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Table 6:

Adopting TupleILP and ExplanationLP constraints in DPP format. For this work we set the hyperparameters w₁ = 2, w₂ = 2, w₃ = 1 and w₄ = 2.

Description	DPP Format	Parameters
TupleILP
Sub graph must have ≤ w₁ active tuples	$\sum_{i \in F} Y_{i i} \leq w_{1} + 1$ (16)	–
Active hypothesis term must have ≤ w₂ edges	$H_{θ} [:, :, i] ⊙ Y \leq w_{2} \forall i \in H^{t}$ (17)	H_θ is populated by hypothesis term matrix H with dimension ((n + k) × (n + k) × l) and the values are given by: $H_{i j k} = \{\begin{array}{l} 1, & \forall k \in H^{t}, i \in H, j \in F, \\ t_{k} \in t r m (h_{i}), t_{k} \in t r m (f_{j}) \\ 1, & \forall k \in H^{t}, i \in F, j \in H, \\ t_{k} \in t r m (h_{j}), t_{k} \in t r m (f_{i}) \\ 0, & otherwise \end{array}$ (18)
Active tuple must have active subject	$Y ⊙ T_{θ}^{S} > = E ⊙ A_{θ}$ (19)	A_θ populated by adjacency matrix A, $T_{θ}^{S}$ by subject tuple matrix T^S with dimension ((n + k) × (n + k)) and the values are given by: $T_{i j}^{S} = \{\begin{array}{l} 1, & i \in H, j \in F, \\ \| t r m (h_{i}) \cap t r m (f_{j}^{S}) \| > 0 \\ 1, & i \in F, j \in H, \\ \| t r m (h_{j}) \cap t r m (f_{i}^{S}) \| > 0 \\ 0, & otherwise \end{array}$ (20)
Active tuple must have ≥ w₃ active fields	$\begin{array}{l} Y ⊙ T_{θ}^{S} + Y ⊙ T_{θ}^{P} + Y ⊙ T_{θ}^{O} \geq w_{3} (Y ⊙ A_{θ}) \end{array}$ (21)	A_θ populated by adjacency matrix A and $T_{θ}^{S}$ ⁠, $T_{θ}^{P}$ ⁠, $T_{θ}^{O}$ populated by subject, predicate and object matrices T^S, T^P, T^O respectively. Predicate and object tuples are converted into T^P,T^O matrices similar to T^S
Active tuple must have an edge to some hypothesis term	Implemented during graph construction by only considering tuples that have lexical overlap with a hypothesis	–
ExplanationLP
Limits the total number of abstract facts to w₄	$d i a g (Y) \cdot F_{θ}^{A B} \leq w_{4}$ (22)	$F_{θ}^{A B}$ is populated by Abstract fact matrix F^AB, where: $F_{i j}^{A B} = \{\begin{array}{l} 1, & i \in H, j \in F^{A} \\ 0, & otherwise \end{array}$ (23)

Description	DPP Format	Parameters
TupleILP
Sub graph must have ≤ w₁ active tuples	$\sum_{i \in F} Y_{i i} \leq w_{1} + 1$ (16)	–
Active hypothesis term must have ≤ w₂ edges	$H_{θ} [:, :, i] ⊙ Y \leq w_{2} \forall i \in H^{t}$ (17)	H_θ is populated by hypothesis term matrix H with dimension ((n + k) × (n + k) × l) and the values are given by: $H_{i j k} = \{\begin{array}{l} 1, & \forall k \in H^{t}, i \in H, j \in F, \\ t_{k} \in t r m (h_{i}), t_{k} \in t r m (f_{j}) \\ 1, & \forall k \in H^{t}, i \in F, j \in H, \\ t_{k} \in t r m (h_{j}), t_{k} \in t r m (f_{i}) \\ 0, & otherwise \end{array}$ (18)
Active tuple must have active subject	$Y ⊙ T_{θ}^{S} > = E ⊙ A_{θ}$ (19)	A_θ populated by adjacency matrix A, $T_{θ}^{S}$ by subject tuple matrix T^S with dimension ((n + k) × (n + k)) and the values are given by: $T_{i j}^{S} = \{\begin{array}{l} 1, & i \in H, j \in F, \\ \| t r m (h_{i}) \cap t r m (f_{j}^{S}) \| > 0 \\ 1, & i \in F, j \in H, \\ \| t r m (h_{j}) \cap t r m (f_{i}^{S}) \| > 0 \\ 0, & otherwise \end{array}$ (20)
Active tuple must have ≥ w₃ active fields	$\begin{array}{l} Y ⊙ T_{θ}^{S} + Y ⊙ T_{θ}^{P} + Y ⊙ T_{θ}^{O} \geq w_{3} (Y ⊙ A_{θ}) \end{array}$ (21)	A_θ populated by adjacency matrix A and $T_{θ}^{S}$ ⁠, $T_{θ}^{P}$ ⁠, $T_{θ}^{O}$ populated by subject, predicate and object matrices T^S, T^P, T^O respectively. Predicate and object tuples are converted into T^P,T^O matrices similar to T^S
Active tuple must have an edge to some hypothesis term	Implemented during graph construction by only considering tuples that have lexical overlap with a hypothesis	–
ExplanationLP
Limits the total number of abstract facts to w₄	$d i a g (Y) \cdot F_{θ}^{A B} \leq w_{4}$ (22)	$F_{θ}^{A B}$ is populated by Abstract fact matrix F^AB, where: $F_{i j}^{A B} = \{\begin{array}{l} 1, & i \in H, j \in F^{A} \\ 0, & otherwise \end{array}$ (23)