T-norm Selection for Object Detection in Autonomous Driving with Logical Constraints

We thank Reviewer for the thoughtful and thorough critique.

• 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.
  A: This article focuses on the selection of t-norms to guide the learning process, expanding upon the base of MOD-CL. 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. Furthermore, the original paper for MOD-CL does only a shallow exploration of t-norms and their potential use. We have addressed these issues, and have done extensive expansion upon MOD-CL:

  • We expand the set of t-norms that can be used from 3 to 14, including non-parameterised t-norms.
  • We expand the evaluation backbones to the latest YOLO model, YOLO11.
  • We consider two recent and challenging datasets with constraints, ROAD-R and ROAD-Waymo-R.

  MOD-ECL represents not only a conceptual evolution, but also robust and broad empirical research on the integration of constraints into neural networks.

• There is no comparison with SOTA on the given datasets, only with vanilla MOD-YOLO.
  A:

  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. While MOD-YOLO provides the underlying detection backbone, it does not support constraint integration by itself. 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.

  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. 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.

  Broader comparisons with methods outside the constrained loss paradigm, e.g., transformer-based detectors, are outside the scope of this work. These models typically do not natively support structured constraints, and adapting them would involve altering their training phase, which goes beyond our scope.

• The method involves many hyper-parameters, making tuning difficult.
  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. 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.

• Experimental results on t-norm selection are not conclusive; different t-norms suit different phases, questioning whether single t-norm selection is optimal.
  A: Effectively, our results show that no single t-norm is optimal, but depends on the trade-off between accuracy and constraint satisfaction. 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. 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.

• In section 3.1, it would be good to give some examples for (o, l, a).
  A: We will include a concrete example to clarify the tuple structure. For example ({Pedestrian}, {Crosswalk}, {Walking, PushObj}) would be one detection, meaning "The pedestrian is on a crosswalk, and is walking while pushing an object".

  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.

• Equation 2 would be easier to understand if it is supported by real examples of constraints and t-norms.
  A: To clarify the semantics of Equation 2, consider the following example constraint:

  "If a pedestrian is at a crosswalk, they must be walking."

  Using standard CNF, this is encoded as:

  ¬Pedestrian ∨ ¬Crosswalk ∨ Walking
  
  

  Suppose the model assigns the following confidence values:

  • Pedestrian: a = 0.9
  • Crosswalk: b = 0.8
  • Walking: c = 0.7

  Using the Product t-norm, the constraint violation is:

  1 - T_P(a, b, 1-c) = 1 - (a × b × (1-c)) = 1 - (0.9 × 0.8 × 0.3) = 1 - 0.216 = 0.784
  

• What is the significance of the multi-label part of MOD-ECL? Does it need to be multi-label?
  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. 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.)

• Why not use multiple t-norms for inference?
  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).

• Merging section 6 into 5.3 might be more intuitive to the reader.
  A: We agree. We will restructure the sections accordingly in the camera-ready version if accepted to improve clarity and flow.

We thank Reviewer bCQh for the helpful comments and suggestions.

• - No specific question was asked, though the reviewer suggests comparison against (1) current SOTA methods and (2) multi-modal language models.
  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. (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.

  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. To this end, ROAD-R and ROAD-Waymo-R provide requirements that the outputs must satisfy, in the form of logical constraints. 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.

We thank the Reviewer for the comments and insightful feedback.

• 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. The contribution of MOD-ECL framework compared with MOD-CL should be clarified in both experiment results sections and method sections. A: This article focuses on the selection of t-norms to guide the learning process. MOD-CL serves as a basis to explore this, as the implementation already could handle multi object detection and constrained loss. 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.

  The novel conceptual contributions of MOD-ECL are as follows:

  • An adaptive algorithm that automatically guides the search of well performing t-norms.
  • A scheduler for constrained loss regularization.
  • A metric for measuring constraint satisfaction.

  Experimentally, our contributions are as follows:

  • We expand the set of t-norms that can be used from 3 to 14, including non-parameterised t-norms.
  • We expand the evaluation backbones to the latest YOLO model, YOLO11.
  • We consider two recent and challenging datasets with constraints, ROAD-R and ROAD-Waymo-R.

• 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
  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. We stress that at inference time, there is no computational overhead. Please see the reply to Reviewer 0bpq.

• 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
  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.

We thank Reviewer for the feedback and valuable suggestions.

• The reported improvements in mAP scores are modest. 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). This margin is considerable, given that architectural improvements (YOLOv11 vs. v8) typically yield smaller gains (e.g., +1.3 mAP).

  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. Our results show that this goal is achieved, and that in fact performance might be even improved.

• 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?
  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. 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. 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.

• 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.
  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. Our method incurs no additional overhead at inference time, as t-norms are used exclusively during training.

  Summary of Runs:
  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.

  Dataset	Base Model	Config	mAP	MPBV	Train Time (s)	FLOPs (G)	Peak Mem (GB)
  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
  		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
  		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
  	yolov11n	Baseline	61.69 ± 0.8 (CI 2.03)	40.18 ± 3.4 (CI 6.6)	1481 ± 32.58	3.24	77.24 ± 0
  		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
  		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
  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
  		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
  		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
  	yolov11n	Baseline	34.65 ± 0.28 (CI 0.79)	46.32 ± 0.40 (CI 1.12)	16254 ± 33.47	3.25	23.44 ± 0
  		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
  		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

• 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?
  A: While we focus on exponential scheduling in this work, the method is compatible with other schedulers. To illustrate this concretely, consider a linear decay alternative to the exponential schedule currently used. The constraint weight decays linearly from λ_0 to 0 over the remaining training epochs:

  λ_t = λ_0 × (t / T) 
  

  Where:

  • λ_0 is the initial constraint weight,
  • T > 1 is the total number of epochs.

  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.

• 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?
  A: While we support the motivation, we are unaware of any publicly available multilabel medical image dataset annotated with formal constraints. 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. Nonetheless, consider an illustrative and simplified example from a surgical scene understanding task:

  Constraint: “Incision actions require holding a scalpel”
  Translated as:

  ¬(Incision ∧ ¬Scalpel) ≡ ¬Incision ∨ Scalpel
  

  This formulation is structurally analogous to our constraints in the autonomous driving domain.