SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Kunxun Qi $^{1}$  Jianfeng Du $^{2,3,*}$  Hai Wan $^{1,*}$
$^{1}$  School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, China
$^{2}$  Guangzhou Key Laboratory of Multilingual Intelligent Processing, Guangdong University of Foreign Studies, Guangzhou, China
<sup>3</sup> Bigmath Technology, Shenzhen, China  
qikx@mail2.sysu.edu.cn, jfdu@gdufs.edu.cn, wanhai@mail.sysu.edu.cn
Learning rule-based systems plays a pivotal role in knowledge graph completion (KGC). Existing rule-based systems restrict the input of the system to structural knowledge only, which may omit some useful knowledge for reasoning, e.g., textual knowledge. In this paper, we propose a two-stage framework that imposes both structural and textual knowledge to learn rule-based systems. In the first stage, we compute a set of triples with confidence scores (called soft triples) from a text corpus by distant supervision, where a textual entailment model with multi-instance learning is exploited to estimate whether a given triple is entailed by a set of sentences. In the second stage, these soft triples are used to learn a rule-based model for KGC. To mitigate the negative impact of noise from soft triples, we propose a new formalism for rules to be learnt, named text enhanced rules or TE-rules for short. To effectively learn TE-rules, we propose a neural model that simulates the inference of TE-rules. We theoretically show that any set of TE-rules can always be interpreted by a certain parameter assignment of the neural model. We introduce three new datasets to evaluate the effectiveness of our method. Experimental results demonstrate that the introduction of soft triples and TE-rules results in significant performance improvements in inductive link prediction.
Knowledge graph (KG) consists of real-world facts and has been widely used in many applications, including question answering [27], recommendation [22] and information retrieval [42]. A fact in KGs is usually represented by a triple of the form (head, relation, tail), where head and tail are entities. In general, KGs are highly incomplete since the complete set of facts is hard to collect. Therefore, knowledge graph completion (KGC), which aims to infer missing facts from observed ones, has become a vital ingredient for practically completing KGs.
In recent years, there are mainly two categories for prevalent approaches to KGC. One is the embedding-based category [3, 45, 33]. Methods in this category usually learn knowledge graph embeddings (KGE) by encoding entities and relations as low-dimensional real-value vectors. They have been shown to be effective for large-scale KGC, but are hard to generalize to the inductive scenario where missing facts involve previously unseen entities [20]. Besides, they can hardly be interpreted by human due to their black-box nature. The other category tackles KGC by learning rule-based systems, based on search algorithms with pruning heuristics [13, 24] or neural models for
Figure 1: Reasoning from both structural and textual knowledge.
approximate rule learning [46, 30, 28]. They excel in explaining why a missing fact is inferred and are able to handle the inductive setting.
Previous work focuses merely on learning rule-based systems from structural knowledge. However, the high incompleteness of structural knowledge imposes a heavy burden for building high-quality rule-based systems. Text corpora can naturally be considered to enrich structural knowledge as they contain much additional knowledge for reasoning. Figure 1 showcases an example about how textual knowledge help to KGC. The sub-figure on the left side shows a logical rule to infer whether  $x$  has wife  $y$ , and we cannot infer that "Barack Obama" has wife "Michelle Obama" since the key fact ("Malia Obama", HasMother, "Michelle Obama") is missing in the background KG. The sub-figure on the right side shows that text corpora can provide supporting texts to this key fact, thereby the target relation HasWife can be correctly inferred.
Motivated by this kind of examples, we aim to learn rule-based systems from both structural and textual knowledge. To this end, we propose a two-stage framework named Learning from Structural and Textual Knowledge or LSTK for short, as illustrated in Figure 2. In the first stage, a distantly-supervised method using a textual entailment model is proposed to extract textual knowledge from texts. Distant supervision [26] posits that if two entities involve a relation, then any sentences that mention these two entities might express that relation. Based on this assumption, we generate a set of triples with their mentioned texts and employ a textual entailment model equipped with a multi-instance learning mechanism to estimate whether a triple is entailed by a set of sentences. Each generated triple has an estimated confidence score denoting the supporting degree of the triple by texts, and thus it is called a soft triple. In the second stage, both the generated soft triples and hard triples (i.e. existing triples) are used to learn a rule-based system for KGC.
In practice, it is challenging to learn rules from soft triples, as the soft triples generated by distant supervision may contain noise. To mitigate the negative impact of noisy data, we propose a new formalism for rules to be learnt, named text enhanced rules or TE-rules for short. In this formalism, textual relations are introduced to provide a flexible mechanism to discard noisy soft triples. In other words, a TE-rule is extended from a chain-like rule by adding atoms or disjunction of atoms possibly with textual relations to the rule body. To effectively learn TE-rules, we propose a neural model named TE-rule Learning Model or TELM for short. We show that an arbitrary set of TE-rules can be encoded by a certain parameter assignment of TELM (see Theorem 1). This theoretical result guarantees a certain degree of faithfulness between TELM and TE-rules, enabling us to extract explainable TE-rules from the parameter assignment of the learnt TELM. Based on the extracted TE-rules, LSTK is able to provide explanations for inferred missing facts by backward reasoning.
We enhance three datasets from the field of relation extraction for empirical evaluation. Experimental results demonstrate significant gains achieve by the proposed method LSTK-TELM in inductive link prediction. Our ablation study and case study further clarify why the introduction of soft triples and TE-rules help to improve the performance.
Knowledge Graph. Given a set of entities  $\mathcal{E}$  and a set of relations  $\mathcal{R}$ , a knowledge graph  $\mathcal{G}$  is a subset of  $\mathcal{E} \times \mathcal{R} \times \mathcal{E}$ . Specifically,  $\mathcal{G} = \{(h_i, r_i, t_i)\}_{1 \leq i \leq N}$ , where  $N$  denotes the number of triples,  $h_i \in \mathcal{E}$  the head entity for the  $i^{\text{th}}$  triple,  $r_i \in \mathcal{R}$  the relation for the  $i^{\text{th}}$  triple and  $t_i \in \mathcal{E}$  the tail entity for the  $i^{\text{th}}$  triple. By  $r^{-}$  we denote the inverse relation of  $r \in \mathcal{R}$ . The set of inverse relations for
$\mathcal{R}$ , namely  $\{r^{-} \mid r \in \mathcal{R}\}$ , is denoted by  $\mathcal{R}^{-}$ . Accordingly, the equivalent knowledge graph for  $\mathcal{G}$  composed by inverse relations, namely  $\{(t, r^{-}, h) \mid (h, r, t) \in \mathcal{G}\}$ , is denoted by  $\mathcal{G}^{-}$ .
Logical Rule. Most existing work focuses on learning chain-like rules (CRs). A CR is essentially a plain datalog rule [1] where all atoms are binary and every body atom shares variables with the previous atom and the next atom. Formally, a CR  $R$  with  $L$  body atoms, simply called an  $L$ -CR, is of the form:
$$
H (x, y) \leftarrow B _ {1} (x, z _ {1}) \wedge B _ {2} (z _ {1}, z _ {2}) \wedge \dots \wedge B _ {L} (z _ {L - 1}, y)
$$
where  $x$  is the head entity,  $y$  the tail entity, and  $z_{1},\ldots ,z_{L - 1}$  variables. The part at the left (resp. right) side of  $\leftarrow$  is called the head (resp. body) of  $R$ . The rule  $R$  is called  $r$ -specific if  $H = r$ . By  $H_{R}$  and  $B_{R}$  we denote the atom in the head of  $R$  and the set of atoms in the body of  $R$ , respectively. An atom or a rule is ground if it does not contain any variable. A rule  $R$  is a fact if  $B_{R}$  is empty and  $H_{R}$  is ground. In this paper, a fact or a ground atom  $r(a,b)$  and a triple  $(a,r,b)$  are used interchangeably. To uniformly represent rules using fixed-length bodies, we introduce the identity relation (denoted by  $I$ ) to rule bodies. For example,  $r(x,y)\gets s(x,y)$  can be converted into a rule with two body atoms, namely  $r(x,y)\gets s(x,y)\land I(y,y)$ .
We say  $\mathcal{G} \models H_R(a, b)$  if there exists a ground instance  $R_g$  of  $R$  such that  $H_R(a, b) = H_{R_g}$  and  $B_{R_g} \subseteq \mathcal{G} \cup \mathcal{G}^- \cup \{I(e, e) \mid e \in \mathcal{E}\}$ . Let  $\Sigma$  be a set of  $r$ -specific CRs and  $(a, r, b)$  a triple. We say  $\mathcal{G} \models_{\Sigma}(a, r, b)$  if there exists a logical rule  $R \in \Sigma$  such that  $\mathcal{G} \models H_R(a, b)$ . We say a triple  $(a, r, b)$  is plausible in  $\mathcal{G}$  if there exists a set of possibly correct  $r$ -specific CRs  $\Sigma$  such that  $\mathcal{G} \models_{\Sigma}(a, r, b)$ .
Link Prediction. Link prediction is the main task of knowledge graph completion that we focus on in this work. Given a knowledge graph  $\mathcal{G}$ , a head query  $(?,r,t)$  or a tail query  $(h,r,?)$ , the task of link prediction aims to find all entities  $e\in \mathcal{E}$  such that  $(e,r,t)$  for  $(?,r,t)$  or  $(h,r,e)$  for  $(h,r,?)$  is plausible in  $\mathcal{G}$ . The inductive learning setting requires that at least one entity in the test set should not appear in the training set, while relations in the test set also appear in the training set.
In this section, we describe the proposed LSTK framework for inductive KGC with text corpus. We first formalize our problem setting as follows.
Problem Setting. Given a set of triples and a corpus for training  $\mathcal{U}_{\mathrm{train}} = (\mathcal{G}_{\mathrm{train}},\mathcal{T}_{\mathrm{train}})$ , a set of triples and a corpus for test  $\mathcal{U}_{\mathrm{test}} = (\mathcal{G}_{\mathrm{test}},\mathcal{T}_{\mathrm{test}})$ , our inductive KGC setting aims to learn a KGC system based on  $\mathcal{U}_{\mathrm{train}}$ , and then evaluate the learnt system on  $\mathcal{U}_{\mathrm{test}}$ . During evaluation, given a head query  $(?,r,t)$  or a tail query  $(h,r,?)$ , the learnt KGC system finds an answer with the highest estimated truth degree to answer this query, based on the background knowledge from  $(\mathcal{G}_{\mathrm{train}}\cup \mathcal{G}_{\mathrm{test}}\setminus \{(h,r,t)\},\mathcal{T}_{\mathrm{train}}\cup \mathcal{T}_{\mathrm{test}})$ .
This setting is motivated by real-world application scenarios where we need to fetch texts from search engines to find evidences to verify a new fact. To address this problem setting, we propose Learning from Structural and Textual Knowledge or LSTK for short, as illustrated in Figure 2. Specifically, the proposed LSTK framework has two stages, where the first stage aims to generate soft triples from a text corpus and the second one aims at training a rule-based system for KGC. Furthermore, LSTK is able to provide an explanation to show why an inferred missing fact is plausible in the background knowledge, by tracking back the reasoning paths from both hard triples and soft triples, where soft triples can further be explained by tracking back the supporting texts.
3.1 Formalization of LSTK
In the first stage, our goal is to find all triples that are possibly entailed by the given text corpus. We achieve this goal by adopting distant supervision. As far as we know, distant supervision is mostly applied in relation extraction, where the relation between any two entities is restricted in a close set. However, we naturally hope to discover more open relations that can be entailed by the text corpus. Considering that textual entailment [31] can deal with open relations and is better at exploiting the semantic information of relational contexts, we impose distant supervision to a textual entailment model to generate soft triples.
Distantly supervised data generation. Given a knowledge graph  $\mathcal{G}$  and a set of sentences (i.e. a text corpus)  $\mathcal{T} = \{s_i\}_{1\leq i\leq N_{\mathrm{sen}}}$ , we construct a training set  $\mathcal{D}_{\mathrm{train}} = \{(\tau_i,S_i,y_i)\}_{1\leq i\leq N_{\mathrm{tra}}}$  by
Figure 2: Overview of the proposed LSTK framework.
finding all sentences in  $\mathcal{T}$  that mention the entity pair  $(h_i, t_i)$  of  $\tau_i$ , where  $N_{\mathrm{sen}}$  denotes the number of sentences,  $N_{\mathrm{tra}}$  the number of training instances,  $\tau_i = (h_i, r_i, t_i)$  a triple,  $S_i$  the set of sentences in  $\mathcal{T}$  mentioning the entity pair  $(h_i, t_i)$ , and  $y_i$  the label. We call a triple  $(h, r, t) \in \mathcal{G}$  a positive triple, whereas a triple  $(h, r, t) \notin \mathcal{G}$  a negative triple. For each positive triple  $(h_i, r_i, t_i)$ , we append to the training set the set of negative triples  $\{(h_i, r_j', t_i) \mid r_j' \in \mathcal{R}, r_j' \neq r_i, (h_i, r_j', t_i) \notin \mathcal{G}\}$  obtained by corrupting  $r_i$ . Accordingly  $y_i$  is set to 1 if  $\tau_i \in \mathcal{G}$  or 0 otherwise. Considering that some entity pairs may appear in a large number of sentences, to confine the number of considering sentences, we exploit the well-known graph partitioning algorithm METIS [17] to evenly partition the original  $S_i$  to make every part no more than 20 sentences, and then treat the densest part defined by Jaccard similarity between bags of words as the ultimate  $S_i$ . Finally, we likewise construct the test set  $\mathcal{D}_{\mathrm{test}} = \{(\tau_i, S_i) \mid \tau_i \in \mathcal{E} \times \mathcal{R} \times \mathcal{E}, S_i \neq \emptyset\}$  for generating soft triples where labels are not given.
Multi-instance learning for RTE. Recognising textual entailment (RTE) [9, 4] aims at determining whether a sentence (called hypothesis) can be inferred by the other sentence (called premise). We employ a textual entailment model based on the pre-trained language model BERT [10] to estimate the confidence score for every triple. Specifically, given  $\mathcal{D}_{\mathrm{train}} = \{(\tau_i,S_i,y_i)\}_{1\leq i\leq N_{\mathrm{tra}}}$ , we create a set of sequences  $X_{i} = \{x_{i,j}\}_{1\leq j\leq N_{S_{j}}}$  for each instance in  $\mathcal{D}_{\mathrm{train}}$  by filling the template "[CLS]<triple>[SEP]<sentence>[SEP]", where [CLS] and [SEP] are special tokens in BERT, <triple> denotes the slot to be filled by the context of  $\tau_{i}$  and <sentence> the slot to be filled by the context of  $s_j\in S_i$ . All the sequences in  $X_{i}$  are fed to BERT to calculate their contextual representations and the contextual embedding of the [CLS] token are used to calculate the entailment probability by a full-connected layer activated by the sigmoid function. By  $\epsilon_{i,j}$  we denote the entailment probability for  $x_{i,j}$ , then the entire entailment probability  $\phi_{i}$  for  $X_{i}$  is calculated by  $\phi_{i} = \max_{1\leq j\leq N_{S_{i}}}\epsilon_{i,j}$ . The entire model is trained by minimizing the cross-entropy loss.
Soft triple generation. The trained model is then applied to  $\mathcal{D}_{\mathrm{test}}$  for generating soft triples. A soft triple is defined as a quadruple  $(h,r,t,\phi)$ , where  $(h,r,t)$  denotes a triple and  $\phi$  its corresponding predicted entailment probability. Given a hyper-parameter  $\delta$  to keep only highly confident soft triples, the set of soft triples that will be used in learning chain-like rules is defined as  $\mathcal{G}_{\mathrm{soft}} = \{(h_i,r_i,t_i,\phi_i) \mid (h_i,r_i,t_i) \in \mathcal{D}_{\mathrm{test}}, \phi_i \geq \delta\}$ .
3.2 Formalization of Text Enhanced Rules
In the second stage, we aim at learning logical rules from both hard (i.e. existing) and soft triples. However, it is challenging to learn rules from the background KG with soft triples, as the generated
soft triples by distant supervision may contain noise, which impairs the quality of learnt rules. The following example illustrates a case where noisy data impact the reasoning ability of a chain-like rule.
Example 1. Considering the chain-like rule  $R$  in Figure 1: HasWife(x,y) ← HasDaughter(x,z) ∧ HasMother(z,y). Suppose we have two soft triples “(A, HasDaughter, B, 0.55)” and “(B, HasMother, C, 0.90)” computed by the textual entailment model, where the former one is incorrect in the real world. From this rule  $R$  the negative triple “(A, HasWife, C)” will be incorrectly inferred to be positive.
To alleviate the negative impact of noise from soft triples, a simple way is to restrict that some atoms in the rule should appear in the structural knowledge as existing triples. To this end, we extend chain-like rules by adding atoms or disjunctions of atoms that may involve textual relations. We call such extended chain-like rules text enhanced rules and simply TE-rules. Let  $r_{\mathrm{text}}$  denote the textual relation corresponding to the relation  $r$ , then a TE-rule can be defined below.
Definition 1. A  $p$ -specific  $L$ -TE-rule  $R$ , simply a TE-rule if  $p$  and  $L$  are clear from the context, is of the form:  $p(x,y) \gets \mathcal{C}_1(x,z_1) \wedge \mathcal{C}_2(z_1,z_2) \wedge \ldots \wedge \mathcal{C}_L(z_{L-1},y)$ , where  $\mathcal{C}_l$  can be an original relation, a textual relation, the identity relation, or a disjunction of an original relation and its corresponding textual relation, i.e.,  $\mathcal{C}_l(u,v)$  is of the form  $r(u,v) \lor r_{\text{text}}(u,v)$ .
The following example explains why TE-rules can be used to resolve the error in Example 1.
Example 2. Considering Example 1 again. By allowing TE-rules to be learnt, a learnt rule  $R'$  can be: HasWife(x,y) ← HasDaughter(x,z) ∧ (HasMother(z,y) ∨ HasMother_text(z,y)). From  $R'$  the negative triple “(A, HasWife, C)” will not be inferred to be positive unless there exists a positive hard triple (A, HasDaughter, B) for some B.
3.3 End-to-end Learning of Text Enhanced Rules
To effectively learn TE-rules, we propose an end-to-end neural model named TE-rule Learning Model (TELM for short). The intuition of TELM is to select appropriate parameters to simulate the inference of TE-rules by gradient descent. Formally, given a set of hard triples  $\mathcal{G}$ , a set of soft triples  $\mathcal{G}_{\mathrm{soft}}$ , the maximum number  $L$  of body atoms in every rule, the maximum number of rules  $N$ , we first extend  $\mathcal{G}$  to  $\mathcal{G}_{\mathrm{ext}} = \{(h,r,t,1.0)|(h,r,t)\in \mathcal{G}\}$ . Suppose  $\mathcal{R} = \{r_i\}_{1\leq i\leq n}$ , its corresponding set of inverse relations  $\mathcal{R}^{-} = \{r_{i}\}_{n + 1\leq i\leq 2n}$ , its corresponding set of textual relations  $\mathcal{R}_{\mathrm{text}} = \{r_i\}_{2n + 1\leq i\leq 3n}$ , its corresponding set of inverse textual relations  $\mathcal{R}_{\mathrm{ext}}^{-} = \{r_i\}_{3n + 1\leq i\leq 4n}$ , as well as  $I = r_{4n + 1}$ . Let  $\mathcal{K}_{\mathrm{mix}} = \mathcal{G}_{\mathrm{ext}}\cup \mathcal{G}_{\mathrm{ext}}^{-}\cup \mathcal{G}_{\mathrm{soft}}\cup \mathcal{G}_{\mathrm{soft}}^{-}\cup \{(e,I,e,1.0)\mid e\in \mathcal{E}\}$ . For  $1\leq k\leq N$ ,  $1\leq l\leq L$ , TELM estimates a truth degree  $\psi_{r,x,y}^{(k,l)}$  for each triple  $(x,r,y)\in \mathcal{G}$ , defined by:
$$
\psi_ {r, x, y} ^ {(k, l)} = \left\{ \begin{array}{l} \sigma \left(\sum_ {i = 1} ^ {4 n + 1} \omega_ {i} ^ {(r, k, L)} \mathbb {I} (\phi \mid (x, r _ {i}, y, \phi) \in \mathcal {K} _ {\mathrm {m i x}})\right), \quad l = 1 \\ \sigma \left(\sum_ {i = 1} ^ {4 n + 1} \omega_ {i} ^ {(r, k, L - l + 1)} \sum_ {z, \phi : (x, r _ {i}, z, \phi) \in \mathcal {K} _ {\mathrm {m i x}}} \psi_ {r, z, y} ^ {(k, l - 1)} \phi\right), \quad 2 \leq l \leq L \end{array} \right. \tag {1}
$$
where  $\sigma(x) = \max(\min(x, 1), 0)$  is an activation function that confines the given value to  $[0, 1]$ ,  $\mathbb{I}(\phi \mid C)$  is a function that returns  $\phi$  if  $C$  is true or 0 otherwise, and  $\omega^{(r,k,l)} \in [0, 1]^{4n+1}$  denotes the trainable relational selection weights for the  $l^{\text{th}}$  body atom of the  $k^{\text{th}}$  rule for the head relation  $r$ . Intuitively, Equation (1) simulates the inference of TE-rules under the background KG with soft triples. The selection weights of all predicates for body atoms are formally defined as:
$$
\omega^ {(r, k, l)} = \left[ \alpha^ {(r, k, l)} w _ {\text {o r i g}} ^ {(r, k, l)}; \beta^ {(r, k, l)} w _ {\text {t e x t}} ^ {(r, k, l)}; w _ {4 n + 1} ^ {(r, k, l)} \right] \tag {2}
$$
where  $[;]$  denotes the concatenation operator, and  $w_{\mathrm{orig}}^{(r,k,l)} \in [0,1]^{2n}$  (resp.  $w_{\mathrm{text}}^{(r,k,l)} \in [0,1]^{2n}$  or  $w_{4n+1}^{(r,k,l)} \in [0,1]$ ) denotes the original relation selection weights (resp. textual relation selection weights or the identity relation selection weight) for the  $l^{\mathrm{th}}$  atom of the  $k^{\mathrm{th}}$  rule for the head relation  $r$ ; further,  $\alpha^{(r,k,l)} \in [0,1]$  (resp.  $\beta^{(r,k,l)} \in [0,1]$ ) denotes the trainable selection weight for determining whether the original (resp. textual) relation is involved in the  $l^{\mathrm{th}}$  atom. Intuitively,  $[w_{\mathrm{orig}}^{(r,k,l)}]_i = 1$  (resp.  $[w_{\mathrm{text}}^{(r,k,l)}]_i = 1$  or  $w_{4n+1}^{(r,k,l)} = 1$ ) denotes that the  $i^{\mathrm{th}}$  original relation  $r_i$  (resp. the
$i^{\mathrm{th}}$  textual relation  $r_{2n + i}$  or the identity relation  $r_{4n + 1})$  is selected as the predicate of the  $l^{\mathrm{th}}$  atom. Note that  $\alpha^{(r,k,l)} = 1\wedge \beta^{(r,k,l)} = 1$  indicates that the disjunction of an atom with an original relation and an atom with a textual relation is involved in the  $l^{\mathrm{th}}$  atom. We confine  $\alpha^{(r,k,l)}$  and  $\beta^{(r,k,l)}$  to [0, 1] by sigmoid layers, normalize both  $w_{\mathrm{orig}}^{(r,k,l)}$  and  $w_{\mathrm{text}}^{(r,k,l)}$  by softmax layers, and restrict  $w_{4n + 1}^{(r,k,l)}$  to be  $\max (0,1 - \alpha^{(r,k,l)} - \beta^{(r,k,l)})$  so as to enforce  $\alpha^{(r,k,l)} = \beta^{(r,k,l)} = 0$  when  $w_{4n + 1}^{(r,k,l)} = 1$ .
The final predicted truth degree is calculated by a weighted sum of predicted degrees for  $N$  rules:
$$
\Psi_ {r, x, y} ^ {(N, L)} = \sum_ {k = 1} ^ {N} u _ {r} ^ {(k)} \psi_ {r, x, y} ^ {(k, L)} \tag {3}
$$
where  $u_r^{(k)} \in [-1, 1]$  is a trainable weight for the  $k^{\text{th}}$  rule for the head relation  $r$ , which is confined to  $[-1, 1]$  by a tanh layer. Then the model is trained by minimizing the objective function
$$
\mathcal {L} = - \sum_ {(x, r, y) \in \mathcal {G}} \log \frac {\exp \left(\Psi_ {r , x , y} ^ {(N , L)}\right)}{\sum_ {e \in \mathcal {E}} \exp \left(\Psi_ {r , e , y} ^ {(N , L)}\right)} \tag {4}
$$
We then show the faithfulness between the formalization of TELM and TE-rules in Theorem 1, which relies on the notion of induced parameters as defined below.
Definition 2. Given a set of  $r$ -specific L-TE-rules  $\Sigma = \{R_k\}_{1 \leq k \leq N}$  for  $R_k$  of the form  $r(x,y) \gets \mathcal{C}_{k,1}(x,z_1) \wedge \ldots \wedge \mathcal{C}_{k,L}(z_{L-1},y)$  with  $\mathcal{C}_{k,l}(u,v) \in \{r_{k,l}(u,v), r_{k,l}'(u,v), r_{k,l}^\dagger(u,v) \vee r_{k,l}'(u,v)\}$ , where  $r_{k,l} \in \mathcal{R} \cup \mathcal{R}^- \cup \{I\}$ ,  $r_{k,l}' \in \mathcal{R}_{\text{text}} \cup \mathcal{R}_{\text{text}}^-$ ,  $r_{k,l}^\dagger \in \mathcal{R} \cup \mathcal{R}^-$ , and  $r_{k,l}^\ddagger$  is the corresponding textual relation of  $r_{k,l}^\dagger$ , we call a parameter assignment of TELM  $\theta_r^{(N,L)} = \{[w_{\text{orig}}^{(r,k,l)}]_i, [w_{\text{text}}^{(r,k,l)}]_i\}_{1 \leq k \leq N, 1 \leq l \leq L, 1 \leq i \leq 2n} \cup \{w_{4n+1}^{(r,k,l)}, \alpha^{(r,k,l)}, \beta^{(r,k,l)}\}_{1 \leq k \leq N, 1 \leq l \leq L} \cup \{u_r^{(k)}\}_{1 \leq k \leq N}$  induced if it satisfies the following conditions for all  $1 \leq k \leq N, 1 \leq l \leq L$ :