SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The implementation of DPL is taken from [21] and follows exactly Eq. (1), where to each world a label is assigned accordingly to the prior knowledge  $\mathsf{K}$ . We implemented SL following the original paper [8], with the only difference that the prediction of the labels  $\mathbf{Y}$  is done on top of the logits of  $p_{\theta}(\mathbf{C} \mid \mathbf{x})$ . This relaxes the conditional independence between labels and concepts while being more in line with the generative process we assumed. The implementation of LTN is adapted from LTN-pytorch [86] and is based on the satisfaction loss introduced in Appendix A.
C.2 Data sets & Count of the Reasoning Shortcuts
Here, we illustrate how to count explicitly the number of deterministic RSs using the equations in Table 1, restricting ourselves to the case where no disentanglement is in place, for simplicity. Similarly to Section 4, we assume A1 and A2 and that  $\mathrm{supp}(\mathbf{G}) = \mathcal{C} = \mathcal{G}$ , and count the total number of deterministic optima (or det-opts for short) under different mitigation strategies. In the following,  $\mathcal{C}_{\mathbf{y}} \subseteq \mathcal{C} = \mathcal{G}$  refers to the set of  $\mathbf{c} \in \mathcal{C}$  or  $\mathbf{g} \in \mathcal{G}$  that are mapped by  $\mathsf{K}$  to label  $\mathbf{y} \in \mathcal{V}$ . Note that  $\sum_{\mathbf{y}} |\mathcal{C}_{\mathbf{y}}| = |\mathcal{G}|$ .
Explicit count for the likelihood. When the concept extractor  $p_{\theta}(\mathbf{C} \mid \mathbf{X})$  is sufficiently expressive, the deterministic mappings  $\alpha$  are essentially arbitrary. Specifically, the set  $\mathcal{A}$  of these  $\alpha$ 's includes all functions from  $\mathcal{G}$  to  $\mathcal{C}$ . These functions can be explicitly enumerated by counting how many ways there are to map each input vector  $\mathbf{g}$  to an arbitrary vector  $\mathbf{c}$ .
Here, we are interested in counting the number of  $\alpha$ 's that attain optimal likelihood. Each such  $\alpha$  has to ensure that each  $\mathbf{g} \in \mathcal{G}$  is mapped to a  $\mathbf{c} \in \mathcal{C}_{\beta_{\kappa}(\mathbf{g})}$  that yields the correct label, or in short,  $\forall \mathbf{g} \cdot \alpha(\mathbf{g}) \in \mathcal{C}_{\beta_{\kappa}(\mathbf{g})}$ . This condition is satisfied if and only if, for every  $\mathbf{y}$  in the data set, every  $\mathbf{g} \in \mathcal{C}_{\mathbf{y}}$  is mapped to a  $\mathbf{c}$  that is also in  $\mathcal{C}_{\mathbf{y}}$ , and there are exactly  $|\mathcal{C}_{\mathbf{y}}|^{|\mathcal{C}_{\mathbf{y}}|}$  ways to do this. This immediately shows that the overall number of det-opts  $\alpha$  is:
$$
\sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {G}} (\beta \circ \alpha) (\mathbf {g}) = \alpha (\mathbf {g}) \right\} = \prod_ {\mathbf {y} \in \mathcal {Y}} | \mathcal {C} _ {\mathbf {y}} | ^ {| \mathcal {C} _ {\mathbf {y}} |} \tag {82}
$$
This associates an explicit number to Theorem 2.
Explicit count for the reconstruction. The effect of adding a reconstruction penalty is that now, in order to achieve optimal loss,  $\alpha$  has to map different g's to distinct concepts c's. Unless this is the case, it becomes impossible to reconstruct the ground-truth concepts from the learned ones, and therefore the input x generated from them.
The resulting computation is the same as above, except that now we have to associate the different  $\mathbf{g}$ 's to different  $\mathbf{c}$ 's without replacement. This yields the following count:
$$
\sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {G}} (\beta \circ \alpha) (\mathbf {g}) = \alpha (\mathbf {g}) \right\} \cdot \mathbb {1} \left\{\bigwedge_ {\mathbf {g}, \mathbf {g} ^ {\prime} \in \mathcal {G}: \mathbf {g} \neq \mathbf {g} ^ {\prime}} \alpha (\mathbf {g}) \neq \alpha \left(\mathbf {g} ^ {\prime}\right) \right\} = \prod_ {\mathbf {y} \in \mathcal {Y}} | \mathcal {C} _ {\mathbf {y}} |! \tag {83}
$$
Explicit count for concept supervision. For the combination of logic and concept supervision, we suppose that for each equivalence class  $\mathcal{C}_{\mathbf{y}}$ , there are  $\nu_{\mathbf{y}}$  ground-truth concepts provided with supervision and that all concept dimensions  $C_i$  receive this supervision. Let  $S \subset \mathcal{G}$  be the set of supervised concepts, such that  $|S| = \sum_{\mathbf{y} \in \mathcal{Y}} \nu_{\mathbf{y}}$ . From above, this means that we are specifying a total of  $|S|$  ground-truth concepts. From Table 1, we have that the number of det-opts amounts to:
$$
\sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {G}} (\beta \circ \alpha) (\mathbf {g}) = \alpha (\mathbf {g}) \right\} \cdot \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {S}} \bigwedge_ {i = 1} ^ {k} \alpha_ {i} (\mathbf {g}) = g _ {i} \right\} = \prod_ {\mathbf {y} \in \mathcal {Y}} | \mathcal {C} _ {\mathbf {y}} | ^ {| \mathcal {C} _ {\mathbf {y}} - \nu_ {\mathbf {y}} |} \tag {84}
$$
Finally, combining together the terms of prediction, reconstruction, and concept supervision, we obtain:
$$
\begin{array}{l} \sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {G}} (\beta \circ \alpha) (\mathbf {g}) = \alpha (\mathbf {g}) \right\} \times (85) \\ \times \mathbb {1} \left\{\bigwedge_ {\mathbf {g}, \mathbf {g} ^ {\prime} \in \mathcal {G}: \mathbf {g} \neq \mathbf {g} ^ {\prime}} \alpha (\mathbf {g}) \neq \alpha \left(\mathbf {g} ^ {\prime}\right) \right\} \times (86) \\ \times \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in S} \bigwedge_ {i = 1} ^ {k} \alpha_ {i} (\mathbf {g}) = g _ {i} \right\} = \prod_ {\mathbf {y} \in \mathcal {Y}} | \mathcal {C} _ {\mathbf {y}} - \nu_ {\mathbf {y}} |! (87) \\ \end{array}
$$
The disentangled case. If the network is disentangled, the enumeration procedure becomes substantially more complicated and cannot be written compactly in closed form. While the number of deterministic optima  $\alpha$  can still be computed exactly using model counting [29, 30], doing so is not necessary for the scope of our paper and therefore left to future work.
C.2.1 Dataset: XOR
This dataset, introduced in Example 3, is a toy data set containing 3 bits  $\mathbf{g} = (g_1, g_2, g_3)$ , for a total of 8 possible combinations. The task consists in predicting the XOR operation among them, namely  $y = (g_1 \oplus g_2 \oplus g_3)$ . The dataset is exhaustive and has no validation and test set. The model performances are evaluated on the training set.
Reasoning shortcut. For this dataset, we have  $|\mathcal{C}_0| = |\mathcal{C}_1| = 4$ . RSs arise depending on the structure of the underlying network. When the ground-truth concepts are processed all together without any mitigation strategy, we obtain that the number of *det-opt* amounts to:
$$
\prod_ {y \in \mathcal {Y}} \left| \mathcal {C} _ {y} \right| ^ {\left| \mathcal {C} _ {y} \right|} = 4 ^ {4} \cdot 4 ^ {4} \tag {88}
$$
The confusion matrices for all methods are reported in Appendix C.5. On the other hand, we show that in the disentangled case, only two combinations suffice to identify the correct solution, e.g.,
$$
\left\{ \begin{array}{l} \alpha (0) \oplus \alpha (0) \oplus \alpha (0) = 0 \\ \alpha (0) \oplus \alpha (0) \oplus \alpha (1) = 1 \end{array} \right. \tag {89}
$$
and here the only viable solution for the two is  $\alpha(0) = 0$  and  $\alpha(1) = 1$ . This condition is met in all our experiments in Table 2.
C.2.2 Dataset: MNIST-Addition
We consider the version introduced in [7], which consists of couples of digits, each ranging from 0 to 9, and the target consists in predicting the correct sum, i.e.,  $y = g_{1} + g_{2}$ . This data set contains all possible combinations, for a total of 100. The training set contains 42k data, the validation set 12k, and the test set 6k.
Reasoning shortcuts. RSs arise only as a result of the joint prediction of both digits. Notice that the number of elements  $|\mathcal{C}_y|$  for each sum  $\mathbf{y}$  can be evaluated as:
$$
\left| \mathcal {C} _ {y} \right| = \left\{ \begin{array}{l l} y + 1, & \text {i f} y \leq 9 \\ (1 8 - y) + 1, & \text {o t h e r w i s e .} \end{array} \right. \tag {90}
$$
Therefore, the total number of det-opts amounts to:
$$
\prod_ {y \in \mathcal {Y}} \left| \mathcal {C} _ {y} \right| ^ {\left| \mathcal {C} _ {y} \right|} = \prod_ {y = 1} ^ {9} y ^ {2 y} \cdot 1 0 ^ {1 0} \tag {91}
$$
When providing disentanglement, the number of RSs reduce to 0, as it sufficient to have the sums:
$$
\alpha (c) + \alpha (c) = 2 \cdot c \tag {92}
$$
to uniquely identify the value of the digit  $c$ .
C.2.3 Dataset: MNIST-EvenOdd
This data set, proposed by [21], is a biased version of MNIST-Addition, where only some combinations of the digits appear. We consider here a more challenging scenario w.r.t. the proposed version, consisting of the sums:
$$
\left\{ \begin{array}{l} \mathbf {0} + \mathbf {6} = 6 \\ \mathbf {2} + \mathbf {8} = 1 0 \\ \mathbf {4} + \mathbf {6} = 1 0 \\ \mathbf {4} + \mathbf {8} = 1 2 \end{array} \right. \quad \wedge \quad \left\{ \begin{array}{l} \mathbf {1} + \mathbf {5} = 6 \\ \mathbf {3} + \mathbf {7} = 1 0 \\ \mathbf {7} + \mathbf {9} = 1 0 \\ \mathbf {3} + \mathbf {9} = 1 2 \end{array} \right. \tag {93}
$$
Overall, the training set contains 6720 data, the validation set 1920, and the test set 960.
Reasoning shortcuts. We describe the RSs that arise even when the architecture incorporates disentanglement. We evaluate the possible RSs empirically noticing that the system of observed sums can be written as a linear system, as done by Marconato et al. [21]:
$$
\left\{ \begin{array}{l} \alpha (0) + \alpha (6) = 6 \\ \alpha (2) + \alpha (8) = 1 0 \\ \alpha (4) + \alpha (6) = 1 0 \\ \alpha (4) + \alpha (8) = 1 2 \end{array} \right. \quad \wedge \quad \left\{ \begin{array}{l} \alpha (1) + \alpha (5) = 6 \\ \alpha (3) + \alpha (7) = 1 0 \\ \alpha (1) + \alpha (9) = 1 0 \\ \alpha (3) + \alpha (9) = 1 2 \end{array} \right. \tag {94}
$$
Now, notice that we can find independent reasoning shortcuts for each of the two sides since they do not share any digits. For the LHS, we consider the sum  $\alpha(2) + \alpha(8) = 10$  and notice that we can find at most 10 different attributions for having a correct sum. Notice that, some of them are not allowed as  $\alpha(8) = 0, 1$  leads to inconsistent values for the fourth sum, and  $\alpha(8) = 3$  leads to an inconsistent sum for the first equation. So in total, for the LHS, we obtain 7 possible solutions and, by symmetry, the same number also for the LHS. In total, the number of  $\det$ -opts is equal to  $7 \cdot 7$ .
Experimentally, we consider limited concept supervision on  $I = \{4, 9\}$  which should be sufficient, in principle, to disambiguate between even and odds digits. This happens because specifying  $\alpha(4) = 4$  and  $\alpha(9) = 9$  admits only the ground-truth concepts as the optimal solution.
C.2.4 Dataset: MNIST-AddMul
In this dataset, we consider fewer combinations of digits, explicitly:
$$
\left\{ \begin{array}{l} \mathbf {0} + \mathbf {1} = 1 \\ \mathbf {0} + \mathbf {2} = 2 \\ \mathbf {1} + \mathbf {3} = 4 \end{array} \right. \tag {95}
$$
and similarly for multiplication. This data set contains 1680 training examples, 480 for the validation, and 240 for the test set.
Reasoning shortcuts. For the case of the addition task, we can have 2 possible solutions, which are:
-  $\alpha(0) = 0$ ,  $\alpha(1) = 1$ ,  $\alpha(2) = 2$ , and  $\alpha(3) = 3$ ;
-  $\alpha(0) = 1$ ,  $\alpha(1) = 0$ ,  $\alpha(2) = 1$ , and  $\alpha(3) = 4$ .
For multiplication, we have that since  $\alpha(1) \cdot \alpha(3) = 3$  it can be either  $\alpha(1) = 1$  or  $\alpha(1) = 3$ . In both cases, it holds that  $\alpha(0) = 0$  and  $\alpha(2)$  can be arbitrary. Hence, there are in total  $2 \cdot 4$  possible det-opts. In MTL, the reasoning shortcut for the addition does not hold since it leads to a sub-optimal solution for multiplication.
C.2.5 Dataset: BDD-OIA
This data set contains frames of driving scene videos for autonomous predictions [54]. Each frame is annotated with 4 binary labels, indicating the possible actions,  $\mathbf{Y} =$  (move_forward, stop, turn_left, turn_right). Each scene is also annotated with 21 binary concepts C, underlying the reasons for the possible actions, see Table 5. The training set contains 16k frames, with full label and concept supervision; the validation and the test set contain 2k and 4.5k annotated data, respectively.
For designing the prior knowledge w.r.t. to the concepts in Table 5, we make use of the following rules for move_forward/stop predictions:
$$
\left\{ \begin{array}{l} \text {r e d l i g h t} \Longrightarrow \neg \text {g r e e n l i g h t} \\ \text {o b s t a c l e} = \text {c a r} \vee \text {p e r s o n} \vee \text {r i d e r} \vee \text {o t h e r} _ {\text {o b s t a c l e}} \\ \text {r o a d} _ {\text {c l e a r}} \Longleftrightarrow \neg \text {o b s t a c l e} \\ \text {g r e e n} _ {\text {l i g h t}} \vee \text {f o l l o w} \vee \text {c l e a r} \Longrightarrow \text {m o v e} _ {\text {f o r w a r d}} \\ \text {r e d} _ {\text {l i g h t}} \vee \text {s t o p} _ {\text {s i g n}} \vee \text {o b s t a c l e} \Longrightarrow \text {s t o p} \\ \text {s t o p} \Longrightarrow \neg \text {m o v e} _ {\text {f o r w a r d}} \end{array} \right. \tag {96}
$$
Forturn_left,andsimilartofturn_right,weuse:
$$
\left\{ \begin{array}{l} \text {c a n _ {t u r n} = l e f t _ {l a n e} \vee l e f t  \vee l e f t _ {f o l l o w}} \\ \text {c a n n o t _ {t u r n} = n o  \vee l e f t _ {o b s t a c l e} \vee l e f t } \\ \text {c a n _ {t u r n} \wedge \neg c a n n o t _ {t u r n} \Rightarrow t u r n _ {l e f t}} \end{array} \right. \tag {97}
$$
Notice that, since the concepts are predicted together, as explained in Appendix C.4, we can count the number of RSs as follows:
- For move.forward and stop, the labels are predicted with the constraints such that (move.forward, stop) = (1, 1) has no support. Hence, we consider only the predictions (0, 0), (0, 1), (1, 0). Next, we identify:
-  $|\mathcal{C}_{0,0}| = 1$ , since it corresponds just to the case where no concepts are predicted.
-  $\left| {\mathcal{C}}_{\left( 1,0\right) }\right|  = {2}^{3} - 1$  ,which is the number of different concepts attribution for forward yielding a positive label;
-  $|\mathcal{C}_{(0,1)}| = 280$  are the combination of the remaining concepts that yield the stop action, in agreement with the constraints. These were counted explicitly from the logic implementation.
Overall, the number of det-opts amount to  $1 \cdot 7^{7} \cdot 280^{280}$ .
- For turn_left and turn_right, we count the cardinality of positive and negative predictions of the two classes  $|\mathcal{C}_0|$  and  $|\mathcal{C}_1|$ :
-  $|\mathcal{C}_1| = 2^3 - 1$ , that are the only concepts attributions for the positive label;
-  $|\mathcal{C}_0| = 2^6 - |\mathcal{C}_1|$ , are all the remaining concept combinations.
For left and right, separately, we obtain that possible optimal solutions amount to  $7^{7} \cdot 57^{57}$ .
Altogether, the count of det-opts for BDD-0IA goes as follows:
$$
\prod_ {\mathbf {y} \in \mathcal {Y}} | \mathcal {C} _ {\mathbf {y}} | ^ {| \mathcal {C} _ {\mathbf {y}} |} = 1 ^ {1} \cdot 7 ^ {7} \cdot 2 8 0 ^ {2 8 0} \cdot 7 ^ {7} \cdot 5 7 ^ {5 7} \cdot 7 ^ {7} \cdot 5 7 ^ {5 7} \tag {98}
$$
Table 5: Concepts annotated in BDD-0IA. Table taken from [54]
Action Category	Concepts	Count
move_forward	green_light	7805
follow	3489
road_clear	4838
stop	red_light	5381
traffic_sign	1539
car	233
person	163
rider	5255
other(obstacle	455
turn_left	left_lane	154
left_green_light	885
left_FOLLOW	365
no_left_lane	150
left(obstacle	666
letf_solid_line	316
turn_right	right_lane	6081
right_green_light	4022
right_FOLLOW	2161
no_right_lane	4503
right(obstacle	4514
right_solid_line	3660
C.3 Optimizer and Hyper-parameter Selection
The Adam optimizer [87] was employed for all experiments, with exponential decay of the learning rate amounting to  $\gamma = 0.95$ , exception made for BDD-01A, where we added a weight-decay with parameter  $\omega = 4 \cdot 10^{-5}$  and  $\gamma$  was set to 0.1, to avoid over-fitting.
The learning rate for all experiments was tuned by searching over the range  $10^{-4} \div 10^{-2}$ , with a total of 5 log steps. We found the SL penalty of 2 and 10 for XOR and MNIST-AddMul, respectively, to work well in our experiments. The strength of the single mitigation strategies  $(\eta)$  for each method was chosen accordingly to a grid-search over  $\eta \in \{0.1, 0.5, 1, 2, 5, 10\}$ , varying the learning rate. The best hyper-parameters were selected, in the first step, based on the highest performances in  $F_{1}$ -score for label accuracy on the validation set and, in the second step, on the lowest mitigation loss, accordingly to the value in the validation set. In particular, for SL we chose those runs that yielded the best trade-off between SL minimization and label prediction. For LTN, we found that concept supervision did interfere with the original training objective, for which the best weight for the mitigation strength was found at  $\gamma = 10^{-2}$ . It was adopted for both LTN+C and LTN+R+C in Q2 of Section 6. The hyper-parameters for the combined mitigation strategies were selected according to the aforementioned criterion, by only searching the best-combined mitigation strength, while keeping the individual best strengths fixed from the previous grid search.
For BDD-0IA, we selected the learning rate ranging in the interval  $10^{-4} \div 10^{-2}$  upon selecting those runs with best  $F_{1}$ -scores on labels in the validation set. The strength of the concept supervision and entropy regularization were varied in between  $\{0.1, 1, 5\}$ .
All best hyper-parameters for our tests are reported in the code in the Supplementary Material.
C.4 Architectures
XOR: For this data set, we adopted two MLPs, one for the encoder  $p_{\theta}(\mathbf{C} \mid \mathbf{G})$  and one for the decoder  $p_{\psi}(\mathbf{G} \mid \mathbf{C})$ , both with a hidden size of 3 and ReLU activations. This architecture is used to empirically validate RSs without forcing disentanglement. For the disentangled case, we considered a linear layer with weight  $\omega$  and bias  $b$ .
For SL only, we added an additional MLP, with a hidden size of 3 and tanh activations, implementing the map from the logits of  $\mathbf{C}$  to  $\mathbf{Y}$ .
MNIST-AddMul: We report here the architectures that has been used for MNIST-Addition, MNIST-Multiplication, and MNIST-EvenOdd. For the joint prediction, i.e., without disentanglement, we used the encoder in Table 6. When considering disentanglement, we processed each digit with the encoder in Table 7 and then stacked the two concepts together. For the reconstruction, we used the decoder in Table 8. For SL only, we added an MLP with a hidden size of 50, taking as input the logits of both concepts and processing them to the label.
Table 6: Double digit encoder for MNIST-Addition
INPUT SHAPE	LAYER TYPE	PARAMETERS	ACTIVATION
(28,56,1)	Convolution	depth=32, kernel=4, stride=2, padding=1	ReLU
(32,14,28)	Dropout	p=0.5
(32,14,28)	Convolution	depth=64, kernel=4, stride=2, padding=1	ReLU
(64,7,14)	Dropout	p=0.5
(64,7,14)	Convolution	depth=128, kernel=4, stride=2, padding=1	ReLU
(128,3,7)	Flatten
(2688)	Linear	dim=20, bias = True
Table 7: Single digit Encoder for MNIST-AddMul
INPUT SHAPE	LAYER TYPE	PARAMETERS	ACTIVATION
(28, 28, 1)	Convolution	depth=64, kernel=4, stride=2, padding=1	ReLU
(14, 14, 64)	Dropout	p = 0.5
(14, 14, 64)	Convolution	depth=128, kernel=4, stride=2, padding=1	ReLU
(7, 7, 128)	Dropout	p = 0.5
(7, 7, 128)	Convolution	depth=256, kernel=4, stride=2, padding=1	ReLU
(3, 3, 256)	Flatten
(2304)	Linear	dim=10, bias = True
Table 8: Decoder for MNIST-Addition
INPUT SHAPE	LAYER TYPE	PARAMETERS	ACTIVATION
(40,1)	Unflatten
(128,3,7)	ConvTranspose2d	depth=64, kernel=(5,4), stride=2, padding=1	ReLU
(64,7,14)	Dropout	p=0.5
(128,3,7)	ConvTranspose2d	depth=32, kernel=(4,4), stride=2, padding=1	ReLU
(32,14,28)	Dropout	p=0.5
(128,3,7)	ConvTranspose2d	depth=1, kernel=(4,4), stride=2, padding=1	Sigmoid
BDD-0IA: Images of BDD-0IA are preprocessed following [55] with a Faster-RCNN [88] pre-trained on MS-COCO and fine-tuned on BDD-100k [54]. Successively, we adopted the pre-trained convolutional layer on [55] to extract linear features, with dimension 2048. These are the inputs for the NeSy model, which is implemented with a fully-connected NN, see Table 9.
Table 9: Fully connected layer for BDD-0IA
INPUT SHAPE	LAYER TYPE	PARAMETERS	ACTIVATION
(2048,1)	Linear + BatchNorm1d	dim=1024	Softplus
(1024)	Linear + BatchNorm1d	dim=512	Softplus
(512)	Linear + BatchNorm1d	dim=256	Softplus
(256)	Linear + BatchNorm1d	dim=128	Softplus
(128)	Linear + BatchNorm1d	dim=21	Softplus
C.5 Confusion Matrices
Following, we report the label and concept-level confusion matrices (CMs) for XOR, MNIST-Even0odd, and MNIST-Multiplication. For all of them, we report those obtained in runs with maximal  $F_{1}$ -score over the labels.
C.6 XOR
Figure 3: CMs for XOR: (Left) All NeSy models fail for RSs without any mitigation. (Right) Providing DIS avoids all RSs.
C.7 MNIST-EvenOdd
Figure 4: NeSy models without mitigation strategies. (Left) DPL picks a RS that uses only 3 digits. (Middle) SL optimizes for label predictions but does not always predict a correct configuration for the digits. (Right) LTN also picks a RS using only 3 digits.
Figure 5: NeSy models with R. (Left) DPL picks a sub-optimal RS. (Middle) SL optimizes for label predictions but through a RS. (Right) LTN also picks a sub-optimal RS. For all runs, we found that R interferes with the standard learning objective of DPL and LTN, respectively.
Figure 6: NeSy models with C. (Left) DPL picks a sub-optimal RS and fails to predict correctly the digits 4 and 9. (Middle) SL predicts correctly the 4's and 9's but does not avoid the RS. (Right) LTN picks a RS that uses only 3 digits.
Figure 7: NeSy models with H. (Left) DPL picks a RS that uses concepts more sparsely. (Middle) SL picks a RS, irrespectively of the mitigation. (Right) LTN tends to align to the diagonal but fails to predict correctly multiple digits. The performance, nonetheless, is sub-optimal.
Figure 8: NeSy models with R and H. (Left) DPL picks a sub-optimal RS by using only three digits. (Middle) SL picks a RS. (Right) LTN learns correctly the odd digits but learns a RS for the even ones.
Figure 9: NeSy models with R and C. (Left) DPL picks a sub-optimal RS by using only three digits. (Middle) SL improves along the diagonal but fails to correctly encode four digits. (Right) LTN picks a RS.
Figure 10: NeSy models with C and H. (Left) DPL correctly retrieves almost all digits but fails with the digit 7. For this method, we also found runs completely recovering the diagonal. (Middle) SL picks a sub-optimal RS. (Right) LTN avoids the RS.
Figure 11: NeSy models with R, C and H. (Left) DPLcorrectly identifies the underlying concepts. (Middle) SL does not avoid RSs. (Right) LTN also identifies the correct digits.
C.8 MNIST-AddMul
Figure 12: CM on the addition task. All models, DPL (Left), SL (Middle), and LTN (Right) fail for the RS introduced in Example 1, while failing also to predict the digit 3.
Figure 13: CM on the product task. (Left) DPL picks a RS where it fails to capture the correct semantics of all digits, except 0, (Middle, Right) SL, and likewise LTN, acquires a RS where it never predicts correctly the digit 2.
Figure 14: CM on MTL. Once the predictors are trained for solving addition and multiplication jointly through MTL, they all successfully acquire the concepts with the intended semantics: all confusion matrices are very close to being diagonal.