Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI
Authors: Luca Andolfi
Eleonora Giunchiglia
Venue: NeurIPS 2025 poster ()
Keywords: neurosymbolic AI
reasoning shortcuts
Primary area: deep_learning
TL;DR: We introduce Prototypical Neurosymbolic models to satisfy the symbolic constraints while learning the correct unsupervised concepts.

Abstract: Neurosymbolic AI is growing in popularity thanks to its ability to combine neural perception and symbolic reasoning in end-to-end trainable models. However, recent findings reveal these are prone to shortcut reasoning, i.e., to learning unindented concepts--or neural predicates--which exploit spurious correlations to satisfy the symbolic constraints. In this paper, we address reasoning shortcuts at their root cause and we introduce Prototypical Neurosymbolic architectures. These models are able to satisfy the symbolic constraints (be right) because they have learnt the correct basic concepts (for the right reasons) and not because of spurious correlations, even in extremely low data regimes. Leveraging the theory of prototypical learning, we demonstrate that we can effectively avoid reasoning shortcuts by training the models to satisfy the background knowledge while taking into account the similarity of the input with respect to the handful of labelled datapoints. We extensively validate our approach on the recently proposed rsbench benchmark suite in a variety of settings and tasks with very scarce supervision: we show significant improvements in learning the right concepts both in synthetic tasks (MNIST-EvenOdd and Kand-Logic) and real-world, high-stake ones (BDD-OIA). Our findings pave the way to prototype grounding as an effective, annotation-efficient strategy for safe and reliable neurosymbolic learning.

Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI
Luca Andolfi
Eleonora Giunchiglia Imperial College London
Imperial College London, Sapienza University of Rome a2ndo1 ac..ul uk
e. g guunchiglia@@impriia..
Abstract
Neurosymbolic is growing in popularity thanks to its ability to combine neural perception and symbolic reasoning in end-to-end trainable models. However, recent findings reveal these are prone to shortcut reasoning, i.e., learning unindented concepts or neural predicates-which which exploit spurious correlations to satisfy the symbolic constraints.
this paper, we address reasoning shortcuts their root cause and we introduce Prototypical Neurosymbolic architectures. These models are able to satisfy symbolic constraints (be right) because they have learnt the correct basic concepts (for the right reasons) and not because of spurious correla- tions, even in extremely low data regimes. Leveraging the theory of prototypicas learning, demonstrate that we effectively avoid reasoning shortcuts by train- ing the models to satisfy the background knowledge while taking into account the similarity of the input with respect to the handful of labelled datapoints. We exten- sively validate our approach on the recently proposed rsbench benchmark suite show significant im-
variety of settings and tasks with very scarce supervision: provements in learning the right concepts both in synthetic tasks MIIST-EEVnnDO and Kand-Logic) and real-world, high-stake ones (BDD-OIA). Our findings pave the way to prototype grounding as an effective, annotation-efficient strategy for safe and reliable neurosymbolic learning.
1 Introduction
Neurosymbolic [36, 12] (NeSy) holds the promise of being able to combine the high adaptability
of neural networks with the reasoning abilities of more traditional AL. This results in methods that are more interpretable [1 1, 4, 8]. able to Iearn from fewer labelled datapoints [2, 49, 35] and are compliant by-design with the given constraints [1, 18]. However, recent studies [28, 29] have shown that even state-of-the-aar NeSy methods can fall prey of reasoning shortcuts, i.e., (intuitively) spurious associations among the learnt concepts a.k.a. the neural predicates [2h]-that satisfy the given
symbolic constraints and yet violate the intended semantics.
Some attempts have already been made to address the problem, however existing mitigation strategies carry practical trade-offs. For example, the solution proposed in [29] involves dense annotation of high number of datapoints which can be costly. On the other hand, unsupervised solutions like training jointly with a reconstruction loss [29] or the Shannon entropy [27] work only in limited settings, but fail when applied in more challenging scenarios like those in rs re-bench [5].
In this paper, we address reasoning shortcuts their root cause and we introduce Prototypical Neurosymbolic architectures. Thanks to the prototypes, our mitigation strategy requires minimal an- notations and delivers extremely positive results even in challenging scenarios like those in rs bench.. The work is based on the simple intuition that whenever we are updating the weights of our neural
'The code is available
r4rr Github page
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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network, we need to take into account two possibly orthogonal factors: (i) the satisfaction of the background knowledge and adherence the available ground truth labels, which NeSy methods naturally do, and (ii) the similarity of the input with respect to the handful of labelled datapoints. We show that if prototypical networks are used in conjunction with NeSy methods, then both factors are taken into account at each weights update step. On the contrary, when using NeSy methods in conjunction with standard neural networks, this cannot happen. We also show that under the assumption of clusterability in the embedding space, the number of deterministic shortcuts gets significantly reduced.
We extensively validate our approach on the recently proposed rsbench [5] benchmark suite in a variety of settings and tasks with very scarce supervision: we show significant improvements in learning the right concepts both in synthetic tasks (MNIST- world, high-stake ones (BDD-QIA). As an example, with just one labelled datapoint for each concept, we are able to increase the F1-score macro over the concepts for the standard DeepProbL.og [26] from 4% to 96% in the
and Kand-Logic) and real-
task and from 25% to 94% on the Kand-Logic task.
Problem Statement
Notation. As customary, we denote scalar constants in lower-case æ, random variables X in upper case, and ordered sets of constants x and random variables X in bold typeface. We also will denote n| as [ni] and the prior knowledge as K.
the set [n]
Setting. In line with previous work [28, 29] we consider an underlying ground-truth data generation process p" (X, Y; K) having form as illustrated in Figure I, where (i) X is the ordered set random variables with domain X representing the datapoints in our learning problem, (ii) Y is the ordered set of r discrete random variables with domain y representing the ground truth labels in our learning problem, (iii) G is the ordered set of & discrete random variables ranging in G = [ha] x ... x [hx]]. which influence the observed datapoint and determine the ground truth label, and (iv) S is finite ordered set of random variables which influence the observed data but not the ground truth labels.
S
G
X
Y
A NeSy predictor is model that infers the labels Y by reasoning over the background knowledge K and a set of discrete concepts C (taking values in C with the assumption that C = 9) extracted from the sub-symbolic input X. The background knowledge is assumed to be given beforehand.
Figure 1: Ground truth generation pro- cess (in black).
[26]) this classic problem, the pair of MNIST images X 2 and their ground truth label 5 is determined by the ground truth concepts g (3, 2). The writing style s only influences the appearance of those digits. The ground truth concepts and the labels are compliant with the background knowledge K (Y G G2).
Example (MNIST-Adde)
Goal. Our goal is to learn the parameters 0 of a NeSy predictor pa (Y) X;K) such that p" (G x). for all E X.
The reasoning shorteut problem. The reasoning shortcut problem arises because we do not have access to the ground truth G but only to proxy of it, i.c., Y.
Let D = yx)) be finite dataset, the we say that NeSy predictor po takes a reasoning shortcut (RS) if po achieves maximal log-likelihood L on the training data but it does not match the ground truth distribution, i.e.,
L(ve, D, K) max C(per, D) and (C X) + (" X), gre
where C(pe, D, K) = 1 istic optimum for L.
DD Exxy))Dlogg (y x; K). Observe every RS corresponds to a determin-
Example 2 (Ex. 1, Cont'd.) Consider a restriction of the dataset where the ground truth for all datapoints is either g (0,6) or g' (). Assume PC G) is a deterministic distribution
so that po (5, 5) (2,8)) R 1.0 and p3, (0,6)) R 1.0. NeSy predictor po whose (i)
distribution (C G) is as shown before and (ii) likelihood L(pe, D, K) is maximal, takes RS.
2

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3
Prototypical Neurosymbolic AI
Our goal is to learn the parameters 0 of NeSy predictor pY X; K) such that po x) P"(G x) for all x E X' However, in absence of any labelled datapoints, the network might take reasoning shortcuts there is nothing more than the loss over the labels Y to guide the network.
For ease of presentation, we first assume we have at least one labelled datapoint for each concept, and then we drop such assumption.
Given a finite dataset D and a concept e, let
S. -x
(2)
be support set for concept This is the available set of datapoints which we can use to "anchor" the representation of each concept, hence avoiding the RS problem. For every concept we create centroid or prototype C, IR" starting from S. i.e., representative vector where all datapoints belonging to concept e should be mapped to. To this end, we use k prototype extractors (one for each set of mutually exclusive concepts in C). Each prototype extractor is an embedding function fā Rd + Im for E [k] with learnable parameters which allows us to compute the centroids:
C.
Σ (x),, XES.
for i [A]
Se
where S. S Se is randomly sampled subset of Se.
For every prototype extractor (x), we define a distance Rm x Rm - [0, +00), and, for every new input x, we decide to which concepts it belongs to by measuring its distance to each of the prototypes. In [42], it was shown that for particular class of distance functions, called the regular Bregman divergences family [3], the prototypical networks algorithm is equivalent to performing mixture density estimation on the support set with exponential family density. In this work choose as distance the squared Euclidean distance-itseli belonging to the Bregman divergences family-as it will allow us meaningfully initialize the centroids of the unspecified classes. It is now possible to produce a distribution over the concepts based on a softmax over distances between z, fī(x) and the prototypes of each class:
exp(- 1z: cell2)
(c x,C E [A.]) Eereb.)) exp( |z Cer
for = [k].
We train the prototype extractors to perform two tasks: (i) correctly classify the datapoints in the query set for class c: Q. S. \ S. if available (i.e., Q. + 9) and (ii) make predictions that are coherent with the background knowledge K for all the datapoints in the training set D which labelled with a concept. An overview of the training procedure can be seen in Algorithm 1.
We now relax the previous assumption that each concept must have least one labelled datapoint, hence we can have a concept e such that S. = 0. Instead, we require only that, within each set of mutually exclusive concepts (i.e., for every [h.] with i E [k]), there exist at least two datapoints labelled with two distinct concepts. This condition ensures that we can express the centroids of concepts without labelled examples functions of the centroids of concepts with at least one labelled instance.
Let c E [h.] for some E [k]] be concept for which there are no labelled datapoints in the training dataset D. Let Hi S [hg] be the set of concepts for which there exists labelled datapoint D. Then the centroid for can be computed as
C₂ M'H +c, with C~N(1 (0.
max.CH IM
coll3
mm
(5)
X2m.p
where Usz = [H. EdEn, Ce' represents the mean of the known centroids in [/s]]. and X2 is the p-th lower-tail quantile of a x" distribution with m degrees of freedom. Unless explicitly stated, the parameter is set to 0.99 in our experiments. This ensures that, for each class e, the newly initialized centroid c, has probability 0.99 (over the random draw of e) of lying within the hyperball whose radius is given by the distance from Mzz to the farthest known centroid. This encourages separation
3

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Algorithm Training episode loss computation. 4 is the number of classes sampled per set H, and training episode. Se (resp. qe) the number of support (resp. query) examples per concept c.
Require: (i) Dataset D. (ii) support set Se for every concept
Ensure: Output the loss L for randomly generated training episode.
C+0
2: for i E (L,....k,, k) do
3: < BAANDOMSAMPLEHHL
Select classes per episode and prototype extractor
4:
for c in & do
5:
S. RANDOMSAMPLE(S, s.)
Select support examples
6:
2c t RANDOMSAMPLE(S, \ Sc.g))
D Select query examples Compute centroids for labelled classes
7:
C. 3, Exes. f((x)
8:
for in Qc do
9:
L 1,91
| - 1 log Σ exp(-11z, c.2 113)
c'E(bil
10:
for E [h,] \ H4 do
11:
Ce - MH. +
D> Compute centroids for unlabelled classes
12: for (x,y) D do
13:
t L + NoSy (y, K)
If necessary for NeSy model, compute NeSy loss
among mutually exclusive classes, while reducing the risk that a zero-shot prototype is placed far from the region spanned by the known embeddings, which could otherwise lead to prototype collapse, where multiple concepts are mapped to single, constraint-satisfying centroid.
In the remainder, we will call a NeSy predictor that uses prototype extractors outlined above prototypical NeSy predictor.
Reasoning over Distances
Thanks to the prototypes, given an input x, for every i E [k] the embedding n, is updated taking into consideration not only the degree of constraints violation but its similarity to each centroid.
Theorem 4.1 Let be the output of a prototypical NeSy predictor reasoning over the knowledge and the set of concepts C [hi] x x... [ha). If the predictor is trained using CNeSy (y, K), then for every [ke] the error calculated with respect to the embedding 2, is equal to:
NCNeSy
Vz, NeSy (y, 22
Σ
Oye Ye (ce
Σ Cyece) d'E(hu)
<E (hi)
Proof in Appendix A. Let call the term Eee(p Cer the centre of belief for datapoint and set of classes [h:] of the model, as it the average over the prototypes representing the classes [h.] weighted by the probability assigned to each class by the model. The embedding z; is thus updated in a way that ensures that-in in the next learning iteration the centre of belief for the datapoint is closer to the weighted average over the distances of the prototype of each class the centre of belief. The weights take into account both: (i) the degree of constraint violation, as captured by the magnitude of the gradient (esy /dyel and (ii) how close the embedding is to prototype, captured by the ye term. The embedding update thus allows us at each step to try to satisfy the constraints while also taking into account the similarity between each class prototype and the embedding of the datapoint.
Notice that the same does not happen when training standard neural networks with a NeSy loss. In that case, even if we have some labelled datapoints, for the unlabelled datapoints the update will obviously only depend on the error done with respect the NeSy loss, thus creating fertile ground for reasoning shorteuts (especially as the number of unlabelled datapoints grows).

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We now show what the embedding update looks like in the specific case of the semantic loss [49]. Given an imput x and its corresponding output y we can write NeSy semantic loss as:
NeSy (y. K) = log Σ II (vc) (1))-y))-0 1-(())
CE [Md
where each v UU [h:] + {0. 1} represents (with a slight abuse of notation) a boolean assignment of the concepts in U: [h.]. We say that assignment v satisfies the knowledge K expressed propositional logic formula, denoted by += K. if and only if the recursive evaluation of formula K under assignment yields 1. For every E U. Ih:] we define an independent Bernoulli random variable Y. ~ Bernouli(ye). This induces a distribution over full truth assignments U, [h:]
{0, 1). Then, we can rewrite the semantic loss as: NeSy (y, K) - log Enp k P(v yy)
Corollary 4.2 Let be the output of a prototypical NeSy predictor reasoning over the knowledge K and the set of concepts = [ha] x... ... x [hg]. If the predictor is trained using semantic loss LNSSy (y, K) defined in Equation 7, then for every E [k] the error calculated with respect the embedding 24 is equal to:
yc E y, K] ye ylc)
NoSy (y, K) 22
MolCe
E'coyel
cE(A.]
Proof Appendix A. The above theorem shows that the embedding moved closer to each centroid C. by value proportional to (i) the distance between z; and C, and (ii) the difference between Ve and the expected value of Y. conditioned on the satisfaction the knowledge.
Example 3 Suppose C = [h]] x [h2] where [h1] represents the concepts (Dog. Cat, Duck, Shark) and [h2] represents the concepts (Mammal, Bird, Fish). Let K (Dog Cat + Manmal) A (Duck - Bird) A (Shark + Fish). Suppose we have an image x of a dog, whose embedding 2₁ is represented by X in Figure 2. 2 's squared distances from the centroids are equal to the numbers next to the dotted lines. The centre of belief for x and [hi] is represented as *. Let 22 have distance 2, 3, 4 from the prototypes of Mammal, Bird and Fish, respectively. The magnitude of the error computed with semantic loss by our method wrt Dog. Cat, Duck and Shark is ~ 2.0, 0.3, 0.25 and ~ 5 x 10 thus mirroring the importance that each centroid has to the class prediction. On the contrary, if we were to use a standard neural network with semantic loss we would get that the errors for each of the above outputs all have magnitude of ~ 1.0.
0,05
Figure 2: Prototype configuration for Example 3.
5 Counting Deterministic Reasoning Shorteuts
We now analyse the number of possible deterministic RSs admitted by prototypical NeSy predictor. To isolate the structural sources of shortcuts, work in an idcalized setting where we assume that [A1] the distribution p" (X G, S) is induced by map o: (g. s) + x, where o is invertible and smooth over s, [A2] the ground-truth process is consistent with the prior knowledge K: the distribution p" p' G; K) is induced by deterministic mapping 5x - y and p"(y g: K) = 0, for all (g.y) y) violating K, and [A3] we have separability in the embedding space. This means that given a bijective assignment
+ g such that ogg); E [h,] with i = [k], it holds that for every
datapoint x ((g.s)
arg min l1 c, (g):- JE(k.]
Intuitively, all datapoints associated to the same concept E [hr], will have the embedding f) closest to centroid but this centroid might be labelled with the wrong concept, i.e., it is not necessary that g = (g):
Observe that, under [A1], each (C G) is induces exactly one mapping : such that (C G) = 1(C = (G)). We denote as A the set of all mappings a. shown in [29], the cardinality of AA can be reduced leveraging disentangled concept extractor architectures: when
5

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model is disentangled then PC G) factorizes as II= Po G.). In prototypical NeSy models the conditional distribution Po X; induced by prototype extractors with embedding function f3: I2 RT. for every i [k]: under independence assumption, then our design enforces disentangiement by construction.
Proposition 5.1 Consider a prototypical NeSy predictor inducing pol (o K). Assume G = [ha]] x x [hg]] and let Hi [ha] be the set of concepts with least labelled datapoint from dataset 'D. Under [A1,2,3] the number of deterministic optima for pa G) is pa G) is
Σ'AA (AΛ
= (a(g): - - - -))) ^ ((())(((())))))) = (x)))(
aEA
gEsupp(G)
VE(A] 8,EM,
a being the mapping induced by po(C G) and A being the set of all possible mappings a.
Remarkably, by leveraging [A3], we achieve the same number of RSs obtained by the dense data-
mitigation strategy proposed in [29]. This assumes that every supervised concept receives annotations for every datapoint in which occurs. Hence, thanks to the prototypes, the labelling effort can be substantially reduced. Indeed, Proposition 5.1 implicitly assumes that for every i E [K] we have one labelled datapoint per class 9 E [hy].
Experimental Analysis
Metrics. We evaluate the models coherently with [28, 5] that are labelled as "final labels" and as "concepts". We measure the accuracy and F1-score wrt. both, denoted resp. as Acc(Y), FI(Y), Acc(C) and FI(C), and the concept collapse Cls(C).
5] and we distinguish between the classes
variant of MIIST-Addition with restricted support. Each input is pair of handwritten digits (concepts) labelled with their sum (final label). (ii) Kand-Logic [34, 30]: features the shape (O, O, A) and color (red, yellow, blue) of geometric primitives (concepts), possibly defining pattern to be predicted (final label). (iii) BDD-OIA [50]: an autonomous driving task proposing frames labelled with the actions to be undertaken by the ego-vehicle (final labels). The goal is to correctly classify the reasons (concepts) behind each action.
Tasks. (i) MNIST-EvenOda [28, 30]:
Models. For all tasks, we consider DeepProbLog [26] (DPL), Logic Tensor Networks [2] (LTN) and Semantic Loss [49] (SL), implemented in [5], and we integrate them with prototypes. For MNIST- we also consider Coherent by Construction Networks (CCN+)[18).
RQI: Can Prototypical Neurosymbolic Al avoid Reasoning Shortcuts?
To answer RQI, we randomly label just one image per concept class in MNIST- and Kand-Logic. We then train each Prototypical NeSy model (SL+PNet, LTN+ DPL+PNet) with support sets of size 10 and for Eventdd and Kand- Logic, respectively. For BDD-OIA annotate 126 images-alme almost the size of training batch-with 2w -ddiimensional binary vectors. We compare our re- sults against: (i) the standard NeSy base- lines (SL, LTN, and DPL), and (ii) the same models whose backbones are pre- trained on annotated support sets en- riched with over 150 additional data- points obtained via data augmentation for both Evenddd and hand-Logic, as detailed in Appendix E (denoted as SL+Pre", LTN+Pre*. and DPL+Pre*). No data augmentations were used for BDD-OIA. This second baseline demonstrates that, even on a simple dataset such as tially outperform extensive data augmentation. Conversely, for Kand Logic, the performance gap is minimal, which can be attributed the high homogeneity of the concepts in this dataset. The results for
(a) SL
(b) SL+Pre"
(c) SL+PNet
Figure 3:
concept confusion matrices.
the use of prototypical networks can substan-
and -Lgic are shown in Tables and 2. We observe that Prototypical NeSy models greatly outperform the standard baselines the MNIST-Even LTN'S Acc(C) jumps from 13% to 95% while Cls(C) drops from 80% to 0%. The enhanced concept alignment even more evident comparing the confusion matrices in Figure for the SI. model. While K the gain wrt. concept metries may appear less stark due to their homogeneity,
task: for example

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Table 2: Concept scores Kand-
Table I: Concept scores on MWIST-EvenOdd.
Acc(C)(1)
F1(C)(1) Cls(C)()
Acc(C)(t)
F1(C)(1) Cls(C)())
LTN
0.36=0.06
0.22=0.06
0.64
LTN+Pre
0.90=0.01
0.90=0.01
0.00
LTN
0.13:10.07
0.05
0.79
LTN+PNet
0.90
0.90
0.00
LTN+Pre*
0.51
0.39
0.45+0.23
LTN+PNet
0.95+0.03
0.95+0.03
0.00
SL
0.35
0.20=0.01
0.73+t..12
SL+Pre"
0.91 10.01
0.90
0.0011.000
SL
0.0810.12
0.04
10.04
SL+PNet
0.92
0.92=0.02
0.00
SL+Pre*
0.42 0.52
28
0.40=20.24
PL+
0.981
0.98 10.01
0.00.0.000
DPL
0.39=0.05
0.25=0.00
0.66:+0.18
DPL+Pre*
0.92
0.91 0..04
0.00
DPL
0.08
0.04
0.67
DPL+PNet 0.94
0.94=0.00
0.00+00000
DPL+Pre
0.64
0.55 28
0.32=0.21
DPL+PNet
0.97=0.003
0.96=
=.00
Table 3: Concept scores on BDD-OIA.
CCN
0.03
0.01 10.03
0.73
FI(C)(1) CIs(C)(L
CCN-+Pre
0.33
0.222
0.57
+PNet
0.9611.00
0.95
1.000
DPL
0.08
0.82=0.06
DPL+Pre" 0.05 0.85
DPL+PNet 0.1620.15 0.35
carries significant improvements for the right prediction of the final label as shown in Table 10 in Appendix F, thus indicating a strong synergy between the symbolic and the neural components. Lastly, the superior ability of prototypical models to mitigate reasoning shortcuts (RSs) is also evident on BDD-OIA. As shown in Table 3, PNet model reduces Cls(C) by more than half compared to both baselines, while achieving an FI(C) score that is more than twice as high as the best performing baseline. A more detailed analysis of RQI is provided in Appendix F.
RQ2: How does Prototypical NeSy compare with other mitigation strategies?
We consider the three mitigation strategies, and combinations thereof, as proposed in [27] and [29]. These are: Shannon-entunn enntoopp regularization (H): regularizing the models using the Shannon- entropy. (ii) partial loss concept supervision (C): taking subset of the concepts and labelling all the corresponding datapoints
the dataset-followinn [29], we label the concepts "4" and "9", (iii) reconstruction loss (R): in-
0.9
0.8
0o.
0.6
Staldard MY
0.4
troducing an encoder-decoder ar- chitecture and training the model to jointly minimise NeSy and the reconstruction loss wrt the in- put. We carry out the comparison with our Prototypical NeSy mod- els on the MNIST-EvenDdd task under the same settings for RQ1. The results obtained with each of these strategies and their combination are shown in Table 4. Crucially, each of the mitigation strategies in isolation does not manage to curb the RS problem, even with a costly solution like labelling all the datapoints where concepts "4" and "9" appear through C. Only by using them all together we get comparable results for DPL and LTN, while only FI(C)= 0.35 for SL.
0.7
PNN
C
UTN+PNet
0.2
UN PC-
DPLSPre*=GIS"*
0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 o'a 0.9
0.1 0.2 0.3 0.4 05 0.6 077 6. a.a
Unsupervised Data Percentage
Unsupervised Data Percentage
Figure 4: FI(C) KNIST-EvenDde across unlabelled data ratios.
RQ3: Are Prototypical NeSy models robust wrt. the number of unlabelled datapoints?
In Section 4 we show how Prototypical NeSy models update their parameters accounting at the same time both for the background knowledge and the proximity to each class centroid. With this RQ we validate amount of weakly supervised datapoints used at training time. To this end, we create 10 training subsets of MNIST-EvenOdd containing the original support sets together with a varying percentage of unlabelled examples (from 10% to 100% with step increase of 10%). Then we train on these
extent to which this update makes them less sensitive than traditional NNs to the
7

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Table 4: FI(C) and Cls(C) of different mitigations on KWIST-EvenDåd. Best results in bold.
F1 (C) (t)
Cls (C) (4)
DPL
SL
LTN
DPL
SL
LTN
+
0.07=0.00
0.00
0.05++..00
0.70=0.00
0.7000..0 00
0. 0.70==0.0
+ H
0.01=0.00
0.01
0.55 14
0.70+0.00
0.08++0.02 0.70+00.0
+
0.01
0.22+0.01
0.12440.00
0.70+0.00
0.60=+0.01
0.71+0.00
+ R+H
0.09=0.02
0.01
0.61+0.11
0.63+00.00
0. 0.70==0.00
0.21
+ R+C
0.01=0.00
0.46:2.07
0.10+0.00
0.70=t0.00
0.47 0.00
0.70= 00
+
H+C
0.96=0.06
0.31
0.98=0.00
0.03
0.50=0.00 .00
0.01
R+H+C
0.98+0.00
0.35
0.95+0.00
0.02
0.50+0.00 01
0.04+0.00
PNet
0.96=0.01
0.98#0.01
0.95+0.03
0.00++0000
0.00 0.00
0.00
subsets (i) our prototypical models and (ii) baseline models whose backbone is pretrained using the support sets (Pre*, c.f. RQ1) and receiving additional supervision from them during training (C(S)"). We report the comparison in Figure 4 wrt. FI(C). The Fig-
0.35
DPL+ C(S)
CPL+
ures show that models employ-
0.25
0.30
DPL+ PNets
0.5
0.6
PNets
ing traditional concept extractors
0.20
struggle in effectively exploit the knowledge at hand. Moreover, they are particularly unstable and sensitive both the network ini- tialization and the number of un- labelled datapoints. On the flip side, prototypical ones use the background knowledge refine their concept understanding and converge to stable solutions as more unlabelled data become avail- able. We run a similar experiment for BDD-OIA as well, where we consider a standard DPI, model employing on extractor per concept that replicates the same disentanglement granularity of the prototypical counterpart. This model is also supervised through the support set. Figure 5 left (resp. right) shows that, the number of unlabelled datapoints grows, the F1(C) (resp. Cls(C)) for standard DPL greatly decreases (resp. increases). On the contrary, the prototypical model is robust wrt. to this percentage and it consistently outperforms in both metrics the standard model.
0.15
0.3
0.10
0.2
0,1
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.2
0.4
0.6
0.8
1.0
Unsupervised Data Percentage
Unsupervised Data Percentage
Figure 5: F1(C) and Cls(C) on BDD-OIA across different percent- ages of unlabelled datapoints. The green dotted line represents the baseline non-supervised DPL.
RQ4: What the impact of having totally unlabelled classes?
Ideally, one would like to have at least one labelled example for each class. In this RQ we test approach for cases in which such assumption does not hold and we initialize the centroids for the unknown classes as in Equation 5. We also test method with different numbers of totally unlabelled classes, running each ex- periment 5 times and randomly sampling each time the classes to leave unlabelled. In ad- dition, for RQ4 and RQ5, increase the support sets variety via standard augmenta- tion techniques, as we found our models perform slightly better in this case. The re- sults obtained on -veenddd in terms of F1(C) and (Y) are shown in Figure 6 and those on Kand-Logic in terms of F1 in Table 5. As expected, from Figure and Table we notice the more classes we are able to specify (i.e., provide examples for) the better the results. Crucially, even with minimal labels provided (e.g. two/four labelled images) Prototypical NeSy methods mitigate RSs.
Table 5: FI(C) when having different numbers of concepts left unspecified on Kand-Logic.
# Unspec.
DPL
SL
LTN
0.94
0.92:0.02
0.9040.01
1 2
0.65
0.73#00.08
0.64+0.00
0.42+0.02
0.55++0.01
0.41+0.02
Baseline
0.25+0.06
0.22+0.01
0.22+0.06
RQS: How important is the centroid initialization for unlabelled classes?
As explained in Section 4, it is important to make sure that the centroids for the unlabelled
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1.0
SL FI(Y)
F1-score
0.8
DPL FI(M)
F1-score
0.8
F1-score
0.6
0.6
0.6
0.4
F1(C)
F1(C)
F1(
0.2
DPL
F1(M)
F1LC)
F1(Y)
[i(C)
FI(Y)
2
Number hidden classes
Number of hidden classes
Number of hidden classes
(a) DPL+PNet
(b) SL++Nett
(c) LTN+PNet
Figure 6: FI(C) and F1(Y) obtained with Prototypical NeSy models trained MNIST-EvenOdd when varying the number of unlabelled classes. The red lines indicate the performance of the corresponding baseline model.
1.0
classes lie among the labelled ones. In this ablation study, we show what happens if this assumption is violated. We the
p=0.2
0.8
p-0.5
these experiments picking different values of p. i.e., for the lower-tail quantiles of the x² distributions. The (C( results obtained with SL for = 0.2, 0.5 and = 0.99 are reported in Figure and those in terms of Acc(C) and Cls(C) Appendix F. We clearly see that as the value of increases the network performs better and better. Interestingly, we register the biggest difference in performance with 4 hidden classes, i.e., when the impact of the unlabelled centroids is high, and the known concepts provide enough information allow for meaningful computation of the centroids of the unlabelled classes.
dataset while 0.5
p=0.99
0.4
Number of hidden classes
Figure 7: F1(C) with SL for 0.2, 0.5 and 0.9.
7 Related Work
NeSy encompasses diverse set of approaches that integrate symbolic reasoning with neural net- works [36, 16, 12, 17]. central challenge in this field lies in reconciling discrete, non-differentiable symbolic reasoning with the gradient-based optimization underlying neural networks. To enable end-to-end training, many NeSy approaches soften the semantics of logical reasoning via probabilis- tie [26, 51, 23, 32, 46, 41, 24, 48, 9] fuzzy logic frameworks [10. 11, 45, 2, 40]. A common strategy involves compiling background knowledge into the loss function to encourage constraint satisfaction during training [49, 13, 21, though this typically offers guarantee that constraints will hold at inference time. In contrast, methods that restructure the model's output space can ensure con- straint satisfaction by design [26, 14, 1, 7,3, 18, 15, 19, 43, 44]. Despite the many advances [25, 47), [28, 29] showed that most NeSy systems are prone to RSs. Some mitigation strategies have been proposed [27, 29]. but they be either costly they might fail in challenging settings like those proposed in rs- bench..h. To obviate to such problems, an interesting alternative direction has been taken by [30], where the authors do not try to avoid the RSs problem, but rather inform the user of when the model is taking a shortcut and facilitates acquiring informative dense annotations for mitigation purposes. For comprehensive overview over the RSs problem in NeSy see [31].
Conclusion
In this paper, we introduce Prototypical NeSy models as a novel strategy for achieving more accurate and robust NeSy learning under very limited supervision. In particular, we showed how to effectively avoid reasoning shortcuts by anchoring neural embeddings prototypes, which allows to exploit handful of labelled examples to train prototype-basee concept representations, that then get continu- ously refined via the symbolic knowledge. We show the precise dynamics of the embedding update behind our models' training and the difference with standard NeSy architectures. Moreover, we provide a theoretical analysis on the number of shorteuts possibly affecting our models. We validate them over large set of experiments on the rsbench benchmark suite, where we show remark- able improvements, both the synthetic tasks -MNIST-EvenDdd and Kand-Logic) and real-world high-stake ones (BDD-OIA)

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A Proof Theorem 4.1
Statement. Let be the output prototypical NeSy predictor reasoning over the knowledge and the set of concepts = [h1] x [ If the predictor is trained using C (y, K), then for every [k] the error calculated with respect to the embedding z; equal to:
OLNeSy
NeSy k) 2 Σ
dye Ye (ce
Σ Vccc) d'Elnul
(10)
cE(hel
Proof.
Let us apply the chain rule. For every i E [k]:
NC (y, K)
V L Ne8y (y = Σ
V Me
dye
cE(N.]
exp(-11ze Cel (2)
22 = Vx,
Ed'E(A.) exp(- - - C.' 113)
In order ease the calculations, we calculate the partial derivative of ye with respect to the jth element of the embedding z,, which we will refer 2/-
Let de = - - - -cel2 then:
Oye
Oye Ode
Σ
0zf
Ode 0zg
d'Elhsl
Ode ye)
Ode
Vc
(1
Ye d'(Mull.cta
0=1
O21
Since 0d. D=7
-22 -e.). eg being the jth element of the vector co. then
Öye ye (1 vo)(-2(( ed))
Σ
Ye (- - ed)) -d))
0=1
2ye
(2) cd) + Σ
Ve (2g etc
since Σ Ye'
d'Elhal
c'E(hu]
2ye
Σ ed
d'((h.l
We can hence conclude:
NLNeSy
NeSy
(y, K) 2 Σ
Oye Ve (co E ve
cE(hv.]
c'E(h.)
B Proof Corollary 4.2
Statement. Let be the output of prototypical NeSy predictor reasoning over the knowledge K and
the set of concepts C = [bs] x ... x [Ix].. If the predictor trained using semantic loss (aey (y, K) as defined in Equation 7, then for every i [k] the error calculated with respect to the embedding 2; equal to:
Vc E Y. K
NeSy K)= Σ cE(h:.
Vo Σ Co.ler)
(11)
Ve
Ne)
20

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Proof. Recall that the semantic loss is defined as:
NeSy (y, K) = - log Σ II (1)))
yo) -m())
CE
where each v U,,hg [h.] + {0, 1} represents boolean assignment of the variables in U hha). Given Theorem 4.1, to prove our statement, we need to calculate the value of (
Σ
II Di
(we) - Ye )
a
(va) - yo)
N (y, K)
M KeK,
Oy
Oye
Σ
11 (1 Vc) v(c)
2 DFKcEEU
where
① Dye
((c)
1 - v(c)
yc) --)c)
(1 -yce)= (e)
31c
- yo
(((c) Dcc
(1 ye) ( (yc)) (10)
Vc(l yc)
Substituting obtain:
Σ == dEEU,)N.)
II
(1ar) Ye
)--u(c)) -
N (y.
ve(I-ve)
Oye
Σ
II (v6)r - 10) 1-v(e)
M= U.
Since each concept E [hs corresponds to an independent random event whose truth assignment modelled by Bernoulli random variable Y, Bernoull(yye), the joint probability distribution over assignments : U,[b,] + 1) factorizes as:
plv 1y)
(c6) (1 vc) (1-ve) vc))- CEU,
II
This allows to compute the expected value of each Y. conditioned on the satisfaction of K as:
E|Ya y. v Eup y) v)))(c)
Eupple y)
We rewrite the error calculated with respect to the output for the class
No (y,
Yec
ElY. K,yl
Oye
Vc ye)
The statement thus follows.
C Proof of Proposition 5.1
Statement Consider prototypical NeSy predictor inducing pe (X; Assume G [hu] x x [fu]] and let H4 S [h,] be the set of concepts with at least one labelled datapoint from a dataset D. Under [A1,2,3] the number of deterministic optima for pC G) is
Σ Σ1AA (A aeA
A (a(g), = Agi - -)) ^ ((oK =3<(s))
gEsupp(G)
VE(M]
being the mapping induced by po (C G) and being the set of all possible mappings a.
Proof. We assume that:
[A1] the distribution p (X G, induced by a map Φ: (g. s) + x, where is invertible and smooth over s;
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[A2] the distribution p' G; is induced by deterministic mapping 5x - and pd (y =) for all (g.y) not compliant with K;
[A3] given bijective assignment that for every datapoint x = ogg, s), jrgmin (l. c = og))
9 such that (g), E [h.] with 1,..., holds
Under [A1], the data distribution is induced by smooth, invertible map (g. + x, so every datapoint can be written as x = ()..s)
Under [A1] and [A3], any deterministic predictor Po X) that is constant over the support of p(X g) corresponds uniquely to a deterministic mapping G - 9, defined by a(g) (g).
Conversely, each induces a deterministic distribution Pa C X); thus, it suffices to characterize possible assignments a E given the available supervisions 'D.
We now proceed to characterise the number of deterministic optima. For any E [K] and supervised label 9, E H4 the cross-entropy loss wrt. the concepts is maximised and only if pa(C; = gj g) = L, which corresponds to deterministic mapping agg); = 95. Hence, given i E [K] for all the deterministic optima must hold:
AH (a)(g) 9; 9: = 9j).
When accounting for all the H4 and E [k), we obtain:
A Agesupp(G)
AVEa (a(())
(a:(g) = 9: A 9: = 9]))
Under [A2], a NeSy predictor achieves zero NeSy loss wrt. the final label and only if it is compliant with K, then it must hold:
(6x o ag) = (gk
Hence, under [A1,2,3] the number of deterministic optima for (C G) is
Σ
A
(A
A (a(g): = g: Ag = =)))) ^ (8x og)(g) = Bx(g))
aEA
gEsupp(G)
oyEH;
D Tasks Details
The tasks belong to the ra-bench² benchmark suite. In this benchmark, all ready-made data sets and generated datasets are distributed under the CC-BY-SA 4.0 license, with the exception of Kand-Logic, which is derived from penddiisk-pptteeree the GPL-3.0 license. Regarding the code, most of the code in the repo is distributed under the BSD license, with the exception of Kand Kagd-Logic, which is derived from Kandinsky mandinnsk--atternss and as such is distributed under the GPL-3.0 license.
[34] and as such is distributed under
NNIST-EvenOdd [28, 30] The dataset presents more complex version of the standard task. Each input is pair of handwritten 28 x 28 digits labelled with their sum (background knowledge). The goal is to correctly classify the digits (concepts) while predicting their sum (final label). Notably, the restricted support is designed to stir up shortcut reasoning, with total 49 distinct RSs a model may take. Since the training support is incomplete, thus shortcuts cannot be avoided by simply processing each digit in isolation. During training only 16 possible pairs of digits out of 100 are given comprising only even digits and only odd digits while at test time the model is presented with out-of-distribution pairs. The list of all possible training combinations and their labels are given in Table 6. Overall, the training set contains 6720 data, the validation set 1920, and the test set 960.
MNIST-
Kand-Logic [34, 30] Inspired by the artworks of Wass- ily Kandinsky, this task involves non-trivial perception to classify shapes and colours, and logical reasoning to infer
Link: http:///wabb...ooo//ommmmmm///ee
Figure 8: Datapoint from Kand-Logic.
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Table 6: Pairs of digits and their label available in MNIST-EvenDdd.
Example Pair
Label
Example Pair
Label
0
6
6
2
10
3
7
10
4
10
9
10
4
12
3
9
12
patterns among them. Each datapoint (x1, X2,X2)
X3) consists of three 64 x 64 images, representing
three geometric primitives each. An example is depicted in Figure 8. Every sample labelled as I whether all images either include the same number of primitives with the same shape or with the same colour, and 0 otherwise. In order to formally specify the background knowledge K, we define six binary classes, deterministically determined from the concept classes:
SameS, (resp. SameC,) capturing whether all the primitives in x; have the same shape (resp. color) or not,
DiffS; (resp. DiffC;) capturing whether primitives x; have different shape (resp. color) or not, and
TwoS; (resp. TwoC,) capturing whether exactly two the three primitives in x; have the same shape (resp. color) not.
The background knowledge is given by
K - ((Dfff), A DiffS, A DiffSa) V (TwoS) A TwoS2 A TwoSs)V
(SameS1 A SameS, A SameS;) V (DiffC, A DiffCe A DiffC3)V (TwoC1 A TwoC2 A TwoCs) V (SameC, SameC2 A SameC3)).
Thus specifying that all images either include the same number of primitives with the same shape with the same colour. For example, Figure 8 shows image whose label is 1 because all the primitives appearing in each image have different shapes and different colours. Overall, the training set consists of 4000 datapoints, while the validation and test of 1000. Each image in turn shows three shapes, for a total of primitives per datapoint.
BDD-OIA [50] This real-world and high-stake autonomous driving dataset originally proposed as an extension of EDDD1000 [52] and contains total ~22.55 frames captured from the ego- vehiele. Fach frame, whose size is 720 1280, receives labels from (MoveForward,, Stop, TurnLeff, TurnRight) to reflect the right action to be undertaken. The goal is to correctly classify the reasons (concepts) behind each action (final label, e.g. given Person V RedLight = Stop, Person caused Stop). Frames are processed using a RaNter-RCNN model [38] and the first pre-trained layer from [39] so to obtain 2048-dimensional embeddings expressing the features present each training scene. These embeddings are then used to train the NeSy models. Each frame (or equivalently, embedding) is annotated with one of 4 possible labels [MoveForward, Stop, TurnLeft, TurnRight), each indicating possible action taken by the eeo-vehicle. The learning task, differently from the previous ones, is multi-label: multiple different actions can be labelled as possible in the provided scene with the only exception of MoveForward and Stop that are clearly mutually exclusive choices. Additionally, each embedding annotated with a subset of the 21 concepts that might cause the action:
(GreenL Redligght, TrafficSign, Follow, RoadClear, Car, Person, Rider, OtherOhstacle, LeftLane, LeftGreenLiiht,, LeftFollow, NoLefiLane, LeftObstacle, LeftSolidLine, RightLane, RightGreenLiggh RightFollow, NoRightLann RightObstacle, RightSolidL.ine).
The rules K specify concept-to-concepl and concept-to-action relations. For example:
RedLight > -GreenLight,
-Car V Person Rider V OtherObstacle ++ RoadClear,
GreenLight V Follow V RoadClear - MoveForward.
For the complete specification of the background knowledge K see Appendix C.2.5 of [29].
This dataset presents with several challenges:
23

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(a) Rider in the scene
(b) Rider no longer in the scene
Figure 9: Both Images in the BDD-OIA dataset are labelled as Rider inducing a Stop action.
(a) Pedestrian caused the ego-vchicle stop (b) Pedestrian did not cause the ego-vehicle to stop
Figure 10: Both Images in the BDD-OIA show instances of the Pedestrian class, yet only (a) is labelled as such since the pedestrians in (b) did not cause the ego-vehicle to halt.
Many concepts are under-representted, making it difficult for the model learn to recognize them effectively. In particular the exact distribution of the concepts is shown in Table 7. As we can see the training data are heavily unbalanced, with some concepts being represented by only a few examples;
The labels are noisy, and may contain errors or inconsistencies. For example some image may be labelled with both the actions MoveForward and Stop, which are mutually exclusive. This is typical of large datasets, especially the real world ones like BDD-OIA, where the labelling process is often done by humans and may be subject to errors inconsistencies; In some frames temporal information may be lost. See Figure 9 for an example where the temporal information is needed to correctly label the concepts. Indeed, in the Figure can see that both the frames are labelled as Rider, but in (a) it is clear that Rider is present and cause for the stop, while in (b) the Rider is already far away and we able to infer the correct concept only by looking at the previous ones; conversely, the embeddings of the scene are generated by looking at single frame at a time. This limitation is known, it was already highlighted by [39];
Single frames may lack contextual information, which can be important for understanding the scene. For example, in Figure 10 we can see that the context of the pedestrians is crucial for understanding the correct action be taken. Indeed, even though in both frames the pedestrians are crossing the street, the context is different: in (a) the pedestrians are crossing the street and they are the only reason for the stop, while in (b) there also traffic light that is red. This means that in (a) the pedestrians are labelled as present because the car stops front of them, while in (b) the pedestrians are not labelled as present because the car stops due to the red light.
E Experimental Setup
Neural Architectures As backbone concept extractors for the baseline models we used the archi- tecture [5]. On the other hand, as a neural prototypical extractor model for the task, we used the same convolutional neural network as in [42] composed of four convolutional
-Odd
blocks: the first one is a x 64 and the others 64 x 64, respectively. Each blocks consists of 3 x 3
24

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convolutional filter, a batch normalization layer [20]. ReLU nonlinearity, and 2 x 2 max- max-pooling layer. One such prototypical network is used classify all digits. The same architecture has also been employed for the Kand-Logic task, with the exception that prototypical networks work on three RGB channels and their inputs are primitives detected in their corresponding images using YOLO [37]. In particular, we adopted YOLOv11 [21] that exhibits enhanced detection capabilities in virtue of the introduction of cross-stage partial blocks, spatial pyramid pooling, and convolutional blocks with Parallel Spatial Attention components.
For running our experiments on Kand-Logic the nano version of YOLO was sufficient for the purpose of primitive detection. The model was trained on the same support set constructed through data augmentation for prototypical networks. We employed two prototypical extractors, sharing the same urchitecture and support set, for the task of classifying primitives shapes and colours. For this tasks, the backhone extractors for baseline are modified so as to process independently each primitive extracted using YOLO. We also employ two distinct concept extractors to process in disentangled manner both shapes and colours, matching exactly the same setup for prototypical models.
As for BDD-OIA, each original frame is processed following [39] by the means of a FaNter-RCNN model that was pre-trained on MS-COCO and then fine-tuned on BDD-100K [52]. Through the Faster-RCNN backbone the features for the objects in the scene are computed (e.g., traffic lights).
Subsequently, we employ the first pre- pre-trained layer from [39] to produce a 2048-a -48--imensiooall which we use to train the NeSy models.
To process the embeddings we created, the standard architecture of prototypical extractors from [42] was adapted to work with ID convolutions. More in details, the prototypical architecture we employed adopts three convolutional blocks, including 1 x 32, 32 x 64 and 64 128 layer respectively, each comprising 5-wide convolutional filters, batch normalization, ReLU activation and a ID kernel size 2 max-pooling layer. The last block replaces max-pooling with adaptive average pooling of length 1. Finally, the output is flattened and further processed through a linear layer, ID batch normalization and ReLU yield 256-dimensional embedding. For BDD-OIA, 21 prototypical extractors were employed, each classifying one concepts. As for the case of Kand- Logic, we modify the standard baseline to use the same number of backbones as the prototypical models.
All of our models were trained via SGD with Adam [22], a learning rate equal to 10-3, and weight decay as the regularization method.
Support sets For most of the experiments we worked with at most one annotation per concept. As we showed in Section 6 RQ1, this is enough to curb the reasoning shortcut problem. Under this premise, in many experiments we decided to use standard data augmentation techniques on the annotated data from the support set to pretrain and supervise standard NeSy models (as shown in Section 6) and/or slightly improve the generalization ability of our prototypical extractors. For example, In WHIST-EvenChe points generated), translation (144 data points), scaling (10 data points), elastic distortion (25 data points) and Gaussian noise addition (4 data points). These transformations try to mimic some of the variability observed in handwriting.
each annotated concept gets augmented through rotations (40 data
Likewise, for Kand-Logic the chosen transformations are scaling, anti- aati-aliasing addition and trans- lation, as the generated data are representative of the primitives the actual dataset contains. In the latter case, we generate 163 data points per (shape, colour) pair. For BDD-OIA, given a concept j. at least 6 samples are drawn with label 1. Then positive (resp. negative) samples are added from samples positive for each class j', + j'. where j also occurs (resp. does not occur). We end up with as 200 samples per class, more than 85% of which are negatives. No data augmentation is used and the prototypical networks are all trained using one training batch randomly sampled at training time, with some classes being significantly under-represented. as shown in Table 7.
Hyperparameters All the experiments were across ten seeds: 0, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. The value of the following hyperparameters is used consistently for all the experiments when training the NeSy models to predict weak labels for each example in D:
exponential decay rate 3: 0.99;
number of epochs: 10 for MNIST-EvenOde and Kand-Logic, 40 for BDD-OIA;
semantic loss weight wa: 10.0;
warm-up steps for learning rate increase: 0.
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Table 7: Concept-level number and percentage occurrences for each class 16080 training frames from the BDDOIA dataset.
Concept
Num. of occurrences
Percentage
Green Light
5478
34.07%
Follow Traffic
2448
15.22%
Road is clear
3422
21.28%
Red Light
3766
23.42%
Traffic Sign
1095
6.81% %
Obstacle Car
171
1.06%
Obstacle Person
116
0.72%
Obstacle Rider
3703
23.03%
Obstacle Other
329
2.05%
No Lane Left
100
0.62%
Obstacle Left
464
2.89%
Solid Line Left
220
1.37%
On Right Turn Lane
115
0.72%
Traffic Light Allows Right
629
3.91%
Front Car Turning Right
270
1.68%
No Lane Right
3182
19.79%
Obstacle on Right
3177
19.76%
Solid Line Right
2587
16.09%
On Left Turn Lane
4303
26.76%
Traffic Light Allows Left
2796
17.39%
Front Car Turning Left
1563
9.72%
On the other hand, these hyperparameters were used across different models and tasks for the episodic training of prototypical networks:
learning rate for prototypical networks training: 10-3;
number of support and query examples for prototypical networks training: 10 (5 each) if the support is expanded with data augmentation and 2 (1 each) otherwise;
number of episodes per training epoch: 100.
As for the model and application specific parameters, for every task report detailed list of all the hyperparameters we used for training NeSy models over the weakly labelled examples. We use B for batch size, ? for the learning rate and for the weight decay. For LTN p refers to the hyperparameter for quantifiers. When the models are trained with Shannon Entropy, the parameter wy indicates its weight. Lastly, m denotes the embedding dimension for the prototypes extracted.
DPL: B = 32, 7 10-3 w 10 w. 1, 64,
LTN: B 64, 7 10-3
10-4, .p w/ 10, AND operation uses Godel, OR and IMPLICATION Product as fuzzy semantics, m = 64,
SL: B 32, y 10 w 10-4, m 64.
CCN+: B = =32 10 10-4 m 64.
ABL: B =
10-1
Kand-Logic.
w/ 1. 1024, LTN: B 64, 7 10 w p= 8, Wi = =0.8, AND operation uses Godel, OR and
DPL: B 3
IMPLICATION Product fuzzy semantics, 1024.
SL: B = 7 3 = -100 m = 1024.
BDD-OIA.
DPL: B 128, 7 10
10 w/ m 256.
Notice that lowering the value and m the case of BDD-OIA was necessary as we observed the prototypical networks can in some cases easily overfit the training embeddings.
26

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Computing Infrastructure All the experiments run on a machine equipped with the following:
Intel@ Xeon CPU E5-2620 v3 @ 2.40GHz, 24 Cores, 48 Threads;
4 NVidia TITAN Xp GPU, 12 GB GDDR5X, 3840 CUDA Cores.
whose operating system is Ubuntu 20.04.5 LTS and CUDA Version 12.2. For realizing the code of our experiments we used Python 3.20 and Pytorch 1.13.0.
F Additional Results
In this Section we include the additional results obtained for each research question and experiment which we could not include in the main body of the paper due to space constraints.
RQ1: Can Prototypical Neurosymbolic Al avoid reasoning shortcuts? The complete results for prototypical NeSy models are reported in Table 8, 10 and 9 for the and BDD-OIA task respectively. In the Tables extractor is denoted as PNet, while PNer* is the variant who employs standard data augmentation techniques to enhance the variety of the support set, explained in Appendix E, on and Kand-Logic. Regarding -veendddd we introduced several additional baselines exploiting the annotations from the prototypical support sets in various ways. In particular, the other considered mitigations are:
Kand-Logic mitigation strategy resorting on prototypical concept
support set S concept supervision: similarly to the mitigation strategy C, proposed in [29], during training we supervise the NeSy models with the annotations S that undergone an additional data augmentation step, so to steer them towards the correct predictions. We denote this variant as C(S)";
pretraining: the models' backbones are trained in supervised fashion on the support set before training them on the target task. In this way the baselines can leverage the support set information to learn better initial parameters. The main text discusses pretraining (Pre") on the support set until convergence before training on the weakly supervised data. We also tested epoch pretraining (EPre*), where the model pretrained on the support for one epoch before each training epoch on the actual task. This second pretraining strategy is designed to closely resemble the execution of Algorithm used to train our prototypical models;
combinations of pretraining and support set concept supervision.
These additional baselines are useful to demonstrate that the performance gains we present are due to our architecture, rather than the support information. From Table 8 two main observations be made. First, Prototypical Neurosymbolic models effectively avoid reasoning shortcuts. Second, none of the baselines introduced is competitive with their prototypical counterpart across all metrics and all the models studied.
Regarding this second point, observe for instance that EPre* performs competitively with solution on Semantic Loss, but it is outperformed by it on DeepProbLog and even more so on Logic Tensor Networks.
Interestingly, Prototypical Neurosymbolic models also maintain often even improve their perfor- mance on label scores compared to the baselines affected by the reasoning shortcuts. This shows that enhancing concept alignment is also greatly beneficial to the overall task performance. The only exception to the above concerns the FI score. This metric is lower than that of their respective baselines in the case of Logic Tensor Networks and Semantic Loss. Conversely we do not observe this drop in performance for DeepProbLog, where the FI(Y) score is actually improved wrt. most basclines. The difference can be explained by the different ways in which these models integrate the reasoning and learning components. Indeed, DeepProbLog leverages constrained-outpl marginal- ization, and it allocates probability mass only to outputs that satisfy the logical constraints: as a consequence, during training gradient updates considered only for digits participating in consistent sum predictions. contrast, the semantic objective of Logic Tensor Networks and Semantic Loss allows for occasional miss-classffication.
Finally, notice that the use of data augmentation plays only a marginal role in avoiding reasoning shorteuts in Prototypical Neurosymbolic models: the difference in both the concept and label scores
27

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Table 8: Different mitigations strategies compared across NeSy models, trained with 100% of MNIST-EvenOdd data. Best results are in bold and second best underlined.
Model Mitigation
(tt(C)(() FI(C)(t) Acc(Y)(t) FI(Y(()) CIs(C)()
Logic Tensor Network
Baseline
0.1310.07
0.05.10.4
0.6110..1
0.6611.11
10.08
0.14+0.07
0.05#+0.02
0.72+0.00
0.71
0.70=0.00
H
0.5
0.55
0..0
0.85#
0.08=0.02
C
0.26:10.00
0..0.000
0.77=00.00
0.78+10.00
0.71 =0.00
C(S)
0.46:0.31
0.37
0.80#10.12
0.61+00110
0,49
R+H
0.65
0.61+ 10
0.74+0.20
0.81#0.20
0.21=0.16
R+C
0.23
0.10 00
0..00
0.75=+0.00
0.70=0.00
H+C
0,98 10.00
0.98 10.00
0.97 10000
0.98
0.01:10.00
R+H+C
0.95
0.95
0.91 10.00
0.94
0.04
Pre
0.511
1..2
0.89
0.70.10.06
0.45
Pre" +C(S)*
0.72:10.27
0..4
0.94
0.71100.05
0.24
Pre
0.27=0.24
0. 0.
0.78+0.09
0.58+00.88
0.62=0.21
PNet (ours)
0.95++.03
0.95+10.03
0.91
0.71
0.00
PNer (ours)
0.98
0.97
0.95+
0.73
0.00
Semantic Loss
Baseline
0.08
0.0410.04
0.99
0.99
0..84
R
1.000
0.00
0.99.10.00
0.90
0.70
H
0.01+0.00
0.01
0.99=0.00
0.99=0.00
0.70=0.00
C
0.30=0.01
0.22+0.01 0.
0.99=0.00
0.99=0.00
0.60=0.01
C(S)
0.56+0.24
0.45+0.22
0.99
0.99=0.01
0.26=0.24
R+H
0.01
0.01
0.99=0.00
0.99=0.00
0.70=0.00
R+C
0.54+0.07
0.46+tu.07
0.99=0.00
0.99
0.47=0.00
H+C
0.38+0008
0..17
=0.00
0.99
0.50=0.00
R+H+C
0.40+0.10
0.35+
=0.000000
0.99
0.50=0.01
Pre"
0.521..25
0.42.0.22
0.98
0.98
0.40
Pre Pre*+C(S)) +C(SS)
0.7810.22
0.71
0.99 10.00
0.99 10.00
0.17
Pre*
0.9510.11
10.4
0.90
0.98
1.00
PNet (ours)
0.98+0.01
0.98=0.01
0.97
0. 0.74+0.00
0.00=00.00
PNet (ours)
0.99
0.98=0.01
0.97
0. 0.74
0.00
DeepProbLog
Baseline
0.08+0.19
0.0440.05
0.87+0.06
0.94+..04
0.67=0.05
R
0.17+0.00
0.07+0.00
0.75.10.00
0.741...0
0.70=0.00
H
0.01-0.00
0.01
0.99
0.99
0.70=00.00
C
0.01
0.01 10.00
0.75-10.00
0.8411..00
0.70=0..00
C(S)
0.26+0.11
0.15++0.0 07
0.84+0.07
0.84++0.88
0.68=0.06
R+H
0.16+0.00
0.09
0.83 06
0.85=
0.63=0.00
R+C
0.01+0.00
0.01+ 00
0.75+00.00
0.84+0.00
0.70=0.00
H+C
0.97=0.05
0.96+0.00 06
0.96=0.02
0.97=0.02
0.03=0.05
R+H+C
0.98
0.98 00.00
0.95
0.97
0.02=0.00
Pre*
0.64:0.22
0.55+0.22
0.95
0.95+10.06
0.32=0.24
Pre +C(S)*
0.86:10.20
0,81+0.26
0.99
0.99
0,11=00118
Pre*
0.84+0.22
0.8141..3
0.95+0.00
0.95.10.08
0.14
PNet (ours)
0.97-10.03
0.96+10.04
0.98+10.02
0.98
0.00
PNet (ours)
0.98
0.98.10.00
0.9910.00
0.99.10.00
0.00
28

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Table 9: Results of different RSs mitigation strategies for models trained on the BDD-OIA task employing 100% of the unsupervised data. Best results are in bold and second best are underlined.
Model / Mitigation
Acc(C)(t)
FI(C)()
Acc(Y)(t) FI(Y(()) CIs(C)()
DeepProbLog
BaselineinD
0.64
0.08=0.14
0.76+0.01
0.58
0.81
BaselineinD.pct
0.62
0.08=0.13
0.77 +0.01
0.61
0.82+
C(S) inD
0.83 #0.01
0.07=0.16
0.74#0.02
0.46:4000
0. 0.70=0.06
C(S) inD,
0.81=0.01
0.11=0.17
0.76#0.01
0.53+0..5
0.51+0.15
Pre* inD
0.83 00.01
0.05=0.14
0.72+0.00 03
0.43+0.05
0.86+0.04
Pre* irD
0.82+0.03
0.05=0.13
0.71
0.52=0.07
0.85+0.04
Pre* + C(S)* inD
0.83 1:0.11
0.07
0.73
0.4110.04
0.75:10.08
PCS)* inD,prD
0..02
0.10=0.16
0.7611..05
0.51:0.04
0.52:1..11
0.83 +0.02
0.06=0.15
0.71#0.00
0.44+00.05
0.82+t0.07
Enre
0.83 #0.01
0.05=0.13
0.73+0..03
0.46:+0.10
0.82:
PNet
0.60=0.03
0.16 10.15
0.63#0.05
0.41#0.08
0.35 #0.11
for the prototypical models with and without data augmentation minimal. Thus data augmentation considered only as a complementary technique to slightly improve concept alignment.
We tested well more recent, state-of- -ft---arttt neurosymbolic model, namely Coherent by Con- struction NNs (CCN+). The results are shown Table 11 and notice they are similar to the ones we previously discussed in Table 8: this evidence confirms that integrating NeSy models with prototypes is greatly beneficial in avoiding reasoning shortcuts, regardless of the particular model considered.
To further expand our study, we then tested for shortcut reasoning the abductive learning (ABL) paradigm [6] as well. The results in Table 12 show that pretraining the ABL model and supervising with the augmented support set at training time (Pre* (C()") allows to successfully reconstruct the unlabelled concepts. However, in learning the concepts the model not synergistically aligned with the final task (sum prediction), as pointed out by the significant drop in F1(Y) score. Conversely, prototypical NeSy models can reconstruct the unlabelled digits, while predicting the right final label.
We run similar experiments for the Kand-Logic task. Here we allow for other two type of baselines:
Input Disentanglement (inD): the architecture of the baseline processes single concept in isolation from the rest.
Predicate Disentanglement (prD): the architecture of the baselines processes single concept with separate branches for each set of mutually exclusive concept classes.
Since each concept is assigned pair of different classes (i.e., class of the shape and colour), distinguishing between inD and prD allows to effectively isolate the contribute of disentanglement from the one of prototypes. The results in Table 10 confirm the enhanced ability of PNet models in successfully retrieving the intended semantics for the intermediate concepts, while using them so comply with background knowledge. Moreover, they indicate that combining both forms of disentanglement with pretraining is generally the second best mitigation strategy across the board.
29

---

Table 10: Different mitigations strategies compared across NeSy models, trained with 100% of Kand-Logic data, Best results are in bold and second best underlined.
Model Mitigation
Acc(C)(() FI(C)(t) Ace(Y)(t) FI(O(() CIs(O))
Logic Tensor Network
Baseline:nD
0.37=0.01
0,32
0.50=0.01
0.36=0.03
0.26+0.22
BaDpl
0.361..06
0.22.10.06
0.5010.01
0.3410.02
1..16
C(S)* inD
0.391...03
0.34
10.01
0.4110.00
0.26110.16
C(S)* pnD
0.60
0.52:10.06
0.501
0.36:10.04
0.54:10.09
Pre" inD
0.57:10.01
0.56:t0.01
0.51=00000
0.43:10.01
0.00 100000
Pre" inD.prD
0.90
0.90++0.01
0.76:20.03
0.74+t0.04
0.00 10.00
Pre +C(S)* inD
0.43
0.40+0.02
0.50=0.00
0.34
0.16+
PC(") inD.prD
0.88
0.87
0.501
0.39
0.00 10000
PNet (ours)
10.01
0.90
1.799
0.78 0.003
0.00 1:000
PNet (ours)
0.93 10.01
0.92 10.01
0.87 10.05
0.87 10.05
0.00 10.00
Semantic Loss
Baseline
0.33
0.22:t0.04
0.50+0.00
0.33
0.57:
BaselineinD.pr
0.35=20.05
0.20#10.04
0.50=0.00
0.33
0.73#t0.12
C(S)* inD
0.39
0.26+
0.50=0.00
0.34+0001
0.56+0.22
C(S)* inD.prD
0.65
0.60++.07
0.50=0.00
0.33
0.36
Pre" inD
0.58 0.00
0.57=0.000
0.51+0.00
0.43+t0.01
0.00 =0.00
Pre" inD,prD
0.901
0.9011..01
0.77
0.76.10.6
0.00 :10.00
Pre" +C(S)* inD
0.42.10.04
0.36110.10
0.51
0.37
0.29
Pre" +C(S)* inD,ppDD
0.8810.02
0.8710.02
0.5010.00
0.33.10.00
0.00 100000
PNet (ours)
0.92 10.02
0.92 =0.02
0.88 10.10
0.88 10.11
0.00 10.00
PNer* (ours)
0.92 #0.01
0.91+0.01
0.82+0.05
0.81+0.05
0.00 10000
DeepProbLog
Baseline
0.35+
0.26+0000
0.50=0.01
0.37=0000
0.52+0.17
BaselinennD.pro
0.39 0.05
0.25+0.06
0.50=0.01
0.33++000
0.66++.18
C(S)* inD
0.37
0.36+00000
0.50=0.01
0.34+0.03
0.47=0.08
C(S)* peD
0.73
0.68.1..0
0.71 10.21
0.62.10.29
0.37
Pre" inD
0.58
0.58++000
0.51
0.44+0.00
0.00 #0.00
Pre" inD. prD
0.92 0.00
0.91-t0.00
t..00
0.80++0.01
0.00 110000
Pre" +C(S)* inD
0.42
0.41:t0.01
0.50=0.01
0.40#t0.02
0.17
Pre" + +C(S)* inD.prD
0.79
0.76:10.09
00.1
0.50+1.22
0.38
PNet (ours)
0.94 +0.00
0.94 #0.01
0.99 +0.00
0.99 + 0.000
0.00 +0.00
PNet* (ours)
0.94 #0.00
0.94 #0.01
0.99 #0.00
0.99 #0.00
0.00 =0.00
Table 11: Mitigation strategies compared for CCN+ on MMIST-EvenOdd, Best results in bold and second best underlined.
Acc(C)(1)
FI(C)(1)
Acc(Y)(t)
FI(Y)(1)
Cls(C)(L)
Baseline
0.03#0.07
0.01
0.9240.10
0.92+0.11
0.7340.05
C(S)*
0.36=0.10
0.24
10.06
0.90+a.07
0.52+0.11
Pre*
0.
0.22=0.10
0.91 40.09
0.90
0.5740.09
Pre" + C(S)*
0.54
0091
0.92+0.09
1.91100.10
0.34
EPre*
0.66#0.14
0.56=0.17
0.97
0.97
0.25+
PNet (ours)
0.96+0.01
0.95=0.05
0.96+0.07
0.96+ 08
0.00 +0.00
PNet (ours)
0.98 40.01
0.97 +0.01
0.96+0.09
0.95+
0.00 +0.00
30

---

0.27
0.10
0.12
0.16
0.24
94
Wilcet
0.00
0:31
3.09
3.10
0.31
0.34
0.00
0.12
0.02
0.11
0.23
0.10
0.08
0.18
0.07
0.83
0.15
0.03
Prwe::
0.90
0.04
0.03
0.02
3.42
0.23
0.02
0.16
0.07
0.5
Squara (hae
0.12
0.11
0.30
0.04
0.03
0.14
0.00
0.04
0.15
0,00
0.06
2.06
2.05
0.28
0.16
0.02
027
0.11
0.09
9.43
0.05
0.27
0.24
0.14
0.19
0.05
0.04
02
100c Wikeet
0.00
6.40
3.09
3.09
2.26
0.05
0.00
0.08
0.01
0.02
0.11
0.03
0.19
0.34
0.13
0.02
0.15
0.02
02
Iaree Mkat
0.00
0.00
3.02
3.04
2.52
0.20
0.03
0.12
0.02
0.03
0.04
0.24
0.13
0.15
0.00
0.03
2.01
2.02
2.34
0.23
0.02
0.11
9.05
Hargi mikuet
0.00
0.06
3.00
3.06
3.66
0.12
0.01
0.08
0.01
-00
(Parge Mkat
0.00
0.03
3.00
3.03
3.68
0.14
0.02
0.08
0.01
0.02
0.02
0.11
0.02
Goans Seat
teeak stal
Clack Wilane
spree Ran
Thmaee Ked
10me
Phroee
(a) SI S
(b) SL
+ +Pre*
0.00
2.00
0.00
0.00
0.70
0.00
PRasee Avd1
0.65
0.00
0.02
0.02
0:01
0.01
0.26
0.01
0.01
Seare, Mired
0.00
2.00
0.00
0.00
0.76
0.00
0.00
0.24
0.00
-0.8
Plpuac Vitwa
0.01
0.51
0.01
0.01
0.12
0.01
0.01
0.32
0.01
0.5
0.00
2.00
0.00
0.00
0.00
0.00
0.00
0.47
0.00
Dipara Bua
0.01
5.01
0.50
0.01
0.02
0.02
0.01
(Der
0.00
3.00
0.00
0.00
0.61
0.00
0.00
0.30
0.00
-0.6
look: Milel
0,00
3.00
0.00
0.00
0.67
0.4
0.02
2.02
0.05
0.00
0.02
0.02
Chingl Aet
0.01
0.01
0.00
0.01
0.00
0.01
0.98
0.02
0.02
0.00
0.00
0.00
0.00
(llaggi Vitws
0.01
9.01
0.01
0.01
0.00
0.01
0.02
0.93
0.02
0.00
2,00
0.00
0.00
0.00
0.00
0.00
Clangle Bual
0..0
0.00
0.01
0.02
Bleam
pline Mefi
Coule Neans
Mlad
mhsees Aeb
medde Beet
Sooee
Phlend
SI -iaD, peD
(d) SI 12D peb +re
Oeark Nowe
0.22
0.79
0.01
0.02
0.25
3.73
0.01
0.51
0.53
0.6
Vlate Bedl
0.03
0.01
0.02
0.52
0.00
2.00
2.00
0.01
0.22
0.00
9.03
0.06
Jiel Boil
0.03
0.01
0.08
0.00
0.01
2.06
001
0.02
0.51
-02
(Tagge Wise
0.00
0.01
0.01
0.03
3.51
0.2
0.47
(e) SL+PNet
Figure 11: Kand-Logic (slape,colour) concept confusion matrices for the standard and pretrained Semantic Loss model disentangled wrt. inputs (inD), inputs and the predicates (inD,prD) and integrating prototypes (PNet).
31

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Table 12: Standard mitigation strategies compared for ABL on ENIsT-EvenOdd.
Acc(C)(1)
F1(C)(1)
Acc(Y)(()
F1(Y)(1)
Cls(C)()
Baseline
0.06+
0.03=0.05
0.70=00.8
0.31
0.56:0.115
C(S)*
0.67:10.16
0.64
0.71
0.15:0.02
0.00+10.00
Pre
0.90 03
0.89=0.02
0.83=0.04
0.15=0.01
0.00++0.00
Pre +C(S)*
0.95+t0.01
0.93=0.01
0.90=0.01
0.17=0.01
0.00++0.00
(a) SL
(b) SL+C(S)"
(c) SL+Pre"
(d) SL4Pre" C(S)*
(e) SL+PNet
Figure 12:
concept confusion matrices for different mitigation strategies.
Crucially, even though concept-level scores for this mitigation are comparable to PNet models, the performance on label-level metrics is lower than the one of our approach. This can be explained by the fact that the embedding update introduced in Section 4 does a better job than the standard NeSy update in aligning the concept representation with the final label prediction task.
The improved concept alignment Nesy models enjoy when paired with prototypical extractors is even more clear looking at the confusion matrices: in Figure 12 we see how different mitigation strategies allow to gradually improve the Semantic Loss model ability correctly reconstruct the unlabelled concepts, with the prototypical model being the better. Similar considerations hold for the confusion matrices reported in Figure 11 for the case of Land-L we see that incorporating the disentanglement wrt. the shape and colour concepts does not help the Semantic Loss model to better disambiguate the concepts, but the difference between the two is evident as soon as the support set is made available for pretraining. Even in the case of pretrained, fully disentangled model however, there concept O is often confused with A. On the other hand, the prototypical NeSy model does not suffer for this confusion.
Lastly, we report the full results for BDD-OIA in Table 9. In this task the data are very unbalanced (c.f. Table 7). Interestingly, the best results our architecture achieves for F1(C) and Cls(C) show that the prototypical model is robust wrt. the unbalance and does not collapse its predictions on the dominant class "0". Conversely, pretraining on the support set not help in mitigating the shortcut reasoning behaviour. Even worse, the score for FC( decreases for pretrained models wrt. the standard baselines. Overall, the most competitive approach with prototypical models is C(S)* in its twofold disentangled version. However, this baseline well only achieves minor improvement wrt. the (C) score (0.11) of the standard DeepProbLog model (0.08). Conversely, the PNet model doubles this score (0.16). In addition, even if the C(S)* baseline succeeds in reducing concept collapse (0.51), it is still far from the lower collapse of the prototypical model (0.35). We observe that label-level metries and Acc(C) decrease in the PNet models: this is expected as, given the unbalanced data, basclines frequently predicting class "0".
RQ2. How does prototypical NeSy compare with other miligation strutegies?
In Table 8 we report an exhaustive picture for the comparison of shortcut mitigation strategies in SWIST-EvenÜKe The Table accounts for the mitigations introduced in [29] already discussed in the main text (Section (RQ2)): these are partial concept supervision (C), input reconstruction (R) via the introduction of a reconstruction loss with an encoder-decoded architecture, Shannon entropy loss (H), and combinations thereof. The scores for baseline models are obtained from [5] and we used the code for all the mitigation strategies and combinations thereof as in [29]. Also, to convey the broader picture, we included the computation for concept (Acc(C)) and label (Ace(Y)) accuracy. As expected, we see that the results wrt. Acc(C) mirror those reported in Table 4, where we reported the FI( for the same experiments. Moreover, our models consistently succeed in minimizing Cls(C), which 0.0 across all the models considered. On the other hand, we see that the mitigation strategies generally
32

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1.0
1.0
0.9
Acc(C)
Acc(C)
0.8
0.8
0.6
Standard NN
0.7
SL+PNet
0.4
PNe+
LIN+Pre*+C1S>
0.6
DPL+PNet
0.2
+CL
0.0
0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Unsupervised Data Percentage
Unsupervised Data Percentage
Acc(C) for PeSy+PNet models
(b) Acc(C) for NeSY+Pre+COS models
1.0
1.0
0.8
0.8
Acc
0.7
Acc(M)
Standafd NN
0.6
0.6
PNet
SL+Pre* +C(S))
0.5
LTN+ PNet
0.4
LTN
DPL+ PNet
Pre+
0.3
0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Unsupervised Data Percentage
Unsupervised Data Percentage
(c) Acc(Y) for Peet+ models
(d) Acc(Y) for NeSY PrSY+Pr)+C(S) models
1.0
0.9
0.8
0.7
Standard NN
0.6
0.6
SL+ PNet
SL+Pre* +C(S)*
0.5
PTN+
0.4
0.2 (c) F1(Y) for PeSy+PNet models
PNet
DPL+Pre*+C(S)
0.1
0.2 0.3 0.4
0.7
Unsupervised Data Percentage
Unsupervised Data Percentage
(f) F1(Y) for N models
0.04
SL+PNet
0.8
LTN+
0.6
Cis(C)
0.02
DPL+PNet
0.4
0.2
SL+Pre*
0.02
Standard
LTN+Pre*<CSS>
0.0
0.04
DPL+Pre*+C(S)*
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Unsupervised Data Percentage
Unsupervised Data Percentage
(g) Cls(C) for NeSY+PNet models
(h) Cls(C) for NeSY+Pic+C(S) models
Figure 13: Extensive Comparison of Prototypical NeSy models (left) Standard Pretrained ones (right) supervised on support sets, across different percentages of unlabelled datapoints on Evenndd
33

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1.0
1.0
Acc(Y)
0.8
Accuracy
DPL
Accuracy
0.8
Accuracy
0.6
6.6
0.5
Acc
0.4
Acc(C)
6.4
Acc|C)
Acc(C)
0.2
DPL Acc(C)
Acc(Y)
0.2
Acc(C)
Acc(Y)
Acc(C
Acc(Y
Number hidden classes
Number hidden classes
Number of hidden classes
(a) DPL+
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Figure 14: Aco(C), Ayc(Y) and Cls(C) obtained with the Prototypical NeSy models on MNIST-EvenOde when varying the number of hidden classes, i.e., the number of concepts for which we have zero examples. The red dotted lines indicate the performance of the corresponding baseline model.
do not have a great impact on the metrics over the final label (i.e., Acc(Y) and FI(Y)), as the NeSy models were already performing well on the unlabelled classes.
RQ3. Are Prototypical NeSy models robust wrt. the number of unlabelled datapoints?
In Figure 13 we report the results obtained in the same experiment reported in Section 6 (RQ3) wrt. the metrics Acc(C), AY)(Y), (C) and FI(Y). Similarly to FI(C), we can see from the Figure (a) that the results obtain for ACC(C) using PNets very stable and improve we add more unsupervised data. On the contrary, as (b) shows, if we use the standard NeSy models taking advantage of the supervisions both in a pretraining stage (Pre) and training time with cross- entropy loss (CTS), the results present a high variance and are very susceptible to the percentage of unsupervised data used for training. Similar considerations hold for ACC(Y) well in (c) and (d), with only the Semantic Loss baseline model maintaining lahel accuracy scores comparable with the PNet approach. Moreover, from (e) and (f) we notice the PNet approach consistently improves the standard NeSy models FI(Y) score as well (with SL being the only exception) for all the percentages of unsupervised data. Crucially, observe from (h) that all the standard models suffer from high concept collapse. How much each model collapses it predictions is again susceptible to the percentage of unsupervised data used for training. Now compare this behaviour with (g): our prototypical models have 0.0 concept collapse across all the percentages of unsupervised data.
RQ4. What the impact of having unlabelled classes?
In Figure 14 and Figure 15 we report the results obtained in the experiments described in Section (RQ4) wrt. Acc(C), Acc(Y) and Cls(C) the MNIST-EvenDdd task and the Kand-Logic task, respectively. As expected, the more classes are labelled the better the results become. However, is interesting to note that-in the better results over the concepts than using the standard NeSy models. The same happens for the Kand-Logic task, where need least two examples for the shape primitives and two for the colour primitives. However, we can again see that with just four labelled examples we get better results than all the standard NeSy models.
task-even by labelling only two images we obtain
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Figure 15: Acc(C), Aco(Y) and Cls(C) obtained with the Prototypical NeSy models on Kand-Logic when varying the number hidden classes, i.e., the number of concepts for which we have zero examples. The red dotted lines indicate the performance of the corresponding baseline model.
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Figure 16: FC( Ace(C) and Cls(C) obtained with PNet+SI. when varying the number of hidden classes and for different values of i.e., for different lower-tail quantiles of the x² distribution.
RQ5:
How important is the centroid initialization for unlabelled classes?
In Figure 16 report the results obtained wrt. FI(C), Acc(C) and Cls(C) for different lower-tail quantiles of the x distribution. Clearly, choosing p = 0.99 improved our results- no matter the metric considered.
35


Reviews:
[{"review_id": "02YIIjL825", "reviewer": "Reviewer_bWcR", "summary": "This paper introduces a method called PNet for utilizing limited labeled data in Neuro-Symbolic tasks under the guiding principle of constructing class prototypes. Specifically, for classes with labeled data, the mean of the (augmented) embeddings is used as the class prototype; for classes without labeled data, a prototype is randomly initialized based on the distances between existing prototypes. By doing so, this paper introduces an additional loss term based on the distance between embeddings and prototypes, alongside the traditional NeSy loss. This approach balances logical constraints and label information, mitigating the issue of reasoning shortcuts. Authors conducted multiple experiments to validate the effectiveness of the proposed method.", "questions": "1. Why does the prototype selection for classes without labeled data require introducing randomness (Equation (3))?\n2. Since this paper considers a classic neuro-symbolic task, it would be beneficial to adopt a more concise and traditional set of symbols to enhance the readability.\n3. If the proposed method can demonstrate a significant performance advantage in the fair comparisons mentioned in the weaknesses, I would consider increasing the score.\n4. If the method can showcase its superiority in more challenging scenarios, I would consider increasing the score.", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\n\n1.Authors considered multiple factors that could affect the effectiveness of the method and demonstrated its performance under different conditions through experiments.\n\n2.Authors demonstrated the limitations of the method through experiments.\n\nWeakness:\n\n1.The main experiment considered in this paper is overly simplistic—a (minor) variant of MNIST add with a small amount of labeled data. It is well-known that simple CNN networks can achieve over 95% accuracy on the MNIST dataset even when each class contains only one labeled image. Therefore, to validate the effectiveness of the proposed method, a baseline that first performs supervised learning using labeled data and then conducts neuro-symbolic learning should be added.\n\n2.The comparison between this method and others is not entirely fair. Other methods target more challenging scenarios without labeled data, whereas in cases where some labeled data is available for supervised pretraining, I believe the comparison methods could achieve similar performance to the proposed method.\n\n3.The paper addresses a classic neuro-symbolic task but introduces a new and relatively complex set of symbols in Section 2, particularly merging the Concept space and Final label space, which makes it difficult to read.\n\n4.The paper contains several typos and imprecisions, such as: two instances of \"(iii)\" in Section 2; the definition of mm on line 87 is not a mapping to the power set, which contradicts Example 2; and Equation (3) includes two $C_c$ symbols with entirely different meanings, further complicating readability.", "was_revised": true, "final_justification": "The authors address some of my concerns. Hence I increase my score. I suggest the authors further improve experiments and writing.", "rebuttal": "We would like to thank the reviewer for their thoughtful and constructive comments. We will incorporate the suggested discussions and clarifications in the revised manuscript, which will strengthen the overall contribution and improve the presentation and interpretation of the experimental results.\n\n> [Q1] Why randomness?\n\nWe chose to retain the well-known benefits of random weight initialization in deep learning—such as effective exploration of the parameter space and robustness across different seeds. In our setting, centroids act as learnable parameters (analogous to weights) and thus benefit from similar initialization strategies.\n\nHowever, initialization is particularly important in our setting due to the presence of reasoning shortcuts. Without any control on their placement, a prototype may be placed far from the relevant embeddings, leading the model to ignore it and instead collapse multiple concepts onto a single, incorrect prototype that still satisfies the constraints. To mitigate this, we sample centroids within a hyperball centered on the mean of known prototypes (with $p=0.99$). This encourages separation among mutually exclusive classes and reduces shortcut-induced failures, while still preserving the flexibility and diversity of random initialization.\n\n> [Q2] Standard notation.\n\nIf we followed the standard notation used e.g., in [1] then we would have made the assumption that there are a set of concepts (or intermediate outputs that are not supervised) and a set of final labels (of final outputs which are fully supervised). However, this assumption is not necessary in many NeSy systems (like SL), as they can seamlessly handle scenarios where each datapoint can be partially labelled with any subset of the labels, and there is not inherent distinction between intermediate concepts and final labels.\n\nAdditionally, the current notation allows us to explicitly associate every atom in the formula with a specific input-output pair of the neural network, making the connection between the two more explicit.\n\n> [Q3] Unfair comparison\n\nRegarding this point we would like to highlight that, in addition to the standard baselines (with no access to labelled datapoints)---indicated with LTN, DPL and SL, we have also compared with: \n1. *Fully supervised CNNs (the same we used as prototypical extractors) that have access to the same amount of supervised datapoints* on the MNIST-EvenOdd task, as reported with “Standard NN” in Figure 3. We agree that we should have run this baseline also on Kand-Logic and we should have included these results in Table 1. We will do so in case of acceptance. Below, we report the performance on the MNIST-EvenOdd dataset in Table A and the performance on Kand-Logic in Table B. As we can see from the baseline, with the small augmented labelled dataset available to our PNets, the fully supervised baselines only managed to get mean accuracy equal to 0.59 on MNIST-EvenOdd and 0.34 on Kand-Logic. We have also run the fully supervised baseline on the autonomous driving dataset (BDD-OIA), see answer below.\n3. *Augmented neurosymbolic AI baselines which are trained with both labelled and unlabelled datapoints* indicated with LTN(Aug), DPL(Aug) and SL(Aug) - these methods are  trained with exactly the same dataset as our methods. As we can see from Table 1, adding the same amount of supervisions helps in improving their performance, but in a much more limited way than using the PNets. This is because, for each unlabeled datapoint, the weights are updated solely based on the degree of constraint satisfaction. In contrast, when using prototypical networks, the update also accounts for the proximity of the embedding to each prototype. This also explains why our method exhibits significantly greater robustness to the number of unlabeled datapoints used during training compared to standard architectures, as demonstrated in RQ3 of the paper.\n\nTable A: Performance on MNIST-EvenOdd for (i) the fully supervised baseline, (ii) the augmented NeSy methods and (iii) our NeSy + prototypical networks. For F1 the higher the score the better---indicated as ($\\uparrow$)---for Cls the lower the score the better---indicated as ($\\downarrow$). Best results are always in bold\n\n| Task|Acc(C) ($\\uparrow$) | F1(C) ($\\uparrow$)|Cls(C) ($\\downarrow$)|\n|--|--|--|--|\n| Fully Supervised| 0.59±0.02 | 0.55±0.02 | 0.01±0.01|\n| LTN(Aug)   | 0.46±0.31| 0.37±0.35| 0.49±0.29|\n| LTN+PNet(Ours)| **0.98±0.00**| 0.97±0.00| **0.00±0.00** |\n| SL(Aug)|0.56±0.24| 0.45±0.27|0.26±0.24|\n| SL+PNet(Ours)|**0.98±0.00**|**0.98±0.01**| **0.00±0.00**|\n| DPL(Aug)|0.26±0.11|0.15±0.07| 0.68±0.06|\n| DPL+PNet(Ours)|**0.98±0.00**|**0.98±0.00**|**0.00±0.00**|\n\nTable B: Performance on Kand-Logic for (i) the fully supervised baseline, (ii) the augmented NeSy methods and (iii) our NeSy + prototypical networks\n\n|Task|Acc(C) ($\\uparrow$)|F1(C) ($\\uparrow$)|Cls(C) ($\\downarrow$)|\n|--|--|--|--|\n|Fully Supervised|0.34±0.01|0.34±0.01| **0.00±0.00** |\n|LTN(Aug)|0.33±0.01|0.31±0.02|0.59±0.17\n|LTN+PNet(Ours)|**0.94±0.00**|**0.94±0.00**|**0.00±0.01** |\n|SL(Aug)|0.36±0.02| 0.22±0.05|0.78±0.11|\n|SL+PNet(Ours)|**0.94±0.01**|**0.93±0.01** |**0.00±0.02**|\n|DPL(Aug)| 0.35±0.01|0.26±0.00|0.70±0.07|\n|DPL+PNet(Ours)|**0.94±0.00**|**0.94±0.00**|**0.00±0.00**|\n\nWe believe this comparison is fair, as every model has access to the same amount of data.\n\n> [Q4] Simple experiments\n\nWe would like to clarify that the MNIST-EvenOdd and Kand-Logic tasks were not introduced by us; rather, they were recently proposed specifically to address the challenge we study. These tasks are part of the “reasoning shortcut benchmark” (rs-bench), introduced at **NeurIPS 2024** (D&B track), with the goal of evaluating the ability of neuro-symbolic methods to overcome the reasoning shortcuts problem.\n\nThese tasks are particularly challenging due to the large number of possible reasoning shortcuts a model can exploit. For example, in the MNIST-EvenOdd task, there are 49 different digit-to-label assignments that perfectly satisfy the logical constraints, yet only one of them corresponds to the correct digit labeling. This arises because only a limited set of digit pairs appears in the training set. In Table C below we give the list of the 8 possible input image pairs together with their associated label (their sum) appearing in the dataset:\n\nTable C. Input image pairs and their labels in MNIST-EvenOdd dataset\n|Possible Input Image Pairs| Label|\n|--|--|\n|(0,6)|6 \n|(2,8)|10\n|(4,6)|10\n|(4,8)|12\n|(1,5)|6\n|(3,7)|10\n|(1,9)|10\n|(3,9)|12\n\nAs a result, the model may converge to a spurious mapping that satisfies the sum constraint on all training examples but mislabels every digit. For instance, interpreting each 0 image as 4 and 6 image as 2 leads to correctly infer the label 6 for every image pair (0,6). Similarly, the interpretation in Table D achieves perfect constraint satisfaction on the training set while failing to correctly classify every single digit image:\n\nTable D. Possible digits assignment for MNIST-EvenOdd\n\n|Actual Digit in the Image| Digit assigned by the model |\n|--|--|\n|0 | 4\n|1 | 5\n|2 | 6 \n|3 | 7\n|4 | 8\n|5 | 1\n|6 | 2 \n|7 | 3\n|8 | 4\n|9 | 5\n\nHence, even though MNIST-EvenOdd and Kand-Logic might appear simple datasets, they are actually not for NeSy models, as the knowledge is not enough to correctly identify the concepts. Additionally, as we have seen in the previous question, few labelled datapoints are *not* enough to achieve ~0.90 accuracy.\n\nHowever, we agree that real world scenarios might present noisier data. To better highlight how our model would perform in noisy and real world scenarios, we re-run the experiments on the autonomous driving dataset (BDD-OIA), but this time we used prototypes for *all concepts*, and not only those that lead to the action Stop. \n\nTo have a fair comparison, we also\n1. trained DPL not only with the unlabelled data but also with the 128 datapoints available to our method (indicated as DPL (Aug)), and \n2. the model in a fully-supervised way (using all the 16k datapoints), thus highlighting the difficulty of the task. \n\nWe report the overall results in Table E.\n\nTable E. Results on the autonomous driving dataset (BDD-OIA)\n\n|Model|F1(C) ($\\uparrow$)|Cls(C) ($\\downarrow$)|\n|--|--|--|\n|DPL|0.04±0.00|0.85±0.01|\n|DPL (Aug)|0.07±0.07|0.75±0.05|\n|Fully Supervised|0.06±0.06|0.78±0.10|\n|DPL+PNet|**0.16±0.15**|**0.35±0.01**|\n \nFinally, we report the results when stratifying the concepts according to the resulting action in Table F. For example, F1(Forward) is calculated as the mean over the F1 of all the concepts that cause the action ''Forward'', i.e., \"Green Light\", \"Follow\" and \"Road Clear\".\n\nTable F: Results on BDD-OIA stratified by the observed action\n|Method| F1(Forward)|Cls(Forward)|F1(Stop)|Cls(Stop)|F1(Left)| Cls(Left)| F1(Right)| Cls(Right)|\n|--|--|--|--|--|--|--|--|--|\n|DPL (Aug)|0.19±0.12| 0.80±0.09 |0.08 ±0.08|0.87±0.03|0.01±0.01|0.50±0.00|0.01±0.01|0.84±0.08| \n|Fully Supervised|0.15±0.15|0.78±0.12|0.04±0.03|0.89±0.05|0.02±0.01|0.50±0.24|0.02±0.02|0.95 ±0.02|\n|DPL+PNet|**0.35±0.07**|**0.23±0.23**|**0.09±0.07**|**0.40±0.11**| **0.08±0.02**|**0.00±0.00**|**0.17±0.01**|**0.77±0.09**|  \t\n\nOur original aim in including the BDD-OIA experiment (over just some concepts) was to demonstrate that our approach remains effective even in more realistic and noisy scenarios, where perfect accuracy—as seen in MNIST-EvenOdd and Kand-Logic—is not attainable. However, we acknowledge that the current presentation may have overemphasized the limitations without sufficiently highlighting the strengths. To address this, we will include the above additional results covering all concepts as well as comparisons with the fully-supervised baseline, which will better showcase the model's robustness and generalization in this setting.\n\n### References\n\n[1] Marconato et al. Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts, *NeurIPS*, 2023"}, {"review_id": "YiBTwhDzzw", "reviewer": "Reviewer_wXk3", "summary": "The paper addresses the problem of \"reasoning shortcuts\" in Neurosymbolic (NeSy) models, where a model might satisfy symbolic constraints syntactically but fail to learn the correct underlying semantics of concepts. The authors propose a Prototype-Augmented NeSy model, which integrates prototypical networks into standard NeSy frameworks. The core idea is to anchor the latent representations of concepts to prototypes learned from a very small number of labeled examples. The authors conduct experiments on the rs-bench benchmark, demonstrating improvements in concept learning accuracy and a significant reduction in reasoning shortcuts.", "questions": "1.How does prototype proximity translate into logical consistency? Can the authors formally explain why minimizing the distance between an embedding and a prototype should lead to better satisfaction of logical constraints?\n2.What is the precise relationship between semantic alignment and symbolic satisfaction? The authors use the term “semantic alignment” extensively, but it is never formally defined. How is it measured, and how does it relate to the logical correctness of concept assignments?", "limitations": "The limitations of this work primarily concern the scope and depth of the evaluation; see details in the weaknesses section.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n1.This paper addresses a real and underexplored problem - reasoning shortcut is NeSY, which contributes to the growing field of trustworthy and interpretable AI.\n2.There are clear empirical improvements across multiple tasks, and three different NeSy frameworks (DPL, SL, LTN).\n3.This paper provides a discussion regarding the limitations, and where the method fails (e.g., in BDD-OIA, a dataset requiring temporal/causal reasoning).\n\nWeakness:\nIntroduction:\n1.The “two-fold intuition” presented as the conceptual motivation lacks clarity and rigor. The first \"intuition\" is essentially a weak assumption (that a user who writes rules knows at least one example), and the second is a methodological design choice rather than a compelling motivation.\n2.The introduction leans heavily on citations to establish context but does not clearly clarify the conceptual novelty of the proposed method. Terms such as semantic alignment and background knowledge satisfaction are introduced early but remain undefined and unquantified.\n\nMethod:\n1.The integration of prototypes appears empirically motivated but conceptually superficial. While prototypical networks are effective in metric-based few-shot learning, symbolic reasoning often involves non-metric, compositional, and hierarchical structures. The paper does not address how these characteristics are preserved by the use of prototypes.\n2.The initialization of prototypes for unspecified classes (Eq. 3) using Gaussian noise added to known centroids is ad hoc and lacks semantic grounding. There is no assurance that these generated prototypes meaningfully represent logically coherent or distinguishable concepts.\n3. A closely relevant work is missing: Visual Recognition with Deep Nearest Centroids, ICLR. \n4.The dependence on augmentation also undermines generality. Despite claiming that “just one example per class is sufficient,” the method relies heavily on aggressive data augmentation (e.g. 200+ augmented samples per concept). This contradicts the narrative of label efficiency.\n\nThere remains a significant conceptual gap between the metric-based mechanism (prototype proximity) and the symbolic reasoning objectives. The assumption that proximity to a prototype leads to logical consistency is not theoretically grounded. The paper would benefit from a clearer, step-by-step formal justification of how and why prototype alignment improves symbolic constraint satisfaction. The paper in its current form feels more like a practical engineering solution effective in constrained synthetic settings, rather than a fundamental advance in neuro-symbolic reasoning.\n\nEvaluation:\n1.The performance gains reported (e.g. MNIST-EvenOdd F1 jump from 5% to 98%) are implausibly large. This raises concerns about the simplicity or synthetic nature of the benchmark tasks. Such gains suggest that these tasks may not meaningfully test for generalizable reasoning.\n2.The impressive results are confined to synthetic, small-scale benchmarks. The far more modest gains on the more realistic BDD-OIA dataset (F1 from 0.04 to 0.13) suggest that the method's effectiveness severely diminishes when faced with real-world ambiguity, visual complexity, and the need for richer contextual reasoning. \n3.The use of the term reasoning is overly broad. The benchmarks primarily involve basic logical structures such as addition or first-order logic. It's unclear how it would help with more complex, multi-step, or abstract reasoning tasks where the concepts themselves are not easily represented by a single visual prototype. \n4.The baselines are dated: DeepProbLog (2018), Semantic Loss (2018), and Logic Tensor Networks (2016). The absence of comparisons with more recent NeSy or hybrid neuro-causal models weakens the empirical case for the method’s state-of-the-art relevance.", "was_revised": true, "final_justification": "I thank the authors for their response. While some concerns have been addressed, this work still requires significant improvement to meet NeurIPS standards. It is not yet ready for publication, as substantial additional work is needed. I will maintain my original score.", "rebuttal": "We thank the reviewer for their insightful comments, as we believe implementing them will strongly improve the narrative of the paper.\n\nDue to space concerns, at times we had refer to answers to other Reviewers.\n\n> [Q1+W.Method1] Prototype proximity and logical consistency\n\nTaking into account the proximity of the embeddings of known datapoints when updating the weights of the neural network does not help in better satisfying the constraints, but rather in better assigning the ground truth label to the unlabeled datapoints. Indeed, for many datasets satisfying the constraints is not enough to assign the correct labels.\n\nTo give a simple example, suppose we have an image multi-label classification problem with three classes [$Zebra, Mammal, Fish$] and the constraint $Zebra \\to Mammal$, stating that if for an image the label $Zebra$ is predicted so should the label $Mammal$.\n\nSuppose we have a dataset with 10 images, 7 portraying fish, 1 portraying a zebra (and hence a mammal) and 2 portraying mammals that are not zebras.\n\nSuppose we have 0 labelled datapoints, then a model always predicting the labels [$Zebra, Mammal$] would achieve perfect logical consistency (and NeSy loss = 0) but it would get accuracy over the labels equal to:\n\nAcc($Fish$) = 0.3, Acc($Zebra$) = 0.1 and Acc($Mammal$) = 0.3\n\nSuppose now we have the same dataset with two labelled datapoints: one portraying a fish and one portraying a zebra (and the rest unlabelled). Then, a model always predicting [$Zebra, Mammal$] for all datapoints, but the one labelled [$Fish$]—for which it predicts [$Fish$]---would achieve loss = 0, but only get accuracy equal to:\n\nAcc($Fish$) = 0.4, Acc($Zebra$) = 0.2 and Acc($Mammal$) = 0.4\n\nNow, if we suppose that the datapoints are clusterable in the embedding space (for a formal analysis see answer to [Q4], Reviewer cxP2), then given the same dataset, a model like ours, if it achieves perfect loss, then it also achieves Acc($Fish$) = Acc($Zebra$) = Acc($Mammal$) = 1.0.\n\n> [Q2] Relationship between semantic alignment and symbolic satisfaction\n\nBy semantic alignment we mean associating to each datapoint the right set of labels. For symbolic satisfaction we mean that the set of labels associated to each datapoint satisfies the constraints. Hence, semantic alignment is a sufficient condition for symbolic satisfaction, and symbolic satisfaction is a necessary condition for semantic alignment. We will make this clearer in the final version of the paper.\n\n> [W.Intro1] Intuition lacks rigour\n\nWe thank the reviewer for raising this point.\n\nWe acknowledge that the first intuition, concerning the availability of randomly annotated datapoints, is indeed an assumption. However, we believe that it is *realistically grounded* in many real-world settings, where practitioners can easily annotate a few datapoints. Thus, our assumption reflects a standard partial supervision regime in which the user is asked to label in most cases just one or at most a handful of datapoints.\n\nRegarding the second intuition, we emphasize that this is not merely a heuristic design choice but a key explanatory mechanism underlying the success of our method, particularly in contrast to prior NeSy approaches.\n\nTo illustrate this, consider the MNIST-EvenOdd task discussed in the paper. In this setting, standard NeSy techniques typically proceed as follows:\n1. For datapoints labelled only with the value of the sum, the model is trained to ensure that the sum of the predicted digits return the correct value\n2. For datapoints labelled with both individual digit values, the standard cross-entropy loss is applied\n\nA core limitation of this approach lies in the nature of the constraint space. In MNIST-EvenOdd, given the pairs present in the dataset, there are 49 distinct combinations of digit values that satisfy the sum constraint (see answer to [Q4] to reviewer bWcR for a detailed explanation). Hence, for the unlabelled datapoints the model is not guided toward correct digit-level predictions, but is only trained to satisfy the constraint, allowing for many incorrect configurations that are constraint consistent.\n\nThe model should instead prefer constraint-satisfying configurations that are similar to those observed in the fully-labelled examples, but, more importantly, this notion of similarity cannot be easily defined in the raw input space, as it would require specifying a highly task-specific metric. Instead, we exploit the learned embedding space—induced by prototypical networks—where similarity to labelled examples emerges naturally through training.\n\nThis mechanism allows every datapoint to contribute to training in a way that is guided by proximity to the few annotated examples. As shown in Figure 3 (RQ3), this mitigates the sensitivity to the number of unlabelled datapoints.\n\nWe will include the above in the introduction.\n\n>[W.Intro2] Conceptual novelty\n\nThe mitigation strategies proposed so far and against which we compare are as follows:\n\n1. In [1], the authors propose to densely annotate a subset of the labels, and they are able to theoretically prove that if some concepts are fully annotated then the number of reasoning shortcuts a perfectly trained model can take goes down, as expected.\n2. Again in [1], the authors propose to use a reconstruction loss in addition to the cross-entropy loss. To use this loss, encoder-decoder architectures are needed: the encoder outputs both the concepts and a representation of the input, which is then used by the decoder to reconstruct the input.\n3. In [2], the authors propose to use the Shannon entropy loss as an additional term in the loss function. Notice that this is equal to 0 only when each distribution $p_\\theta(\\mathcal{C^k})$ is uniform, which is often not the case.\n\nWhen compared with the solutions above, ours:\n1. Does not require dense annotations\n2. Does not require an encoder-decoder architecture\n3. Does not assume that the distribution over each $\\mathcal{C}^k$ is uniform\n\nAdditionally, as shown in Table 2 of the paper, our method greatly outperforms all of the above.\n\nWe will make these aspects clearer in the introduction.\n\n> [W.Method1] Not clear how structures are retained in prototypical networks\n\nIn our approach, prototypes serve to anchor concept embeddings, while the NeSy loss $\\mathcal{L}_{\\text{NeSy}}$ encodes the hierarchical and compositional structure and drives the updates of both embeddings and prototypes accordingly.\n\nSection 4 shows that the gradient of the NeSy loss with respect to an embedding $z_I^k$ (for the $k$-th  group $\\mathcal{C}_k$) is a weighted combination of prototype directions, where the weights correspond to logic-consistent error signals (Theorem 4.1, Corollary 4.2).\n\n> [W.Method2] Initialization is ad-hoc and lacks semantic grounding\n\nCentroids are only initialized this way and then we train them. They will move according to the knowledge satisfaction: which will give them the semantic grounding.\n\n> [W3.Method3] Missing citation\n\nWe thank the reviewer for this citation, we will add it to our discussion. We will also take inspiration from it for our future work, where we will allow for more prototypes for each class.\n\n> [W.Method4] Reliance on data augmentation\n\nData augmentation is not a necessary aspect of our algorithm, just something that, when possible, leads to more stable centroids. To underline this, in the autonomous driving experiment, we have not used any data augmentation, still getting remarkable improvements. See answer to [Q4] of Reviewer bWcR for more details.\n\nPlease notice that data augmentation is a standard technique used in both semi-supervised learning and zero-shot learning, and it upholds the label efficiency narrative, as these annotations are not manually made but rather obtained automatically.\n\n> [W.Evaluation1] Perfomance gains too large, concerns on complexity of the task.\n\nWe would like to clarify that the MNIST-EvenOdd and Kand-Logic tasks (together with BDD-OIA) were not introduced by us; rather, they were recently proposed specifically to address the challenge we study. These tasks are part of the “reasoning shortcut benchmark” (rs-bench), introduced at **NeurIPS 2024** (D&B track), with the goal of evaluating the ability of neurosymbolic methods to overcome the reasoning shortcuts problem. Please see answer to [Q4] for reviewer bWcR for more details.\n\n> [W.Evaluation2] More modest results on BDD-OIA\n\nTo better highlight how much we can improve even on this task we re-run the experiments on BDD-OIA, but this time we used prototypes for *all concepts*, and not only those that lead to the action Stop. Please see answer to [Q4] Reviewer bWcR for more details on this experiment. The results show that our method can outperform all the considered models.\n\n> [W.Evaluation3] Unclear how the method performs on tasks where concepts are not easily represented by one prototype.\n\nThis is surely an interesting question for future work. However, regarding the current paper, please take into account that we manage to significantly improve the results on the the most recent benchmark proposed to evaluate NeSy models, where current methods do not manage to disambiguate between concepts that can be represented as a single prototype.\n\n> [W.Evaluation4] Old baselines\n\nThe baselines still represent sota methods in NeSy, they are used in many papers studying the reasoning shortcuts problems (e.g., rs-bench), and we compared against mitigation strategies that are more recent [1,2].\n\nIn answer to [Q1+W1.b] to Reviewer CSxp we showed that our method is able to greatly outperform ABLKit as well.\n\n### References \n\n[1] Marconato et al Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts, NeuIPS, 2023\n\n[2] Manhaeve et al Neural probabilistic logic programming in deepproblog. Artificial Intelligence, 2021\n\n[3] Yu-Xuan Huang et al., ABLkit: a Python toolkit for abductive learning, Frontiers of Computer Science, 2024"}, {"review_id": "dEjDw37v2F", "reviewer": "Reviewer_cxP2", "summary": "The paper proposes combining neuro-symbolic predictors with prototype-augmented neural networks to mitigate reasoning shortcuts. The central idea is to begin training with a minimal set of annotated prototypes. The approach is validated through experiments that demonstrate its effectiveness and analyze concept collapse as the number of supervised prototypes varies.", "questions": "* The method appears to rely heavily on initialization, particularly on having predefined prototypes. In standard disentanglement settings, wouldn’t this require one prototype per concept configuration? For example, in MNIST-Even-Odd with 10 digits per position, this would imply annotating 100 configurations. Have you explored scenarios where disentanglement does not hold in terms of required annotations?\n\n* If prototypes are learned from data, there’s a risk they may collapse to similar representations—i.e., the model may predict different concepts, but their prototypes end up qualitatively indistinguishable. Have you observed such behavior during training?\n\n* A key assumption seems to be that concepts are clusterable in the representation space. Could you elaborate on this requirement and its implications for generalization and applicability?\n\n* Relatedly, have you considered a theoretical analysis of how the use of prototypes impacts the number of reasoning shortcuts that can affect the model?\n\n* How does the method perform in fully unsupervised settings where no prototypes are available, i.e., the standard NeSy setup? You report some interesting results in the appendix (i.e., hidden class count and the BDD-OIA analysis), which seem impactful enough to highlight in the main text.\n\n* You mention that the solution proposed in [1] requires dense annotations across many datapoints, which may be costly. Could you clarify why this is the case? It would also be interesting to compare its annotation efficiency with that of prototype-based networks as well as for other mitigation strategies.\n\n* The code is currently unavailable. Could you clarify the reason for this? \n\n-----\n\n[1] E. Marconato, BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts, in UAI (2024)", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**: The paper addresses an important problem in the literature with a clear and well-motivated idea. It includes a wide range of experiments and research questions, and it explicitly discusses the limitations of the proposed approach.\n\n**Weaknesses**: Effectively mitigating RSs with a prototype-based network requires a well-initialized prototype for each concept configuration, and assumes that concepts form clear clusters. The zero-annotation setting is explored with less focus (there are interesting experiments in the appendix), and there is no theoretical analysis of how prototypes influence the number of reasoning shortcuts.", "was_revised": true, "final_justification": "Rebuttal", "rebuttal": "We thank the Reviewer for all the comments, and especially for encouraging us to consider a theoretical analysis of how the use of prototypes impacts the number of reasoning shortcuts that can affect the model. We believe this will be a great contribution to the paper.\n\n> [Q1] Reliance on prototype initialization and consideration of scenarios where disentanglement does not hold in terms of required annotations\n\nWe believe there might have been a slight misunderstanding here. We will make sure the below will be made clear in the paper. Regarding the reliance on prototype initialization, our method relies on prototype initialization as much as standard networks rely on weights initialization. In the paper we have simply provided a smart initialization scheme which allows us to avoid the trivial satisfaction of the constraints. This initialization scheme can be applied in any scenario. \n\nAdditionally, please note that our method does not require an example for every possible configuration, but rather one example per concept. For example, in the case of MNIST-EvenOdd, for RQ1 we annotated 10 images—one for every digit).\n\nIf we have one labelled example for every concept, the prototypes get trained and our method is very stable wrt weights initialization (see, e.g., Table 1), and as expected, if we have 0 labelled examples for more than one concept then we can see  in Figure 5  that the method becomes more noisy. (This chart also shows why it is important to pick $p=0.99$ in our initialization). \n\nWhen compared to training standard networks with NeSy methods (see Figure 3) we can see that our approach is much more robust to the number of unsupervised datapoints during training, even though the NeSy methods have been trained with exactly the same amount of labelled datapoints.\n\nRegarding the required annotations, in RQ4 we have studied what happens when we do not provide any labelled examples for a subset of the concepts. For example, in Figure 4 we show how the performance vary on the ground of how many classes we leave unspecified in the MNIST-EvenOdd dataset (i.e., how many classes we have in our dataset for which we have 0 labelled examples.) Notice that when we say that we have 8 specified classes, it means that we have only labelled two images and then we perform data augmentation on those images. \n\n\n> [Q2] Representation collapse\n\nWe have never experienced centroid collapse. We will provide the t-SNE plots in the paper in case of acceptance.\nIn BDD-OIA though, we have seen the representations of all the datapoints clustering around the negative prototypes  (and this is why we have concept collapse $\\not = 0$).\n\n> [Q3] Dataset needs to be clusterable in the representation space\n\nWe thank the reviewer for this comment, which gave us some ideas for future development of this work. While it is true that prototypical networks assume that each class forms a cluster in the representation space, we emphasize that the model is explicitly trained to learn embeddings in which such clusterability emerges, and in our practical tasks this assumption is empirically well-supported. In general, this assumption is particularly suited for standard NeSy methods, which learn “basic concepts” and then use the logical constraints to infer the more complex ones. \n\nHowever, we agree that for example in presence of highly multi-modal distributions one cluster might not be enough. To obviate this problem, we could study how to learn multiple prototypes for each class (as done in [1])  and then use the logical constraints. \n\n\n\n> [Q4] Theoretical analysis on the number of reasoning shortcuts\n\nWe totally agree that adding this an analysis on the number of reasoning shortcuts would be beneficial. Below we conduct such analysis and we will include it in the paper in case of acceptance. \n\nFor clarity, we follow the same notation as [2].\n\nSuppose the conditional distribution $p_\\theta(\\mathbf{Y} \\mid \\mathbf{X}; K)$ is induced by a prototypical network with embedding function $f_\\theta: \\mathbb{R}^d \\to \\mathbb{R}^{m_1 \\times m_2 \\times \\ldots \\times m_k}$. \n\nAssume that the datapoints are clusterable in the embedding space. Now, consider the subset of concepts indexed by $I \\subseteq [k]$ for which we have at least one labeled datapoint available. Let $\\mathcal{S} \\subseteq \\text{supp}(G)$ be the subset of concept assignments in the support of $G$ where concept supervision is available (at least one labeled example per concept $i \\in I$).\n\nBy adding a cross-entropy loss for the concepts in $I$ to the training objective, given the clusterability assumption, and under the Assumption 1 from [2] the number of deterministic $p_\\theta(\\mathbf{C} \\mid \\mathbf{G})$ optima becomes:\n\n$$\n\\sum_{\\alpha \\in \\mathcal{A}} \\mathbb{1}\\{\n\\bigwedge_{g \\in \\mathcal{S}} \\bigwedge_{i \\in I} \\alpha_i(g) = g_i\n\\}.\n$$\n\nNotice that this is *the same number of deterministic optima obtained by the data-based mitigation strategy in [2], which assumes that for every concept* $i \\in I$ and for **every datapoint** *the concept supervision is provided*. Our method achieves this equivalence by leveraging clusterability in embedding space, hence reducing the labeling effort required. Indeed, under our clusterability assumption, it suffices to have only a single labeled datapoint for each concept $i \\in I$ to match the theoretical guarantees obtained by dense supervision in [2].\n\n> [Q5] Tested on totally unsupervised settings\n\n One of the assumptions of this work is that we assume we have at least two datapoints belonging to two different classes (or concepts) labelled. Indeed, as shown above, without such supervision the number of reasoning shortcuts is equivalent to the number of reasoning shortcuts existing in any other model. Also, looking at the results shown in RQ4 and Figure 4 (as well as Figures 13 and 14 in the Appendix), it is clear that the more concepts have at least one labelled example, the better the results. This is a very lightweight assumption in many application domains, as randomly sampling two datapoints and labelling them is not costly. However, we agree that how to drop this assumption is an interesting avenue for the future.\n\n> [Q6] Solution proposed in [3] requires dense annotations\n\nWe believe there might have been a slight misunderstanding here. \n\nWe stated that one of the mitigation strategies proposed in [2] requires dense annotations. In [2], to conclude Proposition 5, the authors need to have dense annotations for the concepts they want to consider annotated, otherwise they cannot make use the fact that the deterministic concept distributions $p_\\theta(\\mathbf{C} |\\mathbf{X})$ of NeSy predictors $p_\\theta(\\mathbf{Y} \\mid \\mathbf{X}; K)$, correspond one-to-one to the deterministic distributions $p_\\theta(\\mathbf{C} \\mid \\mathbf{G})$ yielding label distributions $p_\\theta(\\mathbf{Y} \\mid \\mathbf{G}; K)$.\n\nIn our experimental analysis (RQ2) we have compared with the mitigation strategies proposed so far in the literature and also with their combination. As we can see, using prototypical networks managed to outperform even the combination of the three mitigation strategies proposed so far. \n\nRegarding a study on the efficacy of these methods, we will try to conduct one in the rebuttal time.\n\n> [Q7] Code unavailable\n\nOur paper is not yet publicly available on arxiv. We will make the code and paper publicly available upon publication. \n\n### References: \n\n[1] Allen et al. Infinite Mixture Prototypes for Few-Shot Learning, 2019.\n\n[2] Marconato et al. Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts, 2023\n\n[3] E. Marconato, BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts,  UAI (2024)"}, {"review_id": "r9NVKTIjS4", "reviewer": "Reviewer_CSxp", "summary": "This paper proposes using prototype-augmented neural networks to mitigate reasoning shortcuts. In the proposed method, prototype extractors with learnable parameters first compute class centroids from few labeled examples. A neural network is then trained to classify datapoints based on their proximity to the corresponding centroids while simultaneously satisfying background knowledge constraints.\n\nThe proposed method can be used in conjunction with other NeSy models, such as DPL, LTN, and SL. Experiments show that the proposed method can significantly enhance these NeSy approaches, achieving F1-scores of 98% on MNIST-EvenOdd and 94% on Kand-Logic, even when only one labeled datapoint is provided for each concept.", "questions": "1. Consider a direct, intuitive strategy: first, use an SSL or one-shot learning method to predict labels for the unlabeled data, and then provide these pseudo-labels to a NeSy algorithm. What are the differences and potential advantages of the proposed prototype-augmented strategy compared to this approach?\n\nI would be willing to raise my score if the authors could provide convincing evidence that their algorithm is substantially different from, and outperforms, more straightforward counterparts. \n\nSuch a comparison would be necessary to fully support the claimed novelty of their work.", "limitations": "As mentioned in the weaknesses section, the proposed approach assumes that concepts form well-separated clusters. This property may not hold for more realistic NeSy tasks.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n1. To my knowledge, the idea of introducing prototypical neural networks into NeSy research is novel and interesting.\n2. The problem setting is solid and reasonable. The assumption of having one or a few labeled examples per class is intuitive and practically feasible.\n3. The experimental results are quite impressive, as reasoning shortcuts are almost completely avoided on the tested benchmarks, especially MNIST-EvenOdd and Kand-Logic.\n4. The proposed method can be easily incorporated into other NeSy models, thus extending its practical usefulness.\n\n**Weaknesses**\n\n1. The methodology of this work seems to heavily overlap with existing NeSy research that utilizes Semi-Supervised Learning (SSL) techniques, but this relationship is not discussed in detail in the manuscript.\n    - The proposed method is primarily based on the assumption that \"datapoints close to each other are more likely to belong to the same concept,\" which is known as the cluster assumption in SSL research.\n    - To my knowledge, some existing NeSy research already utilizes SSL techniques, either explicitly or implicitly (e.g., by using distance-based metrics between datapoints in loss functions). However, most experimental results reported in the manuscript (except Table 2) are based only on the proposed method or its ablations; a proper comparison with more relevant methods would be helpful.\n    - From an SSL perspective, the proposed framework is quite similar to a process of: 1) using self-training to assign labels to unlabeled datapoints by first initializing centroids from labeled data, assigning pseudo-labels based on distance, and iterating until convergence; and then 2) using this predicted label information to enhance NeSy models. Consequently, the novelty of the proposed method appears somewhat limited.\n\n2. The crucial cluster assumption may not always hold in typical NeSy application scenarios, which relates to the authors' discussion in Section 7 regarding the method's sensitivity to ambiguities in concept definition.\n    - IMHO, a major motivation for using NeSy methods is for scenarios where data alone is insufficient—for example, with very noisy data. In such cases, one hopes a NeSy system can perform better by utilizing symbolic knowledge. \n    - This leads to a paradox: if data is clean and can be easily classified, neural networks are sufficient, and there's no need for NeSy; if data is noisy or concept definitions are ambiguous, NeSy is needed, but the proposed method would not work well in this case. This raises the question of when we should expect to use it.", "was_revised": true, "final_justification": "The authors provide additional experimental results and introduce new baseline models for comparison during rebuttal. These updates have addressed most of my concerns. However, some areas could still be clarified, such as the specific conditions or assumptions under which PNet is expected to succeed or fail.\n\nGiven the substantial improvements, I am raising my score.", "rebuttal": "Firstly, we would like to thank the reviewer for their comments, which allowed us to highlight better both the novelty of our approach and its broad applicability. Additionally, we are thankful for pointing us towards the abductive learning literature, if we have missed some reference, please let us know.\n\n> [W1.a] Same assumption as SSL+Nesy (abductive learning) methods \n\nOur method relies on a slightly different assumption: namely, that datapoints are similar—i.e., clusterable—in the learned embedding space. The embedding space is not fixed but is shaped dynamically during training via backpropagation from both the supervision provided by labelled datapoints and the neuro-symbolic (NeSy) loss. Consequently, similarity in this space does not necessarily correspond to similarity in the original input (feature) space. Rather, it reflects a form of functional similarity that integrates both the data and the symbolic knowledge expressed in the constraints. In Section 4, we formally derive the gradient of the total loss, showing that the embeddings are updated based on both (i) their proximity to class prototypes and (ii) the error signal stemming from the NeSy component. This confirms that the learned representation is shaped by the interplay between data-driven and knowledge-driven signals.\n\nWhenever pseudo-labels are involved, it is indeed a core problem to decide how and when to commit to a label, as an early pseudo-labelling mistake might propagate further. This is why in the abductive learning community, many researchers have studied how to define consistency measures to update the candidate pseudolabels with the most similar ones which are also compliant with the background knowledge. For example, [1,2] take the consistency score as the size of the largest subset of unlabeled examples with abduced labels that are consistent with the knowledge base, while [4,5,6] measure\nthe confidence of the predicted labels by the machine learning model. \n\nThe most similar work to ours in spirit is ABLSim [7], as they also use the embedding similarity to guide training. However, ABLSim constructs a similarity-based consistency score purely in the embedding space, encouraging samples with identical (pseudo-)labels to form tight clusters and those with differing labels to be well-separated. This score is used to select among logically consistent pseudo-label assignments, and the perception model is trained accordingly. In contrast, our method integrates symbolic knowledge directly into the loss function, such that the embedding space is shaped not only by class-based proximity (via prototype loss) but also by gradients arising from differentiable neuro-symbolic constraints. This results in an embedding space that reflects functional similarity: two points may be embedded nearby not because they share superficial input features, but because they induce similar outcomes with respect to both the downstream task and the symbolic knowledge. As shown in our gradient derivations (Section 4), this joint signal ensures that the learned representation is aligned with the structure of both the data and the knowledge base—beyond what similarity in feature space alone can capture. We will integrate this discussion in the paper.\n\n\n> [Q1+W1.b] Comparison with SSL+NeSy methods or intuitive strategy\n\nWe thank the reviewer for their suggestion, and to answer to their question we compare our method against one abductive learning method [1-7], that we believe is a much stronger baseline than the intuitive strategy. \n\nIn particular, we trained this model using the ABLKit package [3] on MNIST-EvenOdd, (i) first without giving the model any labelled datapoint (indicated with ABL in Table A below) and then (ii) giving the model access to exactly the same augmented labelled datasets as our models (indicated as ABL (Aug) in Table A below).  In Table A we report the results. For comparison, we also report the performance of our DPL+PNet.\n\nTable A. Average results obtained on MNIST-EvenOdd over 5 seeds.\n|Model|Acc(Y)|Acc(C)|\n|--|--|--|\n|ABL|0.70±0.05|0.12±0.12|\n|ABL(Aug)|0.72±0.07|0.51±0.24|\n|DPL+PNet|**0.99±0.00**|**0.98±0.00**|\n\nAs we can see from the results, the semi-supervised method is able to improve its own performance by using the augmented datasets, but is not able to perform as well as the prototypical networks. The much higher standard deviation presented by the abductive learning method is also something worth noticing: when performing pseudo-labelling, the methods have to “commit” to a choice (e.g., by classifying an image of the digit 7 as indeed 7 or as 9):, if the wrong decision is taken early on, then this error will propagate through training. \n\nMore in general, notice that ABLKit obtains very good performance on the standard MNIST-Add dataset, where it achieves Acc(C) = 0.98, thus outperforming both LTN and DeepProbLog. However, MNIST-EvenOdd is a particularly challenging scenario due to the high number of reasoning shortcuts it presents. Indeed, even assuming the model achieves perfect final label accuracy (Acc(Y) = 1.00), there remain 49 possible digit-level assignments it could have produced, of which only one corresponds to the correct underlying reasoning. This highlights the difficulty of the task: neither the knowledge nor the data alone are enough to learn the right concepts. \n\n> [W1.c] The method proposed is very similar to training with SSL or one-shot learning until convergence and then using this predicted label information to enhance NeSy models. \n  \nOur approach is similar in spirit to the method proposed by the reviewer but actually it very presents some fundamental differences highlighted by the following example. Suppose we have (as in our case) a single example for each concept (even augmented) and training a SSL or one-shot learning method until convergence. The probability of the network overfitting on the few available examples will be very high. If then we use the overfitted network to train with the NeSy methods, then each network update will only take into consideration the error with respect to the NeSy loss thus making it very likely that the model will again fall prey to the reasoning shortcuts problem. On the contrary, the weights update done in our method takes into account both embedding similarity and knowledge satisfaction for all datapoints (both labelled and unlabelled) -- see Theorem 4.1 and Corollary 4.2.\n\n> [W2] NeSy methods should perform well in noisy scenarios\n\nWe totally agree with the reviewer. Given your response and those of other reviewers, we realised though that we might have overstated the limitations of the approach. To obviate with such problem, we run the following models on the autonomous driving task (BDD-OIA) and report the results obtained in Table B:\n\n1. A fully supervised model, which has been trained on all images and all their lab (Fully Supervised in the Table),\n2. An augmented version of DeepProbLog, which has been trained with all the unsupervised images and the 128 labelled datapoints we used to train our model (DPL (Aug) in the Table),\n3. A new version of our model, where we use prototypes for all concepts, and not just those that cause the action “Stop” (DPL+PNet in the Table).\n\nFor completeness, we also include the results with the standard DPL method, which has been trained with no labelled images. \n\nTable B. Results on the autonomous driving dataset (BDD-OIA)\n\n|Model|F1(C) ($\\uparrow$)|Cls(C) ($\\downarrow$)|\n|--|--|--|\n|DPL|0.04±0.00|0.85±0.01|\n|DPL (Aug)|0.07±0.07|0.75±0.05|\n|Fully Supervised|0.06±0.06|0.78±0.10|\n|DPL+PNet|**0.16±0.15**|**0.35±0.01**|\n\nAs we can see from the results, even though the improvements are smaller than in the MNIST-EvenOdd and Kand-Logic, using DeepProbLog and our PNets we manage to outperform even the fully supervised model, trained with all the supervised images available in the dataset. To train DPL+PNet on the other hand, we only had to label only 128 images, with no data augmentation performed. \n\nIf we compare the results concept by concept for the fully supervised model and DPL+PNets, we can see that for almost all the concepts DPL+PNet outperforms the fully supervised model. \n\nTable C. Concept-level performance (in terms of F1) on BDD-OIA\n\n|Concept|Fully Supervised|DPL+PNet|\n|--|--|--|\n|Green Light|16.66|**46.01**|\n|Follow Traffic|25.39|**30.34**|\n|Road is clear|3.70|**28.53**|\n|Red Light|**9.41**|5.36|\n|Traffic Sign|9.51|**21.31**|\n|Obstacle Car|0.55|**3.21**|\n|Obstacle Person|0.00|**1.16**|\n|Obstacle Rider|6.48|**23.27**|\n|Obstacle Other|0.29|**2.37**|\n|No Lane Left|0.00|**0.31**|\n|Obstacle on Left|0.00|**3.56**|\n|Solid Line Left|0.00|**1.41**|\n|On Right Turn Lane|0.10|**1.11**|\n|Traffic Light Allows Right|0.74|**7.18**|\n|Front Car Turning Right|0.30|**3.18**|\n|No Lane Right|4.40|**31.72**|\n|Obstacle on Right|3.29|**32.17**|\n|Solid Line Right|3.84|**26.69**|\n|On Left Turn Lane|27.98|**38.57**|\n|Traffic Light Allows Left|5.02|**22.92**|\n|Front Car Turning Left|5.57|**9.59**|\n\nThis is indeed in line with the assumption of the reviewer, that in presence of noisy data a NeSy system with access to very limited data can perform better than a fully data-driven one.\n\n### References:\n\n[1] Wang-Zhou Dai et al., Bridging Machine Learning and Logical Reasoning by Abductive Learning, Neurips, 2019\n\n[2] Zhi-Hua ZHOU, Abductive learning: towards bridging machine learning and logical reasoning, 2019\n\n[3] Yu-Xuan Huang et al., ABLkit: a Python toolkit for abductive learning, Frontiers of Computer Science, 2024\n\n[4] Qing Li et al., Closed loop neural-symbolic learning via integrating neural perception, grammar parsing, and symbolic reasoning. ICML, 2020.\n\n[5] Wang-Zhou Dai and Stephen H. Muggleton. Abductive knowledge induction from raw data. IJCAI, 2021. \n\n[6] Le-Wen Cai et al. Abductive learning with ground knowledge base. IJCAI, 2021\n\n[7] Yu-Xuan Huang et al., Fast Abductive Learning by Similarity-based Consistency Optimization, NeurIPS, 2024"}]

Author rebuttal:

Meta-review: All reviewers agree that the paper tackles a significant problem in neuro-symbolic integration. Most of the issues raised during in the reviewers were addressed by the rebuttal, and in fact two reviewers increased their score. The other two were not particularly responsive. I went through their main gripes with the paper (such as usage of simple data sets, surprisingly large performance gains, lack of fairness in the evaluation), and it seems to me they *have* been addressed by the authors.

For this reason, I am leaning toward acceptance. I recommend the authors to clarify all points of contention in the paper and to implement all changes they promised in their rebuttal.
Decision: Accept (poster)