SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where we denoted  $\tilde{p}_{\varphi}(\mathbf{C}\mid \mathbf{G})$  the normalized probability distribution given by integrating  $p_{\varphi}(\mathbf{C}\mid \mathbf{X})$  on  $\mathcal{O}_{\mathbf{X}}$  solely, and  $\lambda$  is the measure of  $\mathcal{O}_{\mathbf{X}}$ . Notice that the RHS of Eq. (44) is exactly  $\mathbb{1}\{\mathbf{C} = \mathbf{c}\}$  iff  $\lambda = 0$  or  $\tilde{p}_{\varphi}(\mathbf{C}\mid \mathbf{G}) = \mathbb{1}\{\mathbf{C} = \mathbf{c}\}$ , which contradicts the claim. Hence, all probabilities  $p_{\theta}(\mathbf{C}\mid \mathbf{G}) = \mathbb{1}\{\mathbf{C} = \mathbf{c}\}$  are only given by probabilities  $p_{\theta}(\mathbf{C}\mid \mathbf{x}) = \mathbb{1}\{\mathbf{C} = \mathbf{c}\}$ , for all  $\mathbf{x}\in \mathcal{X}$ . This yields the claim.
B.2 Proof of Theorem 2: Counting the Deterministic Optima
We want to count the number of deterministic optima of the log-likelihood in Eq. (1). Recall that, by Lemma 1, under A1 and A2 any optimum  $p_{\theta}(\mathbf{Y} \mid \mathbf{X})$  of Eq. (2) yields an optimum  $p_{\theta}(\mathbf{Y} \mid \mathbf{G})$  of the upper bound in Eq. (5). Following, by point (ii) of Lemma 1 we have that deterministic optima are shared between  $p_{\theta}(\mathbf{C} \mid \mathbf{X})$  and  $p_{\theta}(\mathbf{C} \mid \mathbf{G})$ . This means that we can equivalently count the number of deterministic optima  $p_{\theta}(\mathbf{C} \mid \mathbf{G})$  for the upper bound in Eq. (5). We proceed to do exactly this.
Let  $\mathcal{A}$  be the set of all possible maps  $\alpha : \mathbf{g} \mapsto \mathbf{c}$ , each inducing a candidate concept distribution  $p_{\theta}(\mathbf{C} \mid \mathbf{G}) = \mathbb{1}\{\mathbf{C} = \alpha(\mathbf{G})\}$ . The only  $\alpha$ 's that achieve maximal likelihood are those that satisfy the knowledge for all  $\mathbf{g} \in \mathrm{supp}(\mathbf{G})$  for the learning problem, that is:
$$
\beta_ {K} (\mathbf {g}) = \left(\beta_ {K} \circ \alpha\right) (\mathbf {g}), \tag {45}
$$
i.e., that it is indeed the case that the concepts output by  $\alpha (\mathbf{g})$  predict the ground-truth label  $h_{\mathsf{K}}(\mathbf{g})$ . Notice that, only one of them is correct and coincides with the identity, i.e.,  $\alpha (\mathbf{g}) = \mathbf{g}$
These  $\alpha$  are those that satisfy the knowledge on all examples, or equivalently the conjunction of the knowledge applied to all examples, that is:
$$
\bigwedge_ {\mathbf {g} \in \mathcal {D} _ {\mathbf {G}}} \left(\left(\beta_ {\mathsf {K}} \circ \alpha\right) (\mathbf {g}) = h _ {\mathsf {K}} (\mathbf {g})\right) \tag {46}
$$
This means that only a subset of  $\mathcal{A}$  contains those maps consistent with the knowledge. The total number of these maps is then given by:
$$
\left. \sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {D} _ {\mathbf {G}}} \left(\beta_ {\mathsf {K}} \circ \alpha\right) (\mathbf {g}) = \beta_ {\mathsf {K}} (\mathbf {g})\right) \right\} \tag {47}
$$
B.3 Proof of Proposition 3: Link Between Deterministic and Non-deterministic Optima
Point  $(i)$ . We begin by proving that, for DPL, any convex combination of optima of the likelihood Eq. (1) is itself an optimum. Fix any input  $\mathbf{x}$ . First,  $p^{*}(\mathbf{y} \mid \mathbf{x})$  is the optimal value of the log-likelihood in Eq. (2), according to Eq. (29) from point  $(i)$  of Lemma 1. Now, let  $p^{(1)}(\mathbf{C} \mid \mathbf{x})$  and  $p^{(2)}(\mathbf{C} \mid \mathbf{x})$  be two concept distributions that both attain optimal likelihood, i.e., for  $i \in \{1, 2\}$  it holds that  $\sum_{\mathbf{c}} u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}) \cdot p^{(i)}(\mathbf{c} \mid \mathbf{x}) = p^{*}(\mathbf{y} \mid \mathbf{x})$ . The likelihood term of any convex combination of the two optima is given by:
$$
\begin{array}{l} \sum_ {\mathbf {c}} u _ {K} (\mathbf {y} \mid \mathbf {c}) [ \lambda p ^ {(1)} (\mathbf {c} \mid \mathbf {x}) + (1 - \lambda) p ^ {(2)} (\mathbf {c} \mid \mathbf {x}) ] (48) \\ = \lambda \sum_ {\mathbf {c}} u _ {\mathrm {K}} (\mathbf {y} \mid \mathbf {c}) p ^ {(1)} (\mathbf {c} \mid \mathbf {x}) + (1 - \lambda) \sum_ {\mathbf {c}} u _ {\mathrm {K}} (\mathbf {y} \mid \mathbf {c}) p ^ {(2)} (\mathbf {c} \mid \mathbf {x}) (49) \\ = \lambda p ^ {*} (\mathbf {y} \mid \mathbf {x}) + (1 - \lambda) p ^ {*} (\mathbf {y} \mid \mathbf {x}) (50) \\ = p ^ {*} (\mathbf {y} \mid \mathbf {x}) (51) \\ \end{array}
$$
Hence, the convex combination is also an optimum of the likelihood. Note that the very same reasoning applies to the Semantic Loss (Eq. (11)), again due to linearity of the expectation over  $\mathbf{C}$ .
Remark 7. Since SL and DPL limit  $p_{\theta}(\mathbf{C} \mid \mathbf{X})$  to be factorized as  $\prod_{i} p(C_{i} \mid \mathbf{X})$ , some convex combinations that would be in principle solutions as per point (i) cannot be expressed. Essentially, this translates into an additional constraint for the solutions of SL and DPL that is, given two factorized probabilities  $p^{(1)}(\mathbf{C} \mid \mathbf{x})$  and  $p^{(2)}(\mathbf{C} \mid \mathbf{x})$ , then:
$$
p (\mathbf {C} \mid \mathbf {x}) = \lambda p ^ {(1)} (\mathbf {C} \mid \mathbf {x}) + (1 - \lambda) p ^ {(2)} (\mathbf {C} \mid \mathbf {x}) i s a s o l u t i o n \iff p (\mathbf {C} \mid \mathbf {x}) i s f a c t o r i z e d \forall x \tag {52}
$$
This applies only to models that integrate probabilistic logic by predicting the concepts independently, whereby relaxations of SL and DPL, like [10, 84], could naturally express arbitrary convex combinations of deterministic RSs.
Remark 8. The above result does not hold for LTNs, in general. We show this by constructing a counter-example for the default choice of  $T$ -conorms [6]. That is, a convex combination of deterministic optima that is not itself an optimum. Recall that LTN uses product real logic to define a degree of knowledge satisfaction and consider the prior knowledge  $\mathsf{K} = (C_1 \lor C_2)$ , where  $C_1$  and  $C_2$  are two distinct binary concepts. In LTN, the degree of satisfaction of  $\mathsf{K}$  is given by the  $T$ -conorm of the logical disjunction, which is defined as  $S(a, b) = a + b - ab$ . Fix an input  $\mathbf{x}$  and consider two deterministic distributions  $p^{(1)}(\mathbf{C} \mid \mathbf{x}) = \mathbb{1}\left\{\mathbf{C} = (1,0)^{\top}\right\}$  and  $p^{(2)}(\mathbf{C} \mid \mathbf{x}) = \mathbb{1}\left\{\mathbf{C} = (0,1)^{\top}\right\}$ . It is clear that both distributions satisfy the prior knowledge and as such are optimal. Now take any convex combination  $p(\mathbf{C}) = \lambda p^{(1)}(\mathbf{C}) + (1 - \lambda)p^{(2)}(\mathbf{C})$ . Then,  $a = \lambda p^{(1)}(C_1 = 1) + (1 - \lambda)p^{(2)}(C_1 = 1) = \lambda$  and, for similar reasons,  $b = 1 - \lambda$ . Then, it holds:
$$
\begin{array}{l} S \left(p \left(C _ {1} = 1\right), p \left(C _ {2} = 1\right)\right) = \lambda + (1 - \lambda) - \lambda (1 - \lambda) (53) \\ = 1 - \lambda + \lambda^ {2} \leq 1 (54) \\ \end{array}
$$
where the equality holds iff  $\lambda \in \{0,1\}$ .
Point (ii). Under A1 and A2, we can count all deterministic solutions  $p_{\theta}(\mathbf{C} \mid \mathbf{G})$  via Theorem 2. Here, we show that these deterministic optima constitute a complete basis for all optimal solutions of Eq. (29). From (i), we have that any convex combination of the deterministic optima is also an optimum. We will show that these are the only optimal solutions.
Recall that the optimal solutions are all of the form  $p^{*}(\mathbf{Y} = \beta_{\mathsf{K}}(\mathbf{g})\mid \mathbf{g}) = 1$ , as a consequence of A1 and A2 from Lemma 1, for all  $\mathbf{g}\in \mathrm{supp}(\mathbf{G})$ . Notice that any optimal solution  $p_{\theta}(\mathbf{C}\mid \mathbf{g})$  must place
mass to those concepts that lead to the correct label. Formally, for every  $\mathbf{g}$  and  $\mathbf{y} = \beta_{\mathrm{K}}(\mathbf{g})$ , it holds:
$$
\begin{array}{l} p _ {\theta} (\mathbf {y} \mid \mathbf {g}) = \sum_ {\mathbf {c}} u _ {\mathrm {K}} (\mathbf {y} \mid \mathbf {c}) p _ {\theta} (\mathbf {c} \mid \mathbf {x}) (55) \\ \leq \sum_ {\mathbf {c}} p _ {\theta} (\mathbf {c} \mid \mathbf {g}) = 1 (56) \\ \end{array}
$$
where the equality holds iff  $u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}) = 1$  for all  $p_{\theta}(\mathbf{c} \mid \mathbf{x}) > 0$ . In other words, this means that each  $\mathbf{c} \sim p_{\theta}(\mathbf{C} \mid \mathbf{g})$  must be an optimal solution for the logic. This proves the claim.
Point (iii). If A2 does not hold, there can be optima solutions that are given as convex combinations of non-optimal deterministic probabilities. First, notice that according to Lemma 1, point (i), the optimal solutions for  $p_{\theta}(\mathbf{Y} \mid \mathbf{X}; \mathsf{K})$  minimize the KL term and are equivalent to  $p^{*}(\mathbf{Y} \mid \mathbf{X})$ . Then, consider for a given  $\mathbf{x}$  an optimal solution  $p(\mathbf{C} \mid \mathbf{x}) = \lambda \mathbb{1}\{\mathbf{C} = \mathbf{c}_1\} + (1 - \lambda) \mathbb{1}\{\mathbf{C} = \mathbf{c}_2\}$  that is a convex combination of two deterministic probabilities. From the convexity of the KL it holds:
$$
\begin{array}{l} \mathrm {K L} \left[ p ^ {*} (\mathbf {y} \mid \mathbf {x}) \mid \mid \lambda u _ {\mathrm {K}} (\mathbf {y} \mid \mathbf {c} _ {1}; \mathrm {K}) + (1 - \lambda) u _ {\mathrm {K}} (\mathbf {y} \mid \mathbf {c} _ {2}; \mathrm {K}) \right] (57) \\ \leq \lambda \cdot \mathsf {K L} [ p ^ {*} (\mathbf {y} \mid \mathbf {x}) | | u _ {\mathsf {K}} (\mathbf {y} \mid \mathbf {c} _ {1}; \mathsf {K}) ] + (1 - \lambda) \cdot \mathsf {K L} [ p ^ {*} (\mathbf {y} \mid \mathbf {x}) | | u _ {\mathsf {K}} (\mathbf {y} \mid \mathbf {c} _ {2}; \mathsf {K}) ] (58) \\ \end{array}
$$
where the equality holds iff  $u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}_1; \mathsf{K}) = u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}_2; \mathsf{K}) = p^*(\mathbf{y} \mid \mathbf{x})$ . This shows that there can exist solutions that are convex combinations of non-optimal deterministic probabilities. We proceed to show how the space of optimal solutions is defined. On the converse, if  $u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}_1; \mathsf{K}) \mathbb{1}\{\mathbf{C} - \mathbf{c}_1\}$  is a solution, also  $u_{\mathsf{K}}(\mathbf{y} \mid \mathbf{c}_2; \mathsf{K}) \mathbb{1}\{\mathbf{C} - \mathbf{c}_2\}$  must be a solution:
$$
\sum_ {\mathbf {c}} u _ {K} (\mathbf {y} \mid \mathbf {c}) [ \lambda \mathbb {1} \left\{\mathbf {C} = \mathbf {c} _ {1} \right\} + (1 - \lambda) \mathbb {1} \left\{\mathbf {C} = \mathbf {c} _ {2} \right\} ] = p ^ {*} (\mathbf {y} \mid \mathbf {x}) \tag {59}
$$
$$
\lambda u _ {K} (\mathbf {y} \mid \mathbf {c} _ {1}) + (1 - \lambda) u _ {K} (\mathbf {y} \mid \mathbf {c} _ {2}) = p ^ {*} (\mathbf {y} \mid \mathbf {x}) \tag {60}
$$
$$
\lambda p ^ {*} (\mathbf {y} \mid \mathbf {x}) + (1 - \lambda) u _ {K} (\mathbf {y} \mid \mathbf {c} _ {2}) = p ^ {*} (\mathbf {y} \mid \mathbf {x}) \tag {61}
$$
$$
(1 - \lambda) u _ {K} (\mathbf {y} \mid \mathbf {c} _ {2}) = (1 - \lambda) p (\mathbf {y} \mid \mathbf {x}) \tag {62}
$$
$$
u _ {K} (\mathbf {y} \mid \mathbf {c} _ {2}) = p ^ {*} (\mathbf {y} \mid \mathbf {x}) \tag {63}
$$
We now show that any optimum given by a generic  $p_{\varphi}(\mathbf{C} \mid \mathbf{x})$ , with  $\varphi \in \Theta$ , can be expressed as a convex combination of (1) an optimum that is a convex combination of deterministic optima and (2) an optimum that is a convex combination of deterministic, but non-optimal, probabilities:
$$
\begin{array}{l} p ^ {*} (\mathbf {y} \mid \mathbf {x}) = \sum_ {\mathbf {c}} u _ {K} (\mathbf {y} \mid \mathbf {c}) p (\mathbf {c} \mid \mathbf {x}) (64) \\ = \sum_ {\mathbf {c} \in \mathcal {C} _ {\mathbf {y}}} u _ {K} (\mathbf {y} \mid \mathbf {c}) p (\mathbf {c} \mid \mathbf {x}) + \sum_ {\mathbf {c} \in \mathcal {S} _ {\mathbf {y}} ^ {\mathrm {c}}} u _ {K} (\mathbf {y} \mid \mathbf {c}) p (\mathbf {c} \mid \mathbf {x}) (65) \\ = \lambda \sum_ {\mathbf {c} \in \mathcal {C} _ {\mathbf {y}}} u _ {K} (\mathbf {y} \mid \mathbf {c}) \tilde {p} (\mathbf {c} \mid \mathbf {x}) + (1 - \lambda) \sum_ {\mathbf {c} \in \mathcal {S} _ {\mathbf {y}} ^ {c}} u _ {K} (\mathbf {y} \mid \mathbf {c}) \bar {p} (\mathbf {c} \mid \mathbf {x}) (66) \\ \end{array}
$$
where  $\mathcal{C}_{\mathbf{y}} = \{\mathbf{c} \in \mathcal{C} : u_{\mathsf{K}} = p^{*}(\mathbf{y} \mid \mathbf{x})\}$  and  $\mathcal{S}_{\mathbf{y}}^{c} = \mathcal{C} / \mathcal{C}_{\mathbf{y}}$  are two disjoint sets. In the second line, we rewrote the summation on the two terms considering the two sets, whereas in the third line we introduced:  $\lambda = \sum_{\mathbf{c} \in \mathcal{C}_{\mathbf{y}}} p(\mathbf{c} \mid \mathbf{x})$  and  $\tilde{p}(\mathbf{c} \mid \mathbf{x}) = p(\mathbf{c} \mid \mathbf{x}) / \lambda$ ,  $1 - \lambda = \sum_{\mathbf{c} \in \mathcal{S}_{\mathbf{y}}} p(\mathbf{c} \mid \mathbf{x})$  and  $\bar{p}(\mathbf{c} \mid \mathbf{x}) = p(\mathbf{c} \mid \mathbf{x}) / (1 - \lambda)$ . Since each  $\tilde{p}(\mathbf{c} \mid \mathbf{x})$  leads to an optimum by construction, it must be that also  $\bar{p}(\mathbf{c} \mid \mathbf{x})$  is an optimum, by the previous point. In general, there can be many  $\bar{p}(\mathbf{c} \mid \mathbf{x})$  leading to optima, even when  $\tilde{p}(\mathbf{c} \mid \mathbf{x})$  reduces to only the ground-truth element, i.e.,  $\tilde{p}(\mathbf{c} \mid \mathbf{x}) = 11\left\{\mathbf{C} - f_{1:k}^{-1}(\mathbf{x})\right\}$ . Notice that this must hold for all  $\mathbf{x} \sim p^{*}(\mathbf{X})$ . This proves the claim.
B.4 Proof of Proposition 4: Multi-task Learning
When considering multiple task  $T$ , suppose that, for each  $\mathbf{x} \in \bigcap_{t} \mathcal{D}_{t}$ , we get  $T$  different labels  $\mathbf{y}^{(1)}, \ldots, \mathbf{y}^{(T)}$ , each given in accordance to knowledge  $\mathsf{K}^{(1)}, \ldots, \mathsf{K}^{(T)}$ , respectively. Similarly to Eq. (2), we consider the learning objective for all tasks with the joint log-likelihood term:
$$
\log \prod_ {t = 1} ^ {T} p _ {\theta} \left(\mathbf {y} ^ {(t)} \mid \mathbf {x}; \mathsf {K} ^ {(t)}\right) = \sum_ {t = 1} ^ {T} \log p _ {\theta} \left(\mathbf {y} ^ {(t)} \mid \mathbf {x}; \mathsf {K} ^ {(t)}\right) \tag {67}
$$
Under A1 and A2, Lemma 1 point  $(i)$  holds for each task  $t$ . In the following, we denote with  $\beta_{K}^{(t)}$  the underlying maps for each  $K^{(t)}$ , which by A2 are deterministic. Hence, the learning objective becomes:
$$
\mathcal {L} = \sum_ {t = 1} ^ {T} \mathbb {E} _ {\mathbf {x} \sim p (\mathbf {X})} [ \log p _ {\theta} (\mathbf {Y} ^ {(t)} = \left(\beta_ {\mathrm {K}} ^ {(t)} \circ f _ {1: k} ^ {- 1}\right) (\mathbf {x}) \mid \mathbf {x}; \mathrm {K} ^ {(t)}) ] \tag {68}
$$
and, similarly to point  $(i)$  of Lemma 1, we get the following upper-bound:
$$
\mathcal {L} \leq \sum_ {t = 1} ^ {T} \mathbb {E} _ {\mathbf {g} \sim p (\mathbf {G})} [ \log p _ {\theta} (\mathbf {Y} ^ {(t)} = \beta_ {\mathrm {K}} ^ {(t)} (\mathbf {g}) \mid \mathbf {g}; \mathrm {K} ^ {(t)}) ] \tag {69}
$$
Notice that the optimal values for Eq. (69) are given by those distributions that are one for each term in the summation. Then, following from Theorem 2, we have that the deterministic maps  $\alpha$ 's that optimize Eq. (69) must be consistent with each task  $t$  and for each  $\mathbf{g} \in \mathrm{supp}(\mathbf{G})$  satisfy:
$$
\bigwedge_ {t = 1} ^ {T} \left(\left(\beta_ {\mathbf {K}} ^ {(t)} \circ \alpha\right) (\mathbf {g}) = \beta_ {\mathbf {K}} ^ {(t)} (\mathbf {g})\right) \tag {70}
$$
which, equivalently to Theorem 2, points to the condition that the maps  $\alpha$  must be consistent with solving all tasks  $t$ . This exactly amounts to solving the conjunction of all knowledge  $\mathsf{K} = \bigwedge_{t}\mathsf{K}^{(t)}$ . This yields the claim.
B.5 Proof of Proposition 5: Concept Supervision
Let  $p^*(\mathbf{G} \mid S)$  be the distribution of ground-truth concepts restricted to a subset  $S \subseteq \operatorname{supp}(\mathbf{G})$ :
$$
p ^ {*} (\mathbf {G} \mid \mathcal {S}) = \frac {1}{\mathcal {Z}} p ^ {*} (\mathbf {g}) \mathbb {1} \left\{\mathbf {g} \in \mathcal {S} \right\} \quad \text {w i t h} \mathcal {Z} = \sum_ {\mathbf {g}} p ^ {*} (\mathbf {g}) \mathbb {1} \left\{\mathbf {g} \in \mathcal {S} \right\} \tag {71}
$$
and  $p^*(\mathbf{X} \mid \mathcal{S}) = \mathbb{E}_{(\mathbf{g}, \mathbf{s}) \sim p^*(\mathbf{G} \mid \mathcal{S})} p^*(\mathbf{S}) p^*(\mathbf{X} \mid \mathbf{g}, \mathbf{s})$  be the corresponding restricted input distribution.
Under A1, the expectation of the log-likelihood term for concept supervision in Section 5 is:
$$
\begin{array}{l} \mathbb {E} _ {\mathbf {x} \sim p ^ {*} (\mathbf {X} | \mathcal {S})} \log p _ {\theta} (\mathbf {C} _ {I} = f _ {I} ^ {- 1} (\mathbf {x}) \mid \mathbf {x}) = \mathbb {E} _ {\mathbf {g} \sim p ^ {*} (\mathbf {G} | \mathcal {S})} \mathbb {E} _ {\mathbf {s} \sim p ^ {*} (\mathbf {S})} \log p _ {\theta} (\mathbf {C} _ {I} = \mathbf {g} _ {I} \mid f (\mathbf {g}, \mathbf {s})) \\ \leq \mathbb {E} _ {\mathbf {g} \sim p ^ {*} (\mathbf {G} | S)} \log \mathbb {E} _ {\mathbf {s} \sim p ^ {*} (\mathbf {S})} [ p _ {\theta} (\mathbf {C} _ {I} = \mathbf {g} _ {I} \mid f (\mathbf {g}, \mathbf {s})) ] \\ = \mathbb {E} _ {\mathbf {g} \sim p ^ {*} (\mathbf {G} | S)} \log p _ {\theta} \left(\mathbf {C} _ {I} = \mathbf {g} _ {I} \mid \mathbf {g}\right) \tag {72} \\ \end{array}
$$
where  $p^*(\mathbf{X} \mid \mathcal{S}) = \mathbb{E}_{\mathbf{g} \sim p^*(\mathbf{G} \mid \mathcal{S})} \mathbb{E}_{\mathbf{s} \sim p^*(\mathbf{S})} p_\theta(\mathbf{X} \mid \mathbf{g}, \mathbf{s})$ . In the first line we made use of A1 to write  $\mathbf{x} = f(\mathbf{g}, \mathbf{s})$ , in the second line we used Jensen's inequality, and then we introduced the marginal distribution  $p_\theta(\mathbf{C} \mid \mathbf{G}) = \mathbb{E}_{\mathbf{s} \sim p^*(\mathbf{s})}[p_\theta(\mathbf{C} \mid f(\mathbf{G}, \mathbf{s}))]$ . Recall that  $I$  denotes the subset of the supervised ground-truth concepts and that the log-likelihood is evaluated only w.r.t. those concepts.
Now notice that, given A1, the maximum for the LHS of Eq. (72) is zero, i.e., coincides with maximum log-likelihood. Consistently the RHS of Eq. (72) is zero only when  $p_{\theta}(\mathbf{C}_I \mid \mathbf{g})$  places all mass on  $\mathbf{g}_I$ . We proceed considering those maps  $\alpha : \mathbf{g} \mapsto \mathbf{c}$  which lead to deterministic, optimal solutions for the RHS. For these, it must hold that for each  $\mathbf{g} \in S$ :
$$
\bigwedge_ {i \in I} \alpha_ {i} (\mathbf {g}) = g _ {i} \tag {73}
$$
When taken all together, we can estimate how many  $\alpha \in \mathcal{A}$  (cf. Appendix B.2) satisfy the above condition for all  $\mathbf{g}$ :
$$
\sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \mathcal {S}} \bigwedge_ {i \in I} \alpha_ {i} (\mathbf {g}) = g _ {i} \right\} \tag {74}
$$
B.6 Proof of Proposition 6: Reconstruction
Under A1, the reconstruction penalty can be written as:
$$
\begin{array}{l} \mathbb {E} _ {\mathbf {x} \sim p ^ {*} (\mathbf {X})} [ \mathcal {R} (\mathbf {x}) ] = - \mathbb {E} _ {(\mathbf {g}, \mathbf {s}) \sim p ^ {*} (\mathbf {G}) p ^ {*} (\mathbf {S})} \mathbb {E} _ {\mathbf {x} \sim p ^ {*} (\mathbf {X} | \mathbf {g}, \mathbf {s})} \left[ \mathbb {E} _ {(\mathbf {c}, \mathbf {z}) \sim p _ {\theta} (\mathbf {C}, \mathbf {Z} | \mathbf {x})} \log p _ {\psi} (\mathbf {x} | \mathbf {c}, \mathbf {z}) \right] (75) \\ = - \mathbb {E} _ {(\mathbf {g}, \mathbf {s}) \sim p ^ {*} (\mathbf {G}) p ^ {*} (\mathbf {S})} \left[ \mathbb {E} _ {(\mathbf {c}, \mathbf {z}) \sim p _ {\theta} (\mathbf {C}, \mathbf {Z} | f (\mathbf {g}, \mathbf {s}))} \left[ \log p _ {\psi} (f (\mathbf {g}, \mathbf {s}) \mid \mathbf {c}, \mathbf {z}) \right] \right] (76) \\ \end{array}
$$
From A3, we have that the both encoder and decoder distributions are factorized:
$$
p _ {\theta} (\mathbf {c}, \mathbf {z} \mid f (\mathbf {g}, \mathbf {s})) = p _ {\theta} (\mathbf {c} \mid \mathbf {g}) p _ {\theta} (\mathbf {z} \mid \mathbf {s}) \tag {77}
$$
$$
p _ {\psi} (f (\mathbf {g}, \mathbf {s}) \mid \mathbf {c}, \mathbf {z}) = p _ {\psi} (\mathbf {g} \mid \mathbf {c}) p _ {\psi} (\mathbf {s} \mid \mathbf {z}) \tag {78}
$$
This yields:
$$
\begin{array}{l} \mathbb {E} _ {\mathbf {x} \sim p (\mathbf {X})} [ \mathcal {R} (\mathbf {x}) ] = - \mathbb {E} _ {(\mathbf {g}, \mathbf {s}) \sim p ^ {*} (\mathbf {G}) p ^ {*} (\mathbf {S})} \left[ \mathbb {E} _ {\mathbf {c} \sim p _ {\theta} (\mathbf {C} | \mathbf {g})} \left[ \log p _ {\psi} (\mathbf {g} | \mathbf {c}) \right] + \mathbb {E} _ {(\mathbf {z}) \sim p _ {\theta} (\mathbf {Z} | \mathbf {s})} \left[ \log p _ {\psi} (\mathbf {s} | \mathbf {z}) \right] \right] \\ = - \mathbb {E} _ {\mathbf {g} \sim p ^ {*} (\mathbf {G})} \mathbb {E} _ {\mathbf {c} \sim p _ {\theta} (\mathbf {C} | \mathbf {g})} \log p _ {\psi} (\mathbf {g} | \mathbf {c}) - \mathbb {E} _ {\mathbf {s} \sim p ^ {*} (\mathbf {S})} \mathbb {E} _ {\mathbf {z} \sim p _ {\theta} (\mathbf {Z} | \mathbf {s})} \log p _ {\psi} (\mathbf {s} | \mathbf {z}) \\ \geq - \mathbb {E} _ {\mathbf {g} \sim p ^ {*} (\mathbf {G})} \mathbb {E} _ {\mathbf {c} \sim p _ {\theta} (\mathbf {C} | \mathbf {g})} \log p _ {\psi} (\mathbf {g} | \mathbf {c}) \tag {79} \\ \end{array}
$$
where in the first row we separated the two logarithms and removed the expectations on  $p_{\theta}(\mathbf{Z} \mid \mathbf{s})$  and  $p_{\theta}(\mathbf{C} \mid \mathbf{g})$  for the terms  $p_{\psi}(\mathbf{g} \mid \mathbf{c})$  and  $p_{\psi}(\mathbf{s} \mid \mathbf{z})$ , respectively. In the third line, we discarded the term with the match of the reconstruction of the style, giving the lower bound.
When  $\mathcal{R}$  goes to zero, the lower bound also goes to zero and the last term of Eq. (79) is minimized. This happens whenever, for each  $\mathbf{c} \sim p_{\theta}(\mathbf{c} \mid \mathbf{g})$ , it holds that  $p_{\psi}(\mathbf{g} \mid \mathbf{c}) = 1$ . This condition also implies that if for two different  $\mathbf{g}'$  and  $\mathbf{g}''$  there exist at least one  $\mathbf{c}$  such that  $p_{\theta}(\mathbf{c} \mid \mathbf{g}') \cdot p_{\theta}(\mathbf{c} \mid \mathbf{g}'') > 0$ , then  $p_{\psi}(\mathbf{g}' \mid \mathbf{c})$  and  $p_{\psi}(\mathbf{g}'' \mid \mathbf{c})$  cannot both be optimal.
We restrict now to a deterministic map  $\alpha : \mathbf{g} \mapsto \mathbf{c}$  for  $p_{\theta}(\mathbf{C} \mid \mathbf{G})$  and describe the condition when optimal decoders are attained, i.e.,  $p_{\psi}(\mathbf{g} \mid \mathbf{c}) = 1$  for some  $\mathbf{c}$ . By the previous argument, an optimal map  $\alpha$  that leads to optimal decoders must always map ground-truth concepts  $\mathbf{g}$  to different concepts  $\mathbf{c}$ , that is:
$$
\alpha \left(\mathbf {g} ^ {\prime}\right) \neq \alpha \left(\mathbf {g} ^ {\prime \prime}\right) \quad \forall \mathbf {g} ^ {\prime} \neq \mathbf {g} ^ {\prime \prime} \tag {80}
$$
In particular, this condition must hold for all different  $\mathbf{g} \in \mathrm{supp}(\mathbf{G})$ . The number of all solutions is then given by:
$$
\sum_ {\alpha \in \mathcal {A}} \mathbb {1} \left\{\bigwedge_ {\mathbf {g} \in \operatorname {s u p p} _ {\mathbf {G}}} \bigwedge_ {\mathbf {g} ^ {\prime} \in \operatorname {s u p p} _ {\mathbf {G}}: \mathbf {g} ^ {\prime} \neq \mathbf {g}} \alpha (\mathbf {g}) \neq \alpha \left(\mathbf {g} ^ {\prime}\right) \right\} \tag {81}
$$
C Experimental Details and Further Results
We report here all further details concerning the experiments in Section 6.
The code of the experiments builds on top of nesy-c1 [21] and CBM-AUC [55]. All experiments are implemented with Python 3.8.16 and Pytorch [85] and run over one A100 GPU. The code for the experiments is available at github.com/reasoning-shortcuts.