnn
Yvdin
YlIn
B
57.55
61.43
40.45
39.52
34,88
34.33
47.85
46.04
B
57.55
61.43
40.45
39.52
34.88
34.33
47.85
46.04
0.25.
55.10
56.94
33.27
43.41
60.16
35.06
46.19
(0.1. 0.5)
$3.88
56.99
31,75
37.21
33.67
31.37
44.27
45.58
1.25
57.18
57.74
10.36
19.48
35.04
32.84
12.88
31.40
0.225,
57.19
56.95
31.52
39.39
33.15
31.13
42.78
42.13
1.5
58,50
56.17
10.53
0.019
34.27
25.56
37.52
32.41
(0.25. 0.5)
52.28
58.25
32.28
35.00
33.29
31.40
43.15
42.47
1.75
55.00
55.67
0.698
0.006
32.64
28.70
18.98
35.41
(0.5, 0.5)
58.93
57.37
36.34
34,85
33,72
32.28
42.65
2.0
55.18
55.04
0.722
0.005
31.88
21.06
45.13
43.06
(a) Adaptive algorithm using X = (50, 100) with pa- rameters (8, 8).
(b) À scheduler effect using λo 100) and vary-
ing 7
Table 4: Adaptive algorithm and A-scheduler performance across ROAD-R and ROAD-Waymon
Adaptive Algorithm. The results for our adaptive algorithm are in Table 4a for both datasets. We experimented on Tp with exploration values 3 = {0.1, 0.25, 0.5} and discount constant ó = {0.25, 0.5). The percentual frequency each t-norm was used is presented in Figure 4 À (50, 100) was used for ROAD-R and ROAD- Waymo- respectively.
Varying X. We select just one t-norm to reduce the number of experiments, namely Tp. with [0, 1,5, 10, 50, 100, 500, 1000, 5000). Results are shown in Figure 3
Scheduler. The results of using our scheduler are shown in Table 4b We used Tp with an initial Ao (50, 100) for ROAD-R and KOAD-Waymos respectively, with warm-up tw = 3 and = [1.1, 1 25, 1.5, 1.75, 2.0).
5.4
Discussion of Results
Our results show that the integration of logical constraints through constrained loss leads to significant improvements requirement satisfaction, while in many cases maintaining or even enhancing detection performance. Looking at Table several t-norms, including Tsw: T3s and Ti in most cases achieved higher f-mAP@0.5 than the baseline without constrained loss, suggesting that enforcing logical consistency does not necessarily hinder object detection performance. Focusing on the t-norms Tp. Ta and Ti., we can see an interesting relationship between them. Tp presents the most balanced behavior overall, consistently achieving good tradeoffs for accuracy and reducing violations. Ta focuses on reducing violations over detection performance. Last, Tř works conversely by gaining accuracy at the cost of more violations. Interestingly, I& reduces the violations by the most in the case of ROAD-R. We believe this is because the formula allows the calculation of very low values of violations. For ROAD-Waymo-H this is not the case, where it actually hinders both accuracy and compliance. This shows the difficulties of deciding which t-norms work in a specific setting, indicating the need for a comprehensive framework such ours for evaluating each t-norm.
Beyond static t-norm evaluation, the adaptive algorithm reveals t-norm preferences across training configurations. Table pressents the results for our adaptive algorithm using Tp. For all combinations of (8, 8), we get lower violations than the baseline, which was intended by the decision of using the constrained loss the score. While this comes the consequence of decreased accuracy, the

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ROAD-R: Trorm Usage Trequency
25 162
C1 1.25
Usage (5)
CL. STWdE
CL. 1L337
SYNdS
05. 3.31YU
ROAD-Weymo R: T-norm Usage Frequency
CL. 1.317
Uvage hespancy
62%
10
P72
507'
Tmom
Figure 4: Frequency of t-norm selection by the adaptive algorithm under different (3, 6) configura-
tions. Top: ROAD-R R dataset; Bottom: ROAD- A---Wayo--R R dataset.
reduction in violation is usually higher than the accuracy reduction, depending on the configuration. Using some configurations may reduce the violations more the cost of more accuracy, revealing a tradeoff when using the adaptive algorithm. Figure 4 visualizes the frequency of each t-norm's s selection under varying (3, 5) values. On ROAD-R and ROAD-Waymo-l 7 is dominant, especially under conservative exploration settings (e.g.. 3 = 0.1, = 0.25), being selected over 40% of the time. This is supported by previous results using but not for ROAD-Waymo-R, where we see 73 having a detrimental effect on constraint satisfaction in the static case. These results indicate that while in the static case, some t-norms appear better than others, it is not the case throughout the entire training process. Different t-norms may work better at different stages of the training, shown by having a better tradeoff between violation and accuracy than T in the static case.
fixed manner. This makes sense for ROAD-R
Our study of the regularization term À reveals that its effect is non-linear, shown in Figure As increases, violation rates decrease significantly, but accuracy begins to plateau or decline beyond certain thresholds. For instance, increasing À from 50 to 500 reduces MPBV but leads diminishing accuracy gains. This confirms that excessive emphasis on constraints may limit detection capability, and that À must be tuned the desired balance. To address this, we proposed a scheduler that gradually increases À during training. The results show that moderate growth rates (e.g.. 7 1.25) reduce violations while preserving accuracy, supporting the hypothesis that applying constrained loss is most beneficial in later stages of training. Table 4b shows that for every configuration, the violations are always lower than the bascline has the mildest effect, one can increase the value to trade accuracy for compliance. This is expected as higher values of magnify the effect of the constrained loss. However, we should be careful of excessively increase it, as in cases such as ROAD- ROAD-Waymo-H Waymo-R, the compliance worsens with higher values, e.g. 7
low cost of accuracy. Starting from ? 1.1 which
Conclusion and Future Work
We presented the MOD-ECL framework for object detection in AD that enforces logical constraints using t-norms, along with novel algorithms for adaptive t-norm selection and constrained loss regu- larization. The results for challenging AD datasets show that this approach can significantly reduce logical violations as well as enhance detection performance. Furthermore, adaptive t-norm selection and dynamic regularization optimizes the balance between detection and constraint satisfaction.
Future work will explore the potential of leveraging more powerful models within the YOLO family and longer training periods for enhanced detection performance. Additionally, investigating different scoring mechanisms for the adaptive algorithm and experimenting with alternative functions for the scheduler could further refine the balance between detection accuracy and requirement adherence. Finally, deployed on MOD-ECL, our methodology enabled a structured exploration and effective application of t-norms in AD with constraints. While did not consider other scenarios (e.g., healthcare), we suspect that this approach is also beneficial in them, which remains to be explored.
10

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Acknowledgments
This work has been supported by JSPS KAKENHI Grant Number JP25K03 190, JST CREST Grant Number JPMJCR22D3 and JST SPRING Grant Number JPMJSP2104, Japan. This research was funded in whole or part by the Austrian Science Fund (FWF) 10.55776/C0E12. We also thank the JASEC-NII collaboration program.
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The abstract and introduction clearly state the main contributions, including the design of MOD-ECL, the adaptive t-norm selection algorithm, and the À scheduler. These are substantiated in Sections 4 and 5 with experimental validation on ROAD-R and ROAD- D---aaoo-RR datasets.
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: Assumptions and theoretical results for the adaptive algorithm are stated with theorem in Section 4.3. second theorem relegate the appendix. Both proofs are in the appendix.
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The experiments use the ROAD-R and ROAD-Waymo Waymo-R datasets (see Section 3.1). Dataset splits and evaluation protocols are provided, and links to dataset descriptions are included.
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A Appendix
This appendix contains supplementary material not present in the main body of the document. includes a link containing the logs of our experiments, instructions on how to setup and reproduce our results, additional results and implementation details of the t-norms used.
A.1 Evaluation platform
We used two computational nodes for the experiments to handle the large number of experiments: node, used for ROAD-R, had 4 NVIDIA RTX A5000 GPUs, Intel Xeon Silver 4314 CPU, and a total of 1024GB RAM; the other node, used for ROAD- WaL--W----,,, was equipped with NVIDIA RTX A6000 GPUs, AMD Ryzen Threadripper PRO 5995WX CPU with 64 cores, and total of 256GB RAM.
A.2
Experimental Logs
All the resulting logs of our experiments can be found via the following link: https://drin: www/fileeo//////occcococoeeoaaae balee/dd1PP http:///aaaaaaao00/////////11/11
A.3
Setup and Experimental Reproduction
The instructions can be found in the repository of our submission repository: https: ////n/mmnnnnsssssssm
4open. aopen.sciinn wwcciinnee//eeeiipppiieeseceeeee
B
Additional Tables and Graphs
In this section, we provide additional qualitative results showing how the use of constrained loss influences both the accuracy predictions and the satisfaction of logical constraints. These visual- izations support the findings discussed in Sections 5 5.4 illustrating how the combination of t-norm selection and directly impacts both the model's object detection accuracy and its compliance to logical constraints.
Table contains the values that were graphed in Figure 3 As À increases, models better satisfy constrants (lower MPBV) but at the cost of detection accuracy (f-mAP@0.5). YOLOv8n shows a smoother degradation than YOLO1 In. A balance point occurs around À = 25-100, after which accuracy suffers significantly. T shows the percentual frequencies that were graphed in Figure 4 Across both ROAD-R and ROAD- WAAA-WWym--,, the adaptive algorithm favors the Yager t-norm (T)), consistently selecting it more than any other. In contrast, TD and THP are rarely chosen for ROAD-R and ROAD- DWWamooR respectively, suggesting they are generally less effective for their constraint- driven training dataset.
The expanded t-norm set and adaptive algorithm may introduce additional computational costs. To quantify these, we provide a comparative breakdown of GPU memory usage, FLOPs, total training time, and per-epoch wall-clock time relative to the baseline and previously published constrained-los methods. Our method incurs no additional overhead at inference time, as t-norms are used exclusively during training. Table 7 summarizess mean values standard deviation) and 95% confidence intervals (CI) for mAP and MPBV over three seeds (0, 1, 2) for each configuration. Exceptions are adaptive YOLOv8n and YOLO1 In on ROAD-Wayd -WAa--, where the former was run with seed 0 only and the latter with seeds 0 and 1. Also reported are total training time, FLOOPs, and peak GPU memory usage. should be noted that there are no additional costs incurred at inference time.
The following figures graph mAP@0.5 against MPBV. Here it easy to divide the figure into four subquadrants defined by the position of the baseline: (1) top-left represents the experiments that increased accuracy and reduced violations (best case), (2) top-right represents the experiments that increased accuracy and violations, (3) bottom-left represents the ones that decreased accuracy and violations (trade-off cases), and (4) bottom-right those that decreased accuracy and increased violations (worst case). Figure 5 shows graphs of all model and dataset combinations using static t-norms. Please observe that the faded triangles connecting the t-norms Tp. Ta and Ti. are in similar positions for each model and dataset, as mentioned in Section 5.44 of the main text. Figures and (nd show every configuration of the adaptive algorithm and the À scheduler, respectively. The faded line
21

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that goes through the baseline has slope 1. All points to the left of the line have positive trade-off, ie., we get a bigger improvement violation compared f-mAP@0.5.
ROAD-R
ROAD- ROAD-Waymo-R
A
f-mAP@0.5
MPBV (%)
f-mAP@0.5
MPBV (%)
Yv8n
Y11n
Yv8n
Y11m
Yv8n
Y11n
Yv8n
Y11n
57.55
61.43
41.16
39.52
34.88
34.30
47.85
46.04
58.09
60.40
41.16
40.92
34.80
34.20
47.89
46.31
5
56.72
59.45
40.40
39.17
34.54
34.20
46.33
44.89
10
55.01
58.26
39.63
36.18
34.78
34.40
46.77
46.23
25
60.33
58.44
43.40
37.28
34.82
34.33
46.72
45.36
50
61.00
58.75
35.66
36.57
34.89
34.20
46.66
44.48
100
54.53
60.02
41.12
32.85
35.15
33.70
43.04
41.88
250
57.74
59.05
9.25
16.61
33.72
31.49
41.83
40.86
500
56.32
57.44
2.39
0.38
31.33
27.80
40.62
39.84
1000
56.89
56.84
1.38
0.24
27.11
24.20
19.89
51.03
2500
55.97
55.72
1.19
0.21
24.89
20.64
16.64
54.04
5000
54.45
53.85
1.00
0.18
22.62
14.70
13.38
57.04
Table 5: Comparison of detection accuracy (f-mAP@0.5) and constraint violation (MPBV) for YOLOv8n and YOLO1 across ROAD-R and ROAD BOAD--Wymo-R, with different values and Tp t-norm.
Parsans (P, 6)
TG
TF
TV
TA
T-2
Yvšn Ylla Yvšn Yl1a
Ylin
YvRo Y1la Yvšo Y11a Yvša Yl1n
Yvšn
Yvšn Ylla YvšI
(.1. .25
5.7
5.6
1.9
5.1
4.9
3.4
6.0
4.6
43.4
4.2
8.0
5.6
4.8
4,9
3.5
3.7
(.1.
7.5
64
6.2
12, 225)
7.1
8.5
6.8
7.1
6.5
3.6
3.2
6.8
7.2
8.4
8.1
7.3
26.9
23.1
6.1
6.7
7.8
7.
6.4
7.4
6.0
5.8
(225,.5)
3.4
7.5
8.5
6.6
6.7
5.8
2.5
3.2
6.7
6.0
9.4
7.3
72
6.8
23.0
31.4
6.6
6.0
7.2
7.1
7.0
6,8
5.9
4.9
(.1,-25)
88
30
1.5
1.5
4.5
10.3
733
(51,
11.5
9.8
78
8.2
7.6
3.6
2.2
19
4.0
34
1.1
1.0
5.8
48
11.6
19.4
31.7
8.7
5.5
4.4
10.6
8.5
128
9.0
(,26, .25)
8.3
7.0
7.5
6.9
74
7.1
5.9
5.0
5.9
5.2
3.6
3.9
6.7
22.7
32.3
8.6
7.0
6.7
6.0
822
6.8
844
6.7
(.25,.5)
(.6
78
8.1
7.5
6.0
8.1
7.8
7.7
7.6
Table 6: Adaptive algorithm t-norm usage frequency (%) for YOLOv8n (Yv8n) and YOLO1 (Ylln) across different 8, 6 parameter settings in ROAD-R (top) and ROAD- ROAD-Waymo-t Waymo-R (bottom).
Most used values bolded; least grayed.
Model
Config
mAP
MPBV
Train Time (s)
FLOPs (G)
Peak Mem (GB)
Baseline
(CI
4,12
71.38
3.28
v8n
Product (A = 50)
60.77 1: (C1)
36.56
1.9)
2027 1 30.72
4.12
73.25 0
Adaptive (8 0.5, y 0.5)
59.02 0.8 (C1 2.544
36.29 a 0.9 (C1 1.5)
2800 4: $7.93
4.12
73.25 1: 0
Baseline
(C0))
3.4
77.24
11n
Product (A - 50)
58.06 + (CI 2.50)
36.38 + (CI).)
2319 + 24.27
3.24
77.24 +
Adaptive (6 0.5. 0.5)
57.91 0.9 2.42)
31.67 (C1)
3374 :: 26.89
3.24
77.21 4 0
Baseline Product 100)
34.96 0:27 (CI 0.78)
17.19 (C1 2.48)
15874 70.35
4.12
23.04 +: D
(C30)
23.01
Adaptive (3 0.5, 7 0.5)
33.51
0.000
45.58 0.00 (CI 0.00)
24503 1: 0.00
4.12
23.04 1:
Baseline
34.65 -: 28 (CI 0.79)
46.32
00 (CI 1.12)
16254 33.47
3.25
23.44 D
100)
101
74.67
Adaptive (3 0.5, 7 0.5)
32.91 0.42 1.17)
43.02 0.53 (CI 1.50)
25063 4: 1356.45
3.25
23.44 +:: D
Table 7: Computational overhead comparison of static and adaptive t-norm selection. Top: ROAD-R results. Bottom: ROAD- M-----mm--RR results. Mean + standard deviation and 95% confidence intervals (CI) are shown for mAP and MPBV over three seeds unless otherwise noted. No additional inference- time cost is incurred.
22

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ROAD-R Yv8n
ROAD-R Y11n
XT+
XT
62
XT.
60
XTX
X
XL
X
f-AA@0.5
58
T
fmAP@0.5
59
Xhi
Xb
XT
56
XTим
54
5/
XT4
X
52
56
XTG
50
XTs
XTi
30
35
40
50
55
65
30
35
45
50
60
65
MPBV %
MPBV %
(a) Accuracy against MPBV on ROAD-R using
(b) Accuracy against MPBV on ROAD-R using
YOLOv8n.
YOLO1In.
ROAD-WR Yv8n
ROAD WR Y11n
36
<Tix T4N
XT4
35
XI
36
XTR
f-mAP@0.5
XTe
EmAP@0
5
34
XOBT!
Xlrr
XL
33
XTr
XTG
32
32
31
30
XTa
30
XTG
X
40
50
40.0
42.5
45.0
47.5
50.0
55.0
57.5 60.0
MPBV
MPBV %
(c) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOv8n.
(d) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOlln.
Figure 5: Evaluation of static t-norm comparisons across datasets. The faded triangle connects Tp. To and Tr.
23

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ROAD-R Yvθn
ROAD-R Y11n
59
(0.5.
61
10.25,
f-mAp@0.5
fmAP@0...
55
0.25)
(0:25, 9.51
54
0..1.0.5)
X
10.5. U.51
X 10.25,
57
X
(0.1, 0.25)
(0.1. 0.5)
(0.25, 0.25)
34
36
38
40
36
37
38
39
MPBV %
MPBV %
(a) Accuracy against MPBV on ROAD-R using
(b) Accuracy against MPBV on ROAD-R using
YOLOv8n.
YOLO1In.
ROAD-WR Yv8n
ROAD-WR Y11n
34.5
34.75
33.5
EmAP@0.5
34.25
33.0
34.00
32.5
X
0.55
X
10.5, 0.5)
X
10.1, 0.51
33.50
31.5
10225,
(0.10.5)
33.25
X
(0.25< 0 10.1, 0.25)
31.0
X
(0.25, 025)
X
10.25,
30.5
45
46
47
48
42.0
42.5
43.0
43.5 44.0 44.5 45.0 45.5 46.0 MPBV %
MPBV
(c) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOv8n.
(d) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOlln.
Figure 6: Evaluation of the adaptive algorithm across datasets. The faded line that goes through the baseline has slope 1.
24

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ROAD-R Yv8n
ROAD-R Y11n
XL
60
61
f-AA@0.5
X1.5
fmAP@0
X1.25
56
X1.5
XL.75
55
2.9
X2.0
35
10
15
20
25
MPBV %
MPBV %
(a) Accuracy against MPBV on ROAD-R using
(b) Accuracy against MPBV on ROAD-R using
YOLOv8n.
YOLO1
ROAD-WR Yv8n
ROAD WR Y11n
XL.1
35.0
X1.25
1.25
1.1
EmAP@0.5
34.0
mAAP@O
X
33.5
33.0
X1.75
32.5
24
X2.0
34
38
42
45
MPBV %
MPBV
(c) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOv8n.
(d) Accuracy against MPBV on ROAD-Waymo-R us- ing YOLOlln.
Figure 7: Evaluation of the scheduler across datasets. The faded line that goes through the baseline has slope
25

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C Theoretical Guarantees
Theorem 1. Let (r(n) denote the number of times TT is selected up to iteration For any suboptimal t-norm T whose expected discounted normalized gain po is strictly lower than that of the best t-norm T (ie. pr < prr)) ) the expected number of times T is selected satisfies the following upper bound:
log
E[Nr(n)) S
A.
where AT prr pr is the expected discounted performance gap.
Proof. Let T be the optimal t-norm with the highest expected normalized gain:
PT E[na
T; 7)
and pT- max PT TET
Let T be any suboptimal t-norm such that Ar PT- PT > 0.
We consider the structure of the score update:
scores[[] + 8 scores + (1 8)
This is form discounted average which places more weight on recent normalized improvements. Define:
Σ j=1
(1--)66-- nn.
as the effective score of after i updates, where niy is the normalized gain at time y when T was chosen.
Since n.. are bounded (due normalization by e or clp), and the gain distributions are stationary (or slowly drifting), standard concentration inequalities (e.g.. Sauma-Hoeffding for martingale differences) imply that with high probability, sf" will dominate sf after sufficient number of iterations.
Because greedy selections use arg maxy scores [T), the probability selecting a suboptimal T due to exploitation decays exponentially over time. However, exploration ensures that T is still selected occasionally.
Thus, the algorithm favors the optimal T in the long run while bounding the frequency of suboptimal t-norm use, concluding the proof.
Theorem 2. Let 'T be finite set of t-norms. Suppose the constraint loss for each T E T varies stochastically over time and let n, denote the normalized improvement at iteration as computed by Algorithm 7 Then, under fixed exploration rate B >> O and discount factor 8 E (0, 1), the adaptive I-norm selection algorithm ensures that each T E T selected infinitely often with probability 1 as the number of iterations n + Co.
Proof. At each iteration i, the algorithm chooses t-oor T. either randomly with probability 3 or greedily according to the highest current score. Because T finite and /3 is constant, each T E T has non-zero probability of being selected each iteration, where K |T
8 K
P = T)
Thus, by the Borel- Borel-Cantell Cantelli lemma and standard results in stochastic processes, each sampled infinitely often with probability oc. This ensures that each score estimate receives updates infinitely often.
26

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D Implementation of t-norms using PyTorch
We make use of the PyTorch python module efficiently implement t-norms processed in batches. For tensor T of shape (d1,
slicing it column-wiss along the last dimension:
T :.......: i E (O....,d, -
This selects all elements across the first dimensions for fixed i in the last dimension. Consider all input values are PyTorch tensors. f'v = T the full set of bounding boxes predicted and scores associated to the labels. Otherwise, we slice the input tensor and operate iteratively, which is the case when we have a, as input, representing tensor slices be operated with the iterative algorithm described on Section
Listing 1: Minimum t-norm
def ten_ttorr_teesor (fv): min_value, torch min (fv, axis--1) return min_value
Listing 2: Product t-norm
def product_tnores tensor (fv): product_value torch prod axis=-1) return product_valus value
Listing 3: Lukasiewicz t-norm
def snnecevvcc__toor_teeeeee (fv):
sum value torch. sum (fv, axis--1)
return torch, (slumsuuvvauue fv size 1)
Listing Drastic t-norm
def trastic_tnorn_batt (a, b):
cone_one (a == 1) (b == crastic_value torch. where (none_one, tin b), torch. zeros (oros_like(a))
return drastic_value
Listing 5: Nilpotent minimum t-norm
def nilpotentmin, tnormtnommmm.mmmmm betrm_batcche (a, b): nilpotent_valle value torch, where beros_like return silpotent_value
1, tir b), torch.
Listing 6: Hamacher product t-norm
def hanacherprod_tnorm_ub batchh_batch(a (a, b):
denominator torch. clamp (a
a b, min=1e-6, max=1)
hamacher_ value torch. where (a + b == 0, torch zeroh. lires_like(a), (a b) denominator)
return hamacher_value
Listing 7: Yager t-norm
def www.ttonoomtthhhaeeee (a, b, p=2):
if p == 1: return lukasiewicz,tnoran (a, b)
value torch. relu ((1 - a) (1 b) p)) ** (1 / p)
return yager_valle value
27

---

Listing 8: Frank t-norm
def frank_tnorm (a, b, p-2):
if P == 1: return product_tnoras (a, b)
numerator (a**p 1) (b**p-
denominator -
frank_value wog_with_ base numerator / denominator, p) return tranke
Listing Sugeno-
Mngeno-Weber Weber t-norm
def sugeno_weber. www.weeeeerreeeeeeeeeeeee (a, b, p=1):
augeno_ value torch (torch.. wtt.aeeeesss_ai like (a P b) (1 p)) return sugeno_value
Listing 10: Aczel-Alsina t-norm
def aczel_alsime hat.aaalssaattoombbthhhhhh b, p-2):
if p -- 0: return trastic.tnorm_batchhe b)
>= 1e-6) (b 1e-6) a, b torch clamp (a, min=1e-6), torch. clamp min=1e-6)
none_zere
exponent ((torch, abs (-torch log(a)) b)) p) (1 p))
torch, abs (-torch. log
aczel_vale value torch. where (none_ zero, torch exp (-exponent), torch. zeros la))
return aczel_value
Listing 11: Hamacher t-norm
def hamacher _trommmttoommmoo (a, b, p=2):
denominator P (1 p) (a denominator
b)
hamacher_value return hamacher_valus value
Listing 12: Schweizer-Sklar t-noom
def schveizer_skles sswarrrrsskkar...ee tnorm (a b, p): a+*p b+op 1
tnorm_value torch.rell melu(inee__vauue ** (1 / p)
return tnorn_value
28


Reviews:
[{"review_id": "SFAENm944s", "reviewer": "Reviewer_qbpq", "summary": "The paper proposes MOD-ECL, a neurosymbolic framework that enhances multi-label object-detection networks for autonomous driving by incorporating logical-constraint supervision through three key innovations: a broadened library of t-norms (standard, non-parameterized, and parameterized) for translating first-order rules into differentiable losses, an adaptive t-norm selection algorithm that treats the choice as a multi-armed bandit problem with discounted reward updates, and a λ-scheduler that gradually increases the weight of constrained loss to prioritize object detection learning before rule compliance. Experiments on ROAD-R and ROAD-Waymo-R benchmarks using YOLOv8n/11n backbones demonstrate that careful t-norm and λ selection can reduce mean per-box violations by up to ~30% and occasionally improve frame-mAP by a few points compared to unconstrained baselines, with the adaptive algorithm favoring parameterized Yager t-norms while the scheduler provides controllable trade-offs between accuracy and compliance.", "questions": "- Could the authors run at least 3–5 repetitions using different seeds and report mean ± standard deviation or 95% confidence intervals for mAP and MPBV?\n\n- The expanded t-norm set and adaptive algorithm may introduce additional computational costs. Please quantify the GPU memory usage, FLOPs, and wall-clock time overhead compared to baseline and previously published constrained-loss methods.\n\n- Currently, the exponential λ-scheduler is only compared against a static λ. Could alternative schedules (e.g., linear, cosine) or an automatically tuned final λ value be explored, and the sensitivity of the trade-off curve discussed?\n\n- The paper hypothesizes potential applicability to healthcare and other domains (§7). Could a preliminary experiment with multilabel medical images incorporating simple anatomical constraints be included to demonstrate transferability?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- The formulation of the constrained loss, adaptive t-norm selection, and λ-scheduler is well-defined, and an upper-bound regret theorem is provided.\n\n- The experiments comprehensively cover 14 different t-norms, two datasets, and two distinct model sizes.\n\n- The paper's structure follows a logical progression (background → method → experiments), enhancing readability and comprehension.\n\n- Figures effectively illustrate the trade-offs between constraint violation and accuracy (Fig. 1, Fig. 3), while the inclusion of pseudocode significantly aids reproducibility.\n\nWeaknesses\n\n-  The statistical rigor of the experiments is limited. All results presented are from single experimental runs, without reporting confidence intervals or statistical significance tests (as noted in Checklist Q7). Additionally, the memory and computational time overhead associated with the expanded set of t-norms has not been quantified.\n\n-  The reported improvements in mAP scores are modest.", "was_revised": true, "final_justification": "Thank you for all the responses and comments. I will keep my final score the same", "rebuttal": "We thank Reviewer for the feedback and valuable suggestions.\n\n- **The reported improvements in mAP scores are modest.**\n  **A:** When comparing our model's mAP scores to existing baselines (the first column of each set in Table 2), we see that the improvements are significant. In some cases, the performance nearly doubles the best existing work, indicating the strength of our framework in general. Moreover, when comparing to our given baselines, our t-norm implemented models also have increase in performance (+3.45 for the biggest difference).\n  This margin is considerable, given that architectural improvements (YOLOv11 vs. v8) typically yield smaller gains (e.g., +1.3 mAP).\n\n  However, we emphasise that our main goal is not to increase performance in terms of mAP sorce or similar parameters, but to improve the compliance of the model with the semantic constraints, without significant cost in such performance.\n  Our results show that this goal is achieved, and that in fact performance might be even improved.\n\n- **Could the authors run at least 3–5 repetitions using different seeds and report mean ± standard deviation or 95% confidence intervals for mAP and MPBV?**  \n  **A:** Due to the significant computational cost of full-scale evaluation, repeating all experiments is infeasible in short time. However, we have repeated a representative subset of key configurations (baseline, static product t-norm and adaptive) three times and now report the mean, standard deviation, and 95% confidence intervals. Unfortunately, due to time and resource constraints, we were only able to execute the adaptive configuration once with YOLOv8n, and twice with YOLO11n.\n  The results support the consistency of our findings: variance is generally low, with standard deviations typically below 1.0 for mAP and around 1.0–1.3 for MPBV. For instance, on ROAD-R with yolov8n, the adaptive setup achieved 59.02 ± 0.8 mAP and 36.29 ± 0.9 MPBV, compared to the original reported values of 58.93 mAP and 36.34 MPBV in Table 3a.\n  This suggests that both detection performance and constraint satisfaction remain stable across different seeds. Moreover, the observed improvement of up to +3.45 mAP between baseline and constrained configurations (cf. 56.34->60.77 for yolov8n on ROAD-R) is larger than the gap between model sizes (e.g. YOLO11 baseline vs. next-best, +1.3), reinforcing that the gains are meaningful rather than noise. \n\n\n- **The expanded t-norm set and adaptive algorithm may introduce additional computational costs. Please quantify the GPU memory usage, FLOPs, and wall-clock time overhead compared to baseline and previously published constrained-loss methods.**  \n  **A:** We now provide a comparative breakdown of computational overheads introduced by adaptive and static t-norm selection. These include GPU memory usage, FLOPs, total training time, and per-epoch wall-clock time.\n  Our method incurs no additional overhead at inference time, as t-norms are used exclusively during training.\n\n  **Summary of Runs:**  \n  *Mean ± standard deviation and 95% confidence intervals for mAP and MPBV over three seeds (0,1,2) for each configuration, with the exception of adaptive YOLOv8n and YOLO11n on ROAD-Waymo-R, where the former is only run with seed 0 and the latter with seeds 0,1. Also reported are total training time, FLOPs, and peak GPU memory usage.*\n\n    | **Dataset**     | **Base Model** | **Config**  | **mAP** | **MPBV** | **Train Time (s)** | **FLOPs (G)** | **Peak Mem (GB)** |\n    |-----------------|----------------|-------------|--------:|---------:|-------------------:|---------------:|------------------:|\n    | ROAD-R          | yolov8n        | Baseline               | 56.34 ± 0.6 (CI 1.98) | 41.18 ± 1.3 (CI 2.0) | 1176 ± 31.96 | 4.12 | 71.38 ± 3.28 |\n    |                 |                | Product (λ=50)         | 60.77 ± 1.0 (CI 2.61) | 36.56 ± 1.2 (CI 1.9) | 2027 ± 30.72 | 4.12 | 73.25 ± 0    |\n    |                 |                | Adaptive (β=0.5,γ=0.5) | 59.02 ± 0.8 (CI 2.54) | 36.29 ± 0.9 (CI 1.5) | 2800 ± 87.93 | 4.12 | 73.25 ± 0    |\n    |                 | yolov11n       | Baseline               | 61.69 ± 0.8 (CI 2.03) | 40.18 ± 3.4 (CI 6.6) | 1481 ± 32.58 | 3.24 | 77.24 ± 0 |\n    |                 |                | Product (λ=50)         | 58.06 ± 1.0 (CI 2.50) | 36.38 ± 2.2 (CI 5.1) | 2319 ± 24.27 | 3.24 | 77.24 ± 0 |\n    |                 |                | Adaptive (β=0.5,γ=0.5) | 57.94 ± 0.9 (CI 2.42) | 34.67 ± 1.4 (CI 3.6) | 3374 ± 26.89 | 3.24 | 77.24 ± 0 |\n    | ROAD-Waymo-R    | yolov8n        | Baseline               | 34.96 ± 0.27 (CI 0.78) | 47.19 ± 0.88 (CI 2.48) | 15874 ± 70.35   | 4.12 | 23.04 ± 0 |\n    |                 |                | Product (λ=100)        | 35.12 ± 0.05 (CI 0.14) | 43.63 ± 0.81 (CI 2.30) | 21258 ± 233.18  | 4.12 | 23.04 ± 0 |\n    |                 |                | Adaptive (β=0.5,γ=0.5) | 33.51 ± 0 (CI 0.00)    | 45.59 ± 0 (CI 0.00)    | 24903 ± 0.00    | 4.12 | 23.04 ± 0 |\n    |                 | yolov11n       | Baseline               | 34.65 ± 0.28 (CI 0.79) | 46.32 ± 0.40 (CI 1.12) | 16254 ± 33.47   | 3.25 | 23.44 ± 0 |\n    |                 |                | Product (λ=100)        | 33.64 ± 0.08 (CI 0.24) | 42.99 ± 1.01 (CI 2.84) | 20157 ± 74.87   | 3.25 | 23.44 ± 0 |\n    |                 |                | Adaptive (β=0.5,γ=0.5) | 32.91 ± 0.42 (CI 1.17) | 43.02 ± 0.53 (CI 1.50) | 25983 ± 1356.45 | 3.25 | 23.44 ± 0 |\n\n                            \n- **Currently, the exponential λ-scheduler is only compared against a static λ. Could alternative schedules (e.g., linear, cosine) or an automatically tuned final λ value be explored, and the sensitivity of the trade-off curve discussed?**  \n    **A:** While we focus on exponential scheduling in this work, the method is compatible with other schedulers.\n    To illustrate this concretely, consider a linear decay alternative to the exponential schedule currently used.\n    The constraint weight decays linearly from `λ_0` to `0` over the remaining training epochs:\n    \n    \n    ```\n    λ_t = λ_0 × (t / T) \n    ```\n    \n    Where:\n    - `λ_0` is the initial constraint weight,\n    - `T > 1` is the total number of epochs.\n    \n   Our implementation is able to support such schedulers, and we will clarify citing linear, cosine schedules along adaptive λ-tuning as promising directions for future work.\n\n- **The paper hypothesises potential applicability to healthcare and other domains (S7). Could a preliminary experiment with multilabel medical images incorporating simple anatomical constraints be included to demonstrate transferability?**  \n  **A:** While we support the motivation, we are unaware of any publicly available multilabel medical image dataset annotated with formal constraints.\n  Constructing such a dataset would require meticulous crafting of both label structure and constraint definitions, which is a non-trivial undertaking and would demand significant time and domain expertise.\n  Nonetheless, consider an illustrative and simplified example from a surgical scene understanding task:  \n  > *Constraint:* “Incision actions require holding a scalpel”  \n  Translated as:  \n  ```\n  ¬(Incision ∧ ¬Scalpel) ≡ ¬Incision ∨ Scalpel\n  ```  \n  This formulation is structurally analogous to our constraints in the autonomous driving domain."}, {"review_id": "RESC4PAuYA", "reviewer": "Reviewer_g3uF", "summary": "Since T-norm selection is a critical factor in detection systems with logical constraints to ensure safety, this paper explores the relationship between T-norm selection and model performance. To address the risk of suboptimal or unstable T-norm choices, the authors conduct a systematic search and analysis for T-norm selection. Building on these findings, they propose the MOD-ECL neuromyotonic framework. This framework dynamically and adaptively selects high-performing T-norms and regulate constrained loss function during training.", "questions": "1.What is the computational cost associated with the t-norm selection process? It would be beneficial to report experimental results comparing training time or epoch-level cost with and without t-norm selection.\n\n2.What are the optimal results achieved for subset-based t-norm selection discussed in section 5.1? A more detailed analysis of performance across different t-norm subsets would strengthen the validity of the proposed approach.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n\n1.This paper is well-written and presents a novel insight on guiding t-norm selection for object detection with logical constraints.\n\n2.Extensive experiments on the ROAD-R and ROAD-Waymo-R datasets support the effectiveness and robustness of the proposed t-norm selection.\n\n3.The release of well-documented open-source code and logs enhances the reproducibility and transparency of the work.\n\n**Weaknesses**:\n\n1.There are some little typos, like the mod-ec in Line 156, and the organization of appendix materials. \n\n2.The presentation of paper could be better. For instance, it is hard to follow the difference between the proposed framework MOD-ECL and based framework MOD-CL. Meanwhile, although 5.1 propose a methodology, it is not explained how a subset of t-norm selection chooser and scheduler choose.\n\n3.The contribution of MOD-ECL framework compared with MOD-CL should be clarified in both experiment results sections and method sections.", "was_revised": true, "final_justification": "1.This paper is well-written and presents a novel insight on guiding t-norm selection for object detection with logical constraints.\n\n2.Extensive experiments on the ROAD-R and ROAD-Waymo-R datasets support the effectiveness and robustness of the proposed t-norm selection.\n\n3.The release of well-documented open-source code and logs enhances the reproducibility and transparency of the work.\n\nSince the authors effectively address my concerns, i will maintain my positive evaluation.", "rebuttal": "We thank the Reviewer for the comments and insightful feedback.\n\n- **The presentation of paper could be better. For instance, it is hard to follow the difference between the proposed framework MOD-ECL and based framework MOD-CL.\nThe contribution of MOD-ECL framework compared with MOD-CL should be clarified in both experiment results sections and method sections.**\n **A:** This article focuses on the selection of t-norms to guide the learning process.\n  MOD-CL serves as a basis to explore this, as the implementation already could handle multi object detection and constrained loss.\n  However, the original MOD-CL paper neither experiments with nor provides a detailed discussion on the impact of different t-norm choices or the role of regularisation techniques.\n\n  The novel conceptual contributions of MOD-ECL are as follows:\n  - An adaptive algorithm that automatically guides the search of well performing t-norms.\n  - A scheduler for constrained loss regularization.\n  - A metric for measuring constraint satisfaction.\n\n  Experimentally, our contributions are as follows:\n  - We expand the set of t-norms that can be used from 3 to 14, including non-parameterised t-norms.\n  - We expand the evaluation backbones to the latest YOLO model, YOLO11.\n  - We consider two recent and challenging datasets with constraints, ROAD-R and ROAD-Waymo-R.\n  \n- **What is the computational cost associated with the t-norm selection process? It would be beneficial to report experimental results comparing training time or epoch-level cost with and without t-norm selection**  \n  **A:** We have repeated a representative subset of key configurations (baseline, static product t-norm and adaptive) three times and now report the mean, standard deviation, and 95% confidence intervals, alongside wall-clock time, FLOPs and peak GPU memory usage.\n  We stress that at inference time, there is no computational overhead. \n  Please see the reply to Reviewer 0bpq.\n\n- **What are the optimal results achieved for subset-based t-norm selection discussed in section 5.1? A more detailed analysis of performance across different t-norm subsets would strengthen the validity of the proposed approach**  \n  **A:**  Although we acknowledge the importance of analysing the performance of different t-norm subsets, in this work we deliberately evaluate using the full set of 14 t-norms to avoid introducing selection bias and to ensure a fair comparison across configurations. Exhaustively testing all possible subsets would be computationally infeasible and statistically ambiguous due to the combinatorial explosion (16383 subsets). Moreover, selecting a few specific subsets bears the risk of introducing arbitrary choices that may not generalise or align with domain-agnostic evaluation. Instead, our aim in this work is to assess the effectiveness of adaptive selection across the complete space of standard t-norms. That said, the framework is designed to support arbitrary t-norm subsets and domain-specific prior selection, which we leave as a direction for future targeted studies."}, {"review_id": "ah5oEf9gSC", "reviewer": "Reviewer_bCQh", "summary": "This paper addresses the challenge of integrating logical constraints into object detection models for autonomous driving to enhance safety. It argues that prior work relies on a static selection of t-norms to formulate a \"constrained loss,\" which can be suboptimal. To solve this, the authors propose MOD-ECL, a neuro-symbolic framework featuring two main contributions: an adaptive algorithm that dynamically selects the most effective t-norm during training, and a scheduler that adjusts the weight of the constrained loss. Through extensive experiments on the ROAD-R and ROAD-Waymo-R datasets, the paper demonstrates that this dynamic approach significantly reduces constraint violations, and in some cases, also improves detection accuracy, enabling fine-grained control over the trade-off between performance and rule compliance.", "questions": "See weaknesses.", "limitations": "yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\nThe paper addresses a significant and practical problem in autonomous driving: ensuring object detection models adhere to real-world logical rules to improve safety. The use of intuitive examples (e.g., a pedestrian on a moving vehicle) effectively highlights the importance of the work. The paper proposes a novel adaptive algorithm for dynamically selecting the best-performing t-norm during training, which is a clever application of multi-armed bandit principles. The paper introduces a λ-scheduler to control the influence of the constrained loss over time, based on the sensible intuition that the model should first learn detection and then focus on constraint satisfaction. The proposed MOD-ECL framework is presented as a clear and modular system for integrating logical constraints. The diagrams and descriptions make the architecture easy to understand.\n\n### Weaknesses\n\nThe paper's proposed t-norm method and logic-constrained loss are only compared within the YOLO framework, which demonstrates that different t-norms and λ values indeed influence performance. However, it might be more compelling to compare them against current state-of-the-art (SOTA) methods.\n\nAdditionally, multi-modal large language models have shown remarkable performance in scene understanding and reasoning tasks, with a growing number of works extending these models for object detection tasks. It may be beneficial to include comparisons against these approaches to more effectively highlight the advantages of the proposed methods.", "was_revised": true, "final_justification": "This article has done solid work in the field of constraint-aware multi-label detection.", "rebuttal": "We thank Reviewer bCQh for the helpful comments and suggestions.\n\n- **- No specific question was asked, though the reviewer suggests comparison against (1) current SOTA methods and (2) multi-modal language models.**  \n  **A:** As for (1), MOD-CL is the SOTA for constraint-aware multi-label detection on the ROAD and ROAD-Waymo-R datasets. We also compare against strong competitors such as 3D-RetinaNet and I3D, cited in the original ROAD benchmark.\n  (2) Comparing our framework with larger-scale foundation models or vision transformers is interesting, but was beyond the scope of this work due to their lack of direct support for constrained loss-based learning, as well as multi-label object detection in general.\n\n  In general, object detection models built for autonomous driving train directly on the data without any knowledge of real life constraints. While prior research has shown that their performance for accomplishing the task of object detection is very high, we also observe that these models can and will output predictions that contradict known common-sense rules.\n  To this end, ROAD-R and ROAD-Waymo-R provide requirements that the outputs must satisfy, in the form of logical constraints.\n  Our framework offers seamless integration of these constraints, guiding the training to comply with them, in order to achieve a safer model compared to alternative methods."}, {"review_id": "vAXPjt6MxY", "reviewer": "Reviewer_DvBY", "summary": "The main contributions of the paper are as follows:\n1. T-norm selection: The authors device an algorithm which dynamically selects the best t-norm based on a score. This t-norm selection lets the model explore different t-norms during training and the score helps the user understand why a certain t-norm was selected.\n2. Dynamic scheduler: The model learns to satisfy the t-norms by adding a constrained loss term on top of the object detection loss. To help the model balance both losses, the authors add a scheduler which weighs the constrained loss. Based on the scheduler, the authors hope that the model will learn to do object detection in the early stages of training and then focus on satisfying the t-norms by increasing the weight of the constrained loss term in the later parts of training.\n3. MOD-ECL: A model based on MOD-YOLO, with the addition of adaptive t-norm selection and dynamically scheduled constrained loss to optimize model performance.", "questions": "1. In section 3.1, it would be good to give some examples for (o, l, a). \n2. Equation 2 would be easier to understand if it is supported by real examples of constrains and t-norms.\n3. What is the significance of the multi-label part of MOD-ECL? Does it need to be multi-label?\n4. Why not use multiple t-norms for inference?\n5. Merging section 6 into 5.3 might be more intuitive to the reader.", "limitations": "yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The adaptive t-norm selection provides a good balance of exploration vs exploitations, giving the model a good chance at choosing the right t-norm for the given scenario.\n2. The authors have conducted thorough experimentation and have provided good insights into their finding.\n3. The experiments show clear improvement compared to the baseline which is the same model without the constrained loss.\n4. Good mathematical support provided by the theoretical guarantees.\n\nWeakness:\n1. In terms of novelty, the adaptive t-norm selection is the main novel item, while the constrained loss or the scheduler concepts are not that novel in general in deep learning. MOD-ECL is a novel framework, but the core model part is MOD-YOLO which is not the novelty of this paper. The addition of the adaptive t-norm is the main change to MOD-YOLO.\n2. There is no comparison with SOTA on the given datasets, which might help support or deny the need for the motivation of adaptive t-norm selection. The only comparison is with vanilla MOD-YOLO.\n3. There are a lot of hyper-parameters, making the tuning part the main difficulty in using this method.\n4. The experimental results are not very conclusive on the t-norm selection itself. As pointed by the authors, different t-norms suite different timelines in the training, raising a question, whether a single t norm selection is the right way to go.", "was_revised": true, "final_justification": "Thank you for the final comments. I will keep my final score the same and the experiments and novel aspects of the paper are well justified.", "rebuttal": "We thank Reviewer for the thoughtful and thorough critique.\n\n- **In terms of novelty, the adaptive t-norm selection is the main novel item, while the constrained loss or the scheduler concepts are not that novel in general in deep learning. MOD-ECL is a novel framework, but the core model part is MOD-YOLO which is not the novelty of this paper. The addition of the adaptive t-norm is the main change to MOD-YOLO.**  \n  **A:** This article focuses on the selection of t-norms to guide the learning process, expanding upon the base of MOD-CL.\n  As the reviewer has stated, adaptive t-norm selection is one key contribution of our work. However, we believe that while the concept of constrained loss and scheduler are not particularly new, the combination of the two is novel.\n  Furthermore, the original paper for MOD-CL does only a shallow exploration of t-norms and their potential use.\n  We have addressed these issues, and have done extensive expansion upon MOD-CL:\n  - We expand the set of t-norms that can be used from 3 to 14, including non-parameterised t-norms.\n  - We expand the evaluation backbones to the latest YOLO model, YOLO11.\n  - We consider two recent and challenging datasets with constraints, ROAD-R and ROAD-Waymo-R.\n\n  MOD-ECL represents not only a conceptual evolution, but also robust and broad empirical research on the integration of constraints into neural networks.\n\n- **There is no comparison with SOTA on the given datasets, only with vanilla MOD-YOLO.**  \n  **A:**\n\n  In the context of constrained autonomous driving, MOD-CL is recognised as the state-of-the-art framework for integrating logical requirements into object detection, having won Task 2 of the ROAD benchmark challenge. It extends MOD-YOLO, a multi-label object detection architecture, by incorporating constraint-based loss to enforce semantic compliance with domain rules.\n  While MOD-YOLO provides the underlying detection backbone, it does not support constraint integration by itself.\n  MOD-ECL builds directly on this foundation, enhancing it with adaptive t-norm selection and λ scheduling to improve both constraint satisfaction and detection performance.\n\n  Additionally, our paper compares not only against MOD-ECL baselines that do not use constrained loss, but also against other constrained SOTA methods such as 3D-RetinaNet and I3D (cf. Table 2), as reported in the original ROAD and ROAD-Waymo literature.\n  We stress that the focus of this work is on the integration of logical constraints during training via constrained loss. Accordingly, our experiments are conducted on the constrained variants of the datasets, ROAD-R and ROAD-Waymo-R, which were specifically designed to evaluate such methods.\n\n  Broader comparisons with methods outside the constrained loss paradigm, e.g., transformer-based detectors, are outside the scope of this work.\n  These models typically do not natively support structured constraints, and adapting them would involve altering their training phase, which goes beyond our scope.\n\n- **The method involves many hyper-parameters, making tuning difficult.**  \n  **A:** This is precisely what motivates our adaptive approach. By automatically selecting appropriate t-norms during training, the method reduces the need for manual tuning, and helps discover potentially useful configurations.\n  We also perform a study of the parameters in our article, which becomes an indicator towards what t-norms may be of use for training using constrained loss in the autonomous driving setting.\n\n- **Experimental results on t-norm selection are not conclusive; different t-norms suit different phases, questioning whether single t-norm selection is optimal.**  \n  **A:** Effectively, our results show that no single t-norm is optimal, but depends on the trade-off between accuracy and constraint satisfaction.\n  For example, Gödel minimises violation count while severely reducing mAP, Łukasiewicz favours mAP, disregarding constraint satisfaction, while Product strikes a balance between the two. See Appendix B for a visualisation. \n  This underpins the usefulness of our adaptive algorithm, as it can automatically guide the search of well performing t-norms with respect to some metric.\n  \n- **In section 3.1, it would be good to give some examples for (o, l, a).**  \n  **A:** We will include a concrete example to clarify the tuple structure.\n  For example `({Pedestrian}, {Crosswalk}, {Walking, PushObj})` would be one detection, meaning \"The pedestrian is on a crosswalk, and is walking while pushing an object\".\n  \n  It should be noted that each of o, l, a, can be a conjunction of labels, represented via a subset as in the example, as the datasets we use allow multiple of the same type of labels to be present at the same time.\n\n- **Equation 2 would be easier to understand if it is supported by real examples of constraints and t-norms.**  \n  **A:** To clarify the semantics of Equation 2, consider the following example constraint:\n\n  > *\"If a pedestrian is at a crosswalk, they must be walking.\"*\n\n  Using standard CNF, this is encoded as:\n\n  ```\n  ¬Pedestrian ∨ ¬Crosswalk ∨ Walking\n  \n  ```\n\n  Suppose the model assigns the following confidence values:\n  - Pedestrian: `a = 0.9`\n  - Crosswalk: `b = 0.8`\n  - Walking: `c = 0.7`\n\n  Using the Product t-norm, the constraint violation is:\n\n  ```\n  1 - T_P(a, b, 1-c) = 1 - (a × b × (1-c)) = 1 - (0.9 × 0.8 × 0.3) = 1 - 0.216 = 0.784\n  ```\n\n- **What is the significance of the multi-label part of MOD-ECL? Does it need to be multi-label?**  \n  **A:** Yes. Each bounding box in ROAD-R and ROAD-Waymo-R is associated with multiple labels (e.g., object class, location, and action), necessitating a multi-label object detector. This is because one detection can contain multiple labels of the same class type, as shown in the example for (o, l, a) above.\n  We can not use single label classification, even for each class type, hence the need for multi-object detection. (Encoding technically all subsets of labels as the value domain of an auxiliary class will lead to an exponential blowup and is not feasible.)\n\n- **Why not use multiple t-norms for inference?**  \n  **A:** The t-norms are used only during training, for the computation of the constrained loss, but not for inference (which is done by the neural network).\n\n- **Merging section 6 into 5.3 might be more intuitive to the reader.**  \n  **A:** We agree. We will restructure the sections accordingly in the camera-ready version if accepted to improve clarity and flow."}]

Author rebuttal:

Meta-review: The paper proposes MOD-ECL, a neurosymbolic framework for constraint-aware multi-label object detection in autonomous driving. MOD-ECL extends prior work (MOD-CL) by: (1) Expanding the set of t-norms from 3 to 14, including parameterized and non-parameterized variants. (2) Introducing an adaptive t-norm selection algorithm, framed as a multi-armed bandit, to dynamically guide training. (3) Adding a λ-scheduler to regulate the influence of constraint loss during different training stages.

Experiments on the ROAD-R and ROAD-Waymo-R benchmarks with YOLOv8/11 backbones show that MOD-ECL reduces mean per-box violations (MPBV) by up to ~30% without inference overhead, and occasionally improves detection accuracy (frame-mAP). The approach provides a principled way to balance accuracy and rule compliance, with theoretical guarantees (upper-bound regret for adaptive selection).

Despite borderline ratings (4× borderline accept), the paper demonstrates solid technical depth, clear motivation, and valuable contributions to constrained object detection. The adaptive t-norm selection and λ-scheduler represent a meaningful step toward safer, constraint-aware perception models. The rebuttal period resolved major concerns on statistical rigor and computational overhead, with reviewers acknowledging the additional experiments and clarifications. All four reviewers maintained borderline-accept ratings, with final justifications leaning positive. Consensus emerged that the work is technically solid and contributes to safer object detection, warranting acceptance.
Decision: Accept (poster)