entailment

entailment是指一个事物或概念必然包含另一个事物或概念的关系。在自然语言处理中，entailment通常指一个句子是否暗示另一个句子的真实性。例如，“狗可以叫”暗示“狗有声音”，因此可以说“狗可以叫”是“狗有声音”的entailment。

相关问题

Fix two concept names A and B as well as some role name r. Does the entailment {∀r.A v ∃r.A} |= ∀r.B v ∃r.B hold true?

Yes, the entailment {∀r.A v ∃r.A} |= ∀r.B v ∃r.B holds true.

The left-hand side of the entailment, ∀r.A v ∃r.A, states that for any role r, either A holds for all instances of r or there exists at least one instance of r where A holds.

The right-hand side of the entailment, ∀r.B v ∃r.B, states that for any role r, either B holds for all instances of r or there exists at least one instance of r where B holds.

Since A and B are fixed concept names, the truth of either A or B is not dependent on r. Therefore, if A holds for all instances of r, then B must also hold for all instances of r, and if there exists at least one instance of r where A holds, then there must also exist at least one instance of r where B holds. Hence, the entailment holds true.

Fix two concept names A and B as well as some role name r. Does the entailment {∀r.A ⊆ ∃r.A} |= ∀r.B ⊆ ∃r.B hold true?

Yes, the entailment {∀r.A ⊆ ∃r.A} |= ∀r.B ⊆ ∃r.B holds true.

This can be proven using the principle of subsumption, which states that if A ⊆ B, then ∀r.A ⊆ ∀r.B and ∃r.B ⊆ ∃r.A. Since A ⊆ A, we have ∀r.A ⊆ ∃r.A.

Therefore, by applying the principle of subsumption, we have ∀r.B ⊆ ∀r.A and ∃r.A ⊆ ∃r.B. Combining these two inclusions, we get:

∀r.B ⊆ ∃r.B

Thus, the entailment holds true.