SlowGuess/NeurIPS_2023_OCR huggingface dataset:
that  $G_{1}, u, n \models \varphi$  and  $G_{2}, u, n \models \varphi$  and hence we are done. Suppose now that  $\varphi(x, y) = \neg \psi(x, y)$ . Assume  $G_{1}, u, n \models \varphi$  for  $n \in \{u, v\}$ . Then  $G_{1}, u, n \nmid \psi$ , and by inductive hypothesis, we have  $G_{2}, u, n \nmid \psi$  and then  $G_{2}, u, n \models \varphi$ . The other direction is analogous. Assume now that  $\varphi(x, y) = \psi_{1}(x, y) \wedge \psi_{2}(x, y)$ . Suppose  $G_{1}, u, n \models \varphi$  for  $n \in \{u, v\}$ . Then  $G_{1}, u, n \models \psi_{1}$  and  $G_{1}, u, n \models \psi_{2}$ , and by inductive hypothesis,  $G_{2}, u, n \models \psi_{1}$  and  $G_{2}, u, n \models \psi_{2}$ . Then  $G_{2}, u, n \models \varphi$ . The other direction is analogous. Finally, suppose that  $\varphi(x, y) = \exists^{\geq N} z(\psi(x, z) \wedge r(z, y))$ . Assume that  $G_{1}, u, n \models \varphi$  for  $n \in \{u, v\}$ . Then there exist at least  $N$  nodes  $w \in \mathcal{N}_{r}(n)$  in  $G_{1}$  such that  $G_{1}, u, w \models \psi$ . Since  $\mathcal{N}_{r}(u) = \emptyset$ , we have  $n = v$ . As  $\mathcal{N}_{r}(v) = \{u\}$ , we have  $w = u$ . Since the neighborhood  $\mathcal{N}_{r}(v)$  is the same in  $G_{1}$  and  $G_{2}$ , and by the inductive hypothesis, we have that there are at least  $N$  nodes  $w \in \mathcal{N}_{r}(v)$  in  $G_{2}$  such that  $G_{2}, u, w \models \psi$ . This implies that  $G_{2}, u, v \models \varphi$ . The other direction is analogous.
Now consider the  $\mathsf{erFO}_{\mathrm{cnt}}^3$  formula  $\varphi (x,y) = \exists^{\geq 2}z(\mathrm{Green}(x,z)\wedge \neg r(z,y))$ . We claim that there is no  $\mathsf{rFO}_{\mathrm{cnt}}^3$  formula equivalent to  $\varphi (x,y)$ . By contradiction, suppose we have such an equivalent formula  $\varphi^\prime$ . As shown above, we have  $G_{1},u,v\models \varphi^{\prime}$  iff  $G_{2},u,v\models \varphi^{\prime}$ . On the other hand, by definition, we have that  $G_{1},u,v\models \varphi$  and  $G_{2},u,v\models \varphi$ . This is a contradiction.
Now we are ready to prove Theorem 5.3. We follow the same strategy as in the proof of the backward direction of Theorem A.11.
Let  $\varphi(x, y)$  be a formula in  $\mathsf{erFO}_{\mathrm{cnt}}^3$ , for relation types  $R$  and pair colors  $\mathcal{C}$ . Let  $\varphi_1, \ldots, \varphi_L$  be an enumeration of the subformulas of  $\varphi$  such that if  $\varphi_i$  is a subformula of  $\varphi_j$ , then  $i \leq j$ . In particular,  $\varphi_L = \varphi$ . We will construct a C-MPNN  $\mathcal{A}_{\varphi}$  with  $L$  layers computing  $L$ -dimensional features in each layer. At layer  $\ell \in \{1, \ldots, L\}$ , the  $\ell$ -th component of the feature  $h_{v|u,q}^{(\ell)}$  will correspond to 1 if  $\varphi_\ell$  is satisfied on  $(u, v)$ , and 0 otherwise. The query relation  $q$  plays no role in the construction, that is, for any possible  $q \in R$  the output of  $\mathcal{A}_{\varphi}$  is the same. Hence, for simplicity, in the remaining of the proof we shall write  $h_{v|u}^{(t)}$  instead of  $h_{v|u,q}^{(t)}$ . We add an additional final layer that simply outputs the last component of the feature vector.
We use models of C-MPNNs of the following form:
$$
\boldsymbol {h} _ {v | u} ^ {(t + 1)} = \sigma \Big (\boldsymbol {W} _ {0} \boldsymbol {h} _ {v} ^ {(t)} + \sum_ {r \in R} \sum_ {w \in \mathcal {N} _ {r} (v)} \boldsymbol {W} _ {r} \boldsymbol {h} _ {w | u} ^ {(t)} + \boldsymbol {W} _ {1} \sum_ {w \in V} \boldsymbol {h} _ {w | u} ^ {(t)} + \boldsymbol {b} \Big),
$$
where  $\mathbf{W}_0, \mathbf{W}_1 \in \mathbb{R}^{L \times L}$  are parameter matrices and  $\mathbf{b} \in \mathbb{R}^L$  is the bias term. As message function  $\mathrm{Msg}_r$  we use  $\mathrm{Msg}_r(\mathbf{h}) = \bar{\mathbf{W}}_r\mathbf{h}$ , where  $\mathbf{W}_r \in \mathbb{R}^{L \times L}$  is a parameter matrix. For the non-linearity  $\sigma$  we use the truncated ReLU function  $\sigma(x) = \min(\max(0, x), 1)$ . The  $\ell$ -th row of  $\mathbf{W}_0, \mathbf{W}_1$  and  $\mathbf{W}_r$ , and the  $\ell$ -th entry of  $\mathbf{b}$  are defined as follows (omitted entries are 0):
1. If  $\varphi_{\ell}(x,y) = a(x,y)$  for a color  $a\in \mathcal{C}$ , then  $(W_0)_{\ell \ell} = 1$
2. If  $\varphi_{\ell}(x,y) = \neg \varphi_{k}(x,y)$  then  $(W_0)_{\ell k} = -1$  , and  $b_{\ell} = 1$
3. If  $\varphi_{\ell}(x,y) = \varphi_{j}(x,y)\wedge \varphi_{k}(x,y)$  then  $(W_0)_{\ell j} = 1$ $(W_0)_{\ell k} = 1$  and  $b_{\ell} = -1$
4. If  $\varphi_{\ell}(x,y) = \exists^{\geq N}z(\varphi_k(x,z)\wedge r(z,y))$  then  $(W_{r})_{\ell k} = 1$  and  $b_{\ell} = -N + 1$
5. If  $\varphi_{\ell}(x,y) = \exists^{\geq N}z\left(\varphi_k(x,z)\wedge \neg r(z,y)\right)$  then  $(W_{r})_{\ell k} = -1$ $(W_{1})_{\ell k} = 1$  and  $b_{\ell} = -N + 1$
Let  $G = (V, E, R, \eta)$  be a knowledge graph with pair colors from  $\mathcal{C}$ . In order to apply  $\mathcal{A}_{\varphi}$  to  $G$ , we choose the initialization INIT such that  $L$ -dimensional initial features  $\pmb{h}_{v|u}^{(0)}$  satisfy  $(\pmb{h}_{v|u}^{(0)})_{\ell} = 1$  if  $\varphi_{\ell} = a(x,y)$  and  $\eta(u,v) = a$  and  $(\pmb{h}_{v|u}^{(0)})_{\ell} = 0$  otherwise. That is, the  $L$ -dimensional initial feature  $\pmb{h}_{v|u}^{(0)}$  is a one-hot encoding of the color of  $(u,v)$ . Using the same arguments as in the proof of Theorem A.11, we have  $(\pmb{h}_{v|u}^{(t)})_{\ell} = 1$  if  $G, u, v \models \varphi_{\ell}$  and  $(\pmb{h}_{v|u}^{(t)})_{\ell} = 0$  otherwise, for all  $u, v \in V$  and  $t \in \{\ell, \dots, L\}$ . In particular, after  $L$  layers,  $\mathcal{A}_{\varphi}$  calculates  $\pmb{h}_{v|u}^{(L)}$  such that  $(\pmb{h}_{v|u}^{(L)})_L = 1$  if  $G, u, v \models \varphi$  and  $(\pmb{h}_{v|u}^{(L)})_L = 0$  otherwise. As layer  $L + 1$  extracts the  $L$ -th component of the feature vector, the result follows.
Remark A.16. We note that in Barceló et al. [3], it is shown that a more expressive graded modal logic (with more expressive modalities) can be captured by R-MPNNs in the context of single-relation
graphs. It is by no means obvious that an adaption of this logic to the multi-relational case is captured by our C-MPNNs. We leave as an open problem to find more expressive logics that can be captured by C-MPNNs.
A.5 Proofs of the results from Section 5.3
First, recall the definition of  $\mathsf{rawl}_2$ . Given a knowledge graph  $G = (V,E,R,c,\eta)$ , we have
$$
\operatorname {r a w l} _ {2} ^ {(0)} (u, v) = \eta (u, v),
$$
$$
\mathsf {r a w l} _ {2} ^ {(t + 1)} (u, v) = \tau \big (\mathsf {r a w l} _ {2} ^ {(t)} (u, v), \{\{\big (\mathsf {r a w l} _ {2} ^ {(t)} (u, w), r \big) | w \in \mathcal {N} _ {r} (v), r \in R) \} \} \big)
$$
Note that  $\mathsf{rawl}_2$ , and hence C-MPNNs, are one-directional: the neighborhood  $\mathcal{N}_r(v)$  only considers facts in one direction, in this case, from neighbors to  $v$ . Hence, a natural extension is to consider bi-directional neighborhoods. Fortunately, we can define this extension by simply applying the same test  $\mathsf{rawl}_1$  to knowledge graphs extended with inverse relations. We formalize this below.
For a test  $\mathsf{T}$ , we sometimes write  $\mathsf{T}(G,u,v)$ , or  $\mathsf{T}(G,v)$  in case of unary tests, to emphasize that the test is applied over  $G$ , and  $\mathsf{T}(G)$  for the node/pairwise coloring given by the test. Let  $G = (V,E,R,c,\eta)$  be a knowledge graph. We define its augmented knowledge graph to be  $G^{+} = (V,E^{+},R^{+},c,\eta)$ , where  $R^{+}$  is the disjoint union of  $R$  and  $\{r^{-} \mid r \in R\}$ , and
$$
E ^ {+} = E \cup \left\{r ^ {-} (v, u) \mid r (u, v) \in E, u \neq v \right\}.
$$
We define the augmented relational asymmetric local 2-WL test on  $G$ , denoted by  $\mathsf{rawl}_2^+$ , as
$$
\operatorname {r a w l} _ {2} ^ {+ (t)} (G, u, v) = \operatorname {r a w l} _ {2} ^ {(t)} (G ^ {+}, u, v),
$$
for all  $t \geq 0$  and  $u, v \in V$ . As we show below,  $\mathsf{rawl}_2^+$  is strictly more powerful than  $\mathsf{rawl}_2$ .
Proposition A.17. The following statements hold:
1. For all  $t \geq 0$  and all knowledge graphs  $G$ , we have  $\mathsf{rawl}_2^{+(t)}(G) \preceq \mathsf{rawl}_2^{(t)}(G)$ .
2. There is a knowledge graph  $G$  and nodes  $u, v, u', v'$  such that  $\mathsf{rawl}_2^{+(1)}(G, u, v) \neq \mathsf{rawl}_2^{+(1)}(G, u', v')$  but  $\mathsf{rawl}_2^{(t)}(G, u, v) = \mathsf{rawl}_2^{(t)}(G, u', v')$  for all  $t \geq 1$ .
Proof. To prove  $\mathrm{rawl}_2^+(G) \preceq \mathrm{rawl}_2(G)$  first, we consider induction on iteration  $t$ . The base case for  $t = 0$  is trivial by the assumption, so it is enough to consider the inductive step. We need to show that for some  $k$ ,
$$
\operatorname {r a w l} _ {2} ^ {+ (k + 1)} (u, v) = \operatorname {r a w l} _ {2} ^ {+ (k + 1)} \left(u ^ {\prime}, v ^ {\prime}\right) \Rightarrow \operatorname {r a w l} _ {2} ^ {(k + 1)} (u, v) = \operatorname {r a w l} _ {2} ^ {(k + 1)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
By the definition of  $\mathsf{rawl}_2^{+(k + 1)}(u,v),\mathsf{rawl}_2^{+(k + 1)}(u',v')$  and the injectivity of  $\tau$ , it holds that
$$
\operatorname {r a w l} _ {2} ^ {+ (k)} (u, v) = \operatorname {r a w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
$$
\{\{(\mathsf {r a w l} _ {2} ^ {+ (k)} (u, x), r) \mid x \in \mathcal {N} _ {r} ^ {+} (v), r \in R ^ {+} \} \} = \{\{(\mathsf {r a w l} _ {2} ^ {+ (k)} (u ^ {\prime}, x ^ {\prime}), r ^ {\prime}) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} ^ {+} (v ^ {\prime}), r ^ {\prime} \in R ^ {+} \} \}
$$
Because all inverse relations  $r^{-}$  are newly introduced, so it is impossible to be mixed with  $r$ , we can split the second equation into the following equations:
$$
\{\left\{\left(\operatorname {r a w l} _ {2} ^ {+ (k)} (u, x), r\right) \mid x \in \mathcal {N} _ {r} (v), r \in R \right\} \} = \{\left\{\left(\operatorname {r a w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, x ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(v ^ {\prime}\right), r ^ {\prime} \in R \right\}
$$
$$
\{\left\{\left(\operatorname {r a w l} _ {2} ^ {+ (k)} (u, y), r ^ {-}\right) \mid y \in \mathcal {N} _ {r ^ {-}} (v), r \in R \right\} \} = \{\left\{\left(\operatorname {r a w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, y ^ {\prime}\right), r ^ {\prime -}\right) \mid y ^ {\prime} \in \mathcal {N} _ {r ^ {\prime -}} \left(v ^ {\prime}\right), r \in R \right\}
$$
By the inductive hypothesis and unpacking the first equation, we can further imply that
$$
\{\{\left(\mathbf {r a w} \mid_ {2} ^ {(k)} (u, x), r\right) \mid x \in \mathcal {N} _ {r} (v), r \in R \} \} = \{\{\left(\mathbf {r a w} \mid_ {2} ^ {(k)} \left(u ^ {\prime}, x ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(v ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
Thus, By definition of  $\mathsf{rawl}_2^{(k + 1)}(u,v),\mathsf{rawl}_2^{(k + 1)}(u',v')$  it holds that
$$
\mathsf {r a w l} _ {2} ^ {(k + 1)} (u, v) = \mathsf {r a w l} _ {2} ^ {(k + 1)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
For the counterexample, we consider a relational graph with two types of relation  $G' = (V', E', R', c, \eta)$  such that  $V' = \{u, v, v'\}$ ,  $E' = \{r_1(v, u), r_2(v', u)\}$ , and  $R' = \{r_1, r_2\}$ . Let the initial pairwise labeling  $\eta$  for node pairs  $(u, v)$  and  $(u', v)$  satisfy  $\mathsf{rawl}_2^{+(0)}(u, v) = \mathsf{rawl}_2^{+(0)}(u, v')$  and  $\mathsf{rawl}_2^{(0)}(u, v) = \mathsf{rawl}_2^{(0)}(u, v')$ . For such graph  $G'$ , we consider node pair  $(u, v)$  and  $(u, v')$ . For  $\mathsf{rawl}_2^{(t)}$  where  $t \geq 0$ , we show by induction that  $\mathsf{rawl}_2^{(t)}(u, v) = \mathsf{rawl}_2^{(t)}(u, v')$ . The base case is trivial by assumption. The inductive step shows that by the inductive hypothesis,
$$
\begin{array}{l} \left. \mathbf {r a w l} _ {2} ^ {(t + 1)} (u, v) = \tau \left(\mathbf {r a w l} _ {2} ^ {(t)} (u, v), \{\{\}\}\right) \right. \\ = \tau (\operatorname {r a w l} _ {2} ^ {(t)} (u, v ^ {\prime}), \{\{\}) \\ = \mathsf {r a w l} _ {2} ^ {(t + 1)} (u, v ^ {\prime}) \\ \end{array}
$$
$$
\begin{array}{l} \mathsf {r a w l} _ {2} ^ {+ (1)} (u, v) = \tau (\mathsf {r a w l} _ {2} ^ {+ (0)} (u, v), \{\{\left(\mathsf {r a w l} _ {2} ^ {+ (0)} (x, v), r _ {1} ^ {-}) \right\rbrace \}) \\ \neq \operatorname {r a w l} _ {2} ^ {+ (1)} (u, v ^ {\prime}) = \tau (\operatorname {r a w l} _ {2} ^ {+ (0)} (u, v ^ {\prime}), \{\{\left(\operatorname {r a w l} _ {2} ^ {+ (0)} (x, v ^ {\prime}), r _ {2} ^ {-}\right)) \} \}) \\ \end{array}
$$
We can also extend C-MPNNs with bi-directionality in an obvious way, obtaining augmented C-MPNNs. By applying Theorem 5.1 to the augmented graph  $G^{+}$ , we obtain the equivalence between augmented C-MPNNs and the test  $\mathrm{rawl}_2^+$  directly. In turn, Proposition A.17 implies that augmented C-MPNNs are strictly more powerful than C-MPNNs in distinguishing nodes in a graph.
Recall the definition of  $\mathsf{rw}l_2$ . Given a knowledge graph  $G = (V, E, R, c, \eta)$ , we have
$$
\begin{array}{l} \left. \mathsf {r w l} _ {2} ^ {(t + 1)} (u, v) = \tau \big (\mathsf {r w l} _ {2} ^ {(t)} (u, v), \{\{\left(\mathsf {r w l} _ {2} ^ {(t)} (w, v), r\right) \mid w \in \mathcal {N} _ {r} (u), r \in R) \} \}, \right. \\ \left. \left\{\left(\operatorname {r w l} _ {2} ^ {(t)} (u, w), r) \mid w \in \mathcal {N} _ {r} (v), r \in R) \right\} \right\}\right) \\ \end{array}
$$
We define the augmented version  $\mathsf{rw}|_2^+$  in an obvious way, that is,
$$
\operatorname {r w l} _ {2} ^ {+ (t)} (G, u, v) = \operatorname {r w l} _ {2} ^ {(t)} (G ^ {+}, u, v),
$$
for all  $t\geq 0$  and  $u,v\in V$
We will drop the notation of  $G$  during the proof when the context is clear for simplicity.
Proposition A.18. For all  $t \geq 0$  and all knowledge graph  $G$ , let  $\mathsf{rw}|_2^{(0)}(G) \equiv \mathsf{raw}|_2^{(0)}(G)$ . The following statements hold:
1. For every  $t > 0$ , it holds that  $\mathsf{rwI}_2^{(t)}(G) \preceq \mathsf{rawI}_2^{(t)}(G)$ .
2. There is a knowledge graph  $G$  and pair of nodes  $(u, v)$  and  $(u', v')$  such that  $\mathsf{rwl}_2^{(t)}(G, u, v) = \mathsf{rwl}_2^{(t)}(G, u', v')$  for all  $t \geq 0$  but  $\mathsf{rwl}_2^{(1)}(G, u, v) \neq \mathsf{rwl}_2^{(1)}(G, u', v')$ .
Proof. First, we show  $\mathsf{rw} \mathsf{l}_2^{(t)}(G) \preceq \mathsf{rw} \mathsf{l}_2^{(t)}(G)$  and proceed by induction on  $t$ . The base case for  $t = 0$  is trivial by assumption. For the inductive step, given that we have  $\mathsf{rw} \mathsf{l}_2^{(k + 1)}(u,v) = \mathsf{rw} \mathsf{l}_2^{(k + 1)}(u',v')$ , by the definition of  $\mathsf{rw} \mathsf{l}_2^{(k + 1)}(u,v)$  and  $\mathsf{rw} \mathsf{l}_2^{(k + 1)}(u',v')$ , and as  $\tau$  is injective, it holds that
$$
\operatorname {r w l} _ {2} ^ {(k)} (u, v) = \operatorname {r w l} _ {2} ^ {(k)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
$$
\{\{\left(\mathsf {r w} | _ {2} ^ {(k)} (x, v), r\right) \mid x \in \mathcal {N} _ {r} (u), r \in R \} \} = \{\{\left(\mathsf {r w} | _ {2} ^ {(k)} \left(x ^ {\prime}, v ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(u ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
$$
\{\{\left(\mathsf {r w} | _ {2} ^ {(k)} (u, x), r\right) \mid x \in \mathcal {N} _ {r} (v), r \in R \} \} = \{\{\left(\mathsf {r w} | _ {2} ^ {(k)} \left(u ^ {\prime}, x ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(v ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
Now, by the inductive hypothesis we have  $\mathsf{rawl}_2^{(k)}(u,v) = \mathsf{rawl}_2^{(k)}(u',v')$ . We can further transform the last equation by applying the inductive hypothesis again after unpacking the multiset. This results in
$$
x \xrightarrow {r} u ^ {\prime}
$$
Figure 3: Graph  $G'$  as the counter-example in Proposition A.18. It is also shown in Proposition A.20 to prove  $\mathsf{rawl}_2^{+(t)}(u, v) = \mathsf{rawl}_2^{+(t)}(u', v')$  for all  $t \geq 0$  but  $\mathsf{rw} \mathsf{l}_2^{(1)}(u, v) \neq \mathsf{rw} \mathsf{l}_2^{(1)}(u', v')$ .
Figure 4: Two counterexamples shown in Proposition A.17 and A.19. The left graph  $G'$  is to show  $\mathsf{rwl}_2^{(t)}(u,v) = \mathsf{rwl}_2^{(t)}(u',v')$  for all  $t \geq 0$  but  $\mathsf{rwl}_2^{+(1)}(u,v) \neq \mathsf{rwl}_2^{+(1)}(u',v')$ , whereas the right graph  $G''$  is to show  $\mathsf{rwl}_2^{(t)}(u,v) = \mathsf{rwl}_2^{(t)}(u',v')$  for all  $t \geq 0$  but  $\mathsf{rwl}_2^{+(1)}(u,v) \neq \mathsf{rwl}_2^{+(1)}(u',v')$ .
Thus, it holds that  $\mathsf{rawl}_2^{(k + 1)}(u,v) = \mathsf{rawl}_2^{(k + 1)}(u',v')$  by definition of  $\mathsf{rawl}_2^{(k + 1)}(u,v)$  and  $\mathsf{rawl}_2^{(k + 1)}(u',v')$ .
For counter-example, we show the case for  $t \geq 0$ . Consider a relational graph  $G' = (V', E', R, c, \eta)$  such that  $V' = \{u, u', v, x\}$ ,  $E' = \{r(x, u')\}$ , and  $R = \{r\}$  with the initial labeling  $\eta$  for node pairs  $(u, v)$  and  $(u', v)$  satisfies  $\mathsf{rw} \mathsf{l}_2^{(0)}(u, v) = \mathsf{rw} \mathsf{l}_2^{(0)}(u', v)$  and  $\mathsf{raw} \mathsf{l}_2^{(0)}(u, v) = \mathsf{raw} \mathsf{l}_2^{(0)}(u', v)$ . For such graph  $G'$ , we consider node pair  $(u, v)$  and  $(u', v)$ . For  $\mathsf{raw} \mathsf{l}_2^{(t)}$  where  $t \geq 0$ , we show by induction that  $\mathsf{raw} \mathsf{l}_2^{(t)}(u, v) = \mathsf{raw} \mathsf{l}_2^{(t)}(u', v)$ . The base case is trivial by assumption. The inductive step shows that by the inductive hypothesis:
$$
\begin{array}{l} \left. \operatorname {r a w l} _ {2} ^ {(t + 1)} (u, v) = \tau \left(\operatorname {r a w l} _ {2} ^ {(t)} (u, v), \{\{\}\}\right) \right. \\ = \tau (\operatorname {r a w l} _ {2} ^ {(t)} (u ^ {\prime}, v), \{\{\} \}) \\ = \operatorname {r a w l} _ {2} ^ {(t + 1)} \left(u ^ {\prime}, v\right) \\ \end{array}
$$
$$
\begin{array}{l} \mathsf {r w l} _ {2} ^ {(1)} (u, v) = \tau (\mathsf {r w l} _ {2} ^ {(0)} (u, v), \{\{\}, \{\{\} \}) \\ \neq \mathsf {r w l} _ {2} ^ {(1)} (u, v ^ {\prime}) = \tau (\mathsf {r w l} _ {2} ^ {(0)} (u, v ^ {\prime}), \{\{\left(\mathsf {r w l} _ {2} ^ {(0)} (x, v ^ {\prime}), r)\right) \}, \{\{\} \}) \\ \end{array}
$$
Proposition A.19. For all  $t \geq 0$  and all knowledge graph  $G$ , let  $\mathsf{rw}|_2^{+(0)}(G) \equiv \mathsf{rw}|_2^{(0)}(G)$ . The following statements hold:
1. For every  $t > 0$ ,  $\mathsf{rwl}_2^{+(t)}(G) \preceq \mathsf{rwl}_2^{(t)}(G)$
2. There is a knowledge graph  $G''$  and pair of nodes  $(u, v)$  and  $(u', v')$  such that  $\mathsf{rw} \mid_{2}^{(t)}(G'', u, v) = \mathsf{rw} \mid_{2}^{(t)}(G'', u', v')$  for all  $t \geq 0$  but  $\mathsf{rw} \mid_{2}^{+(1)}(G'', u, v) \neq \mathsf{rw} \mid_{2}^{+(1)}(G'', u', v')$ .
Proof. As before, we first prove  $\mathsf{rw}|\_2^+(G) \preceq \mathsf{rw}|\_2(G)$  by induction on iteration  $t$ . The base case for  $t = 0$  is trivial by the assumption. By the inductive hypothesis,
$$
\mathsf {r w l} _ {2} ^ {+ (k)} (u, v) = \mathsf {r w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, v ^ {\prime}\right) \Longrightarrow \mathsf {r w l} _ {2} ^ {(k)} (u, v) = \mathsf {r w l} _ {2} ^ {(k)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
for some  $k$ . Thus, assuming  $\mathsf{rw} \mathsf{l}_2^{+(k+1)}(u, v) = \mathsf{rw} \mathsf{l}_2^{+(k+1)}(u', v')$ , by the definition of  $\mathsf{rw} \mathsf{l}_2^{+(k+1)}(u, v)$  and  $\mathsf{rw} \mathsf{l}_2^{+(k+1)}(u', v')$  and by the injectivity of  $\tau$ , it holds that
$$
\mathrm {r w l} _ {2} ^ {+ (k)} (u, v) = \mathrm {r w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
$$
\{\{(r w l _ {2} ^ {+ (k)} (x, v), r) \mid x \in \mathcal {N} _ {r} ^ {+} (u), r \in R ^ {+} \} \} = \{\{(r w l _ {2} ^ {+ (k)} (x ^ {\prime}, v ^ {\prime}), r ^ {\prime}) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} ^ {+} (u ^ {\prime}), r ^ {\prime} \in R ^ {+} \}
$$
$$
\{\left\{\left(\mathsf {r w l} _ {2} ^ {+ (k)} (u, x), r\right) \mid x \in \mathcal {N} _ {r} ^ {+} (v), r \in R ^ {+} \right\} = \{\left\{\left(\mathsf {r w l} _ {2} ^ {+ (k)} \left(u ^ {\prime}, x ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} ^ {+} (v ^ {\prime}), r ^ {\prime} \in R ^ {+} \right\}
$$
By the similar argument in proving  $\mathsf{rawl}_2^+\preceq \mathsf{rawl}_2$ , we can split the multiset into two equations by decomposing  $\mathcal{N}_r^+ (u) = \mathcal{N}_r(u)\cup \mathcal{N}_{r^-}(u)$  and  $\mathcal{N}_r^+ (v) = \mathcal{N}_r(v)\cup \mathcal{N}_{r^-}(v)$ . Thus, we have
$$
\{\{(r w l _ {2} ^ {+ (k)} (x, v), r) \mid x \in \mathcal {N} _ {r} (u), r \in R \} \} = \{\{(r w l _ {2} ^ {+ (k)} (x ^ {\prime}, v ^ {\prime}), r ^ {\prime}) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} (u ^ {\prime}), r ^ {\prime} \in R \}
$$
$$
\{\{\left(\mathsf {r w l} _ {2} ^ {+ (k)} (y, v), r ^ {-}\right) \mid y \in \mathcal {N} _ {r ^ {-}} (u), r \in R \} \} = \{\{\left(\mathsf {r w l} _ {2} ^ {+ (k)} \left(y ^ {\prime}, v ^ {\prime}\right), r ^ {\prime -}\right) \mid y ^ {\prime} \in \mathcal {N} _ {r ^ {\prime -}} \left(u ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
$$
\{\{(r w l _ {2} ^ {+ (k)} (u, x), r) \mid x \in \mathcal {N} _ {r} (v), r \in R \} \} = \{\{(r w l _ {2} ^ {+ (k)} (u ^ {\prime}, x ^ {\prime}), r ^ {\prime}) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} (v ^ {\prime}), r ^ {\prime} \in R \} \}
$$
$$
\{\{(r w | _ {2} ^ {+ (k)} (u, y), r ^ {-}) \mid y \in \mathcal {N} _ {r ^ {-}} (v), r \in R \} \} = \{\{(r w | _ {2} ^ {+ (k)} (u ^ {\prime}, y ^ {\prime}), r ^ {\prime -}) \mid y ^ {\prime} \in \mathcal {N} _ {r ^ {\prime -}} (v ^ {\prime}), r ^ {\prime} \in R \} \}
$$
By the inductive hypothesis and unpacking the first and the third equations, we can further imply that
$$
\mathrm {r w l} _ {2} ^ {(k)} (u, v) = \mathrm {r w l} _ {2} ^ {(k)} \left(u ^ {\prime}, v ^ {\prime}\right)
$$
$$
\{\{\left(\operatorname {r w l} _ {2} ^ {(k)} (x, v), r\right) \mid x \in \mathcal {N} _ {r} (u), r \in R \} \} = \{\{\left(\operatorname {r w l} _ {2} ^ {(k)} \left(x ^ {\prime}, v ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(u ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
$$
\{\{\left(\mathsf {r w l} _ {2} ^ {(k)} (u, x), r\right) \mid x \in \mathcal {N} _ {r} (v), r \in R \} \} = \{\{\left(\mathsf {r w l} _ {2} ^ {(k)} \left(u ^ {\prime}, x ^ {\prime}\right), r ^ {\prime}\right) \mid x ^ {\prime} \in \mathcal {N} _ {r ^ {\prime}} \left(v ^ {\prime}\right), r ^ {\prime} \in R \} \}
$$
This would result in  $\mathsf{rw}l_{2}^{(k + 1)}(u,v) = \mathsf{rw}l_{2}^{(k + 1)}(u^{\prime},v^{\prime})$  by the definition of  $\mathsf{rw}l_{2}^{(k + 1)}(u,v)$  and  $\mathsf{rw}l_{2}^{(k + 1)}(u^{\prime},v^{\prime})$
For the counter-example, we consider a relational graph  $G'' = (V'', E'', R'', c, \eta)$  such that  $V'' = \{u, u', v, v', x, x'\}$ ,  $E'' = \{r_1(u, x), r_2(u', x')\}$ , and  $R'' = \{r_1, r_2\}$ . For such graph  $G''$ , we set the initial labeling  $\eta$  for node pairs  $(u, v)$  and  $(u', v')$  to satisfy  $\mathsf{rwI}_2^{+(0)}(u, v) = \mathsf{rwI}_2^{+(0)}(u', v')$  and  $\mathsf{rwI}_2^{(0)}(u, v) = \mathsf{rwI}_2^{(0)}(u', v')$ . For such graph  $G''$ , we consider node pair  $(u, v)$  and  $(u', v')$ .
Consider  $\mathsf{rw}l_2^{(t)}$  where  $t\geq 0$ , we show by induction that  $\mathsf{rw}l_2^{(t)}(u,v) = \mathsf{rw}l_2^{(t)}(u',v')$ . The base case is trivial by assumption. The inductive step shows that by inductive hypothesis,
$$
\begin{array}{l} \mathsf {r w l} _ {2} ^ {(t + 1)} (u, v) = \tau (\mathsf {r w l} _ {2} ^ {(t)} (u, v), \{\{\}, \{\{\} \}) \\ = \tau (\mathsf {r w l} _ {2} ^ {(t)} (u ^ {\prime}, v ^ {\prime}), \{\{\} \}, \{\{\} \}) \\ = r w l _ {2} ^ {(t + 1)} \left(u ^ {\prime}, v ^ {\prime}\right) \\ \end{array}
$$
$$
\begin{array}{l} \mathsf {r w l} _ {2} ^ {+ (1)} (u, v) = \tau (\mathsf {r w l} _ {2} ^ {+ (0)} (u, v), \{\{\{\left(\mathsf {r w l} _ {2} ^ {+ (0)} (x, v), r _ {1} ^ {-}\right) \} \}, \{\{\{\} \}) \\ \neq \mathsf {r w l} _ {2} ^ {+ (1)} (u ^ {\prime}, v ^ {\prime}) = \tau (\mathsf {r w l} _ {2} ^ {+ (0)} (u ^ {\prime}, v ^ {\prime}), \{\{\{\left(\mathsf {r w l} _ {2} ^ {+ (0)} (x ^ {\prime}, v ^ {\prime}), r _ {2} ^ {-}\right)) \} \}, \{\{\{\} \}) \\ \end{array}
$$